), and since the maxi
mum shifting of the image need never exceed 1 millim. (if
comparison is made on a miliim. scale), the corresponding
correction to the tangent value is only 3^ millim. and may
be disregarded. In this case we may read off the value of A
directly on a straight scale 5 s by means of a pointer, w, which
consists of a thin plate of glass or mica on which is ruled a
fine radial line. The distance ro from 0, to~the point of inter
section of this line with the longitudinal line s s on the scale,
equals c'o tang , and hence is directly proportional to A. If
we make c'o equal to 17J t in millim., then each millim. on
the scale s s corresponds to a shifting of the image through
i^y millim. For an angle of 30° A=i£, hence for a shifting
of 1 millim. corresponding to this angle, the plate P must be
h millim. thick, and the distance c'o therefore about 89 millim.
as laid off in fig. 6. The scale §5 of fig. 5 is graduated in
Fig. 6.
2 millim. intervals so that such intervals correspond to ^
millim., but it is easy to set and read the position of the
pointer to j 1 ^ div. or jIq millim.
In the case of glass of refractive index 1*5 the proportion
ality between the scalereadings obtained in this manner and
the value of A is not so exact, the error amounting in case of
an angle of 30° to nearly j of 1 per cent, or to nearly O'Ol
millim. This is a quantity too large to be neglected. Poyn
ting suggests a very ingenious system of linkwork, whereby
the readings on the scale may be made directly proportional
K 2
132 A Simple and Accurate Cathetometer.
to A for all values of . If we
draw a line no (fig. 6), inclined at a small angle a, in each
direction from o, to the longitudinal line on the scale s s, and
read in each case to the intersection of this line with the
radial line u on the pointer, — we increase the scalereading by
an amount «r=o« tang a tang
. [1 + tang « tang ].
Hence we have only to make tang a tang =$, in order
to make os the new scalereading directly proportional to A
as before. To find the inclination a of the line w, o, to the
axis of graduation we have only to put
tang u tang <£ = S
for some particular value of (p. Suppose we do this for
= 30°. Then we find
tang a= vyy or a =42'.
Using this value of a to calculate the values of the scale
correction, rs, at other points, we find the valnes given in the
5th column of Table I. As will be seen, they differ on the
average from the corresponding values of S by less than ^ per
cent. ; or only about 0*0008 millim. at the maximum for
<£ = 25°. By this simple method, therefore, the necessity for
making any correction to the scalereading, even in the most
accurate work, is entirely avoided.
The exact constant of the scalereading for any particular
value of the index, differing from those given above, may be
either calculated from the above formula or determined experi
mentally.
In order to always make the value of a 1 millim. scale
division correspond to some convenient fractional part of a
millim., the support for the scale is made adjustable in height so
n — 1 t
that the value of c'o may be always made equal to ,
J  J  1 n a
where a is the fractional value desired. One advantage which
Straining of the Earth resulting from Secular Cooling, 133
the rotating plato has over the ordinary eyepiece micrometer,
is that its constant remains the same for all distances of the
scale from the telescope*.
The above form of parallelplate micrometer may also be
advantageously substituted for the eyepiece micrometer in
the first instrument described. In this case it should be
placed between the objective and the reflecting mirror A, or
else mounted on the mirrorframe itself so as to rotate with it.
The last position enables the same instrument to be used
either in the method of comparison or in the method of coin
cidence, but is objectionable both on account of the increased
weight of the moving parts and because of the liability of dis
turbing the adjustment of the axis or mirror while manipulating
the micrometer. It is therefore better to place the rotating
plate in the first position indicated and adapt it to either
method of use if desired, by making it cover only onehalf
the field of the telescope, that half of course which is
opposite the unsilvered half of the mirror when the coinci
dence method is used, and opposite the silvered half when the
comparison method is employed.
In closing I wish to express my thanks to Messrs. Francis
and Kathan, the mechanicians of the laboratory, for the care
exercised in the mechanical execution of these designs.
University of Chicago, September, 1895.
XVII. On the Straining of the Earth resulting from Secular
Cooling. By Charles Davison, M.A., F.G.S., Mathema
tical Master at King Edward's High School, Birmingham f.
ESTIMATES of the depth of the surface of no strain have
hitherto been founded on the assumptions that the con
ductivity and the coefficient of dilatation are constant J. In
the present paper, I propose to calculate the depth on the
* The great practical advantage of this form of micrometer over the
ordinary form is its much greater simplicity and cheapness. For
these reasons it would have been adopted in all of the above instruments,
had it not happened that we already had on hand a number of micrometer
eyepieces which were available for this purpose.
t Communicated by the Author. A paper with the same title was
read before the Koyal Society on Feb. 15, 1894. The present paper
contains the substance of the former, but has been rewritten.
X Phil. Trans. 1887 A, pp. 231249; Phil. Mag. vol. xxv. 1888,
pp. 720.
134  Mr. G. Davison on the Straining of the
supposition that the coefficient of dilatation increases with the
temperature, being e + e'v, where v is the temperature. In
assuming this law to hold true up to a temperature as high as
7000° F., it is evident that the numerical results here obtained
cannot be regarded as reliable. They are given for their
qualitative rather than for their quantitative value.
Let r be the internal radius of a thin spherical shell con
centric with the earth *, its thickness Br, density p, and
coefficient of linear dilatation e. In any given time, let the
temperature of this shell be increased by ; then, if the
shell were isolated, r would be increased by red. But, in
consequence of the expansion of the mass inside it, let the
internal radius be further increased by rk, so that r becomes
r(l + «), where eighteen
determinations for noncrystallized bodies are given as especi
ally trustworthy. Making use of these only, and taking V
at 7000° F., the average value of /3 is *1. With this value,
the depth of the surface of no strain after 100 million years
is 7*79 miles. At the same time, with a constant coefficient
of dilatation, it would be 2*17 miles.
It is evident from equation (4) or (5) that there is no simple
relation between the depth of the surface of no strain and
the time. If e'/e be small, the depth varies nearly as t: if
eVe be large, the depth varies nearly as t*.
Volume of Folded Hock above the Surface of No Strain, —
Since k and e are both small fractions, the volume of that
part of the shell of radius r and thickness Br which is stretched
or folded in unit time is
47r6Y . r 1 — 4:7r8r(r Q + 2r . kr) = $tt(c— a)8x . kr,
where kr is given by the expression (1) or (2). Denoting
this by SU, we have
dU _ IQeYs/JTTic)
dy /3{catf)y/(tyW)>
* Phil. Trans. 1887 A, p. 246 ; Phil. Mag. vol. xxv. 1888, p. 14.
f Jamin's Cours de Physique, vol. ii. 1878, pp. 8081.
u=
Earth resulting from Secular Cooling, 137
where
/8vw Jo ca/ v > : w
where s is the value of y at the depth of the surface of no
strain.
Hence, if 8W be volume of rock crushed in time 8t above
the surface of no strain, U = dW/dt, and therefore
^n/W Jojocay *
This gives the total volume of the rockfolding since consolida
tion, while equation (8) gives the rate at which rockfolding
takes place at time t.
For a first approximation, we may put
1 1
c— ay c
and
f(y)=Ac*ay/3c 3 y*c%
where
A=3(v/2^ + 4/3)/8.
We thus get
P {~ dy = Acasi/3c*s*ic*s\
where
* = ¥[£+'
and, lastly, of the difference
a
B
VSirg
in iron and steel.
Table II. contains the results of my former experiments
(I. c. p. 265), which are here reproduced in order to show the
complete range experimented over.
The results are also exhibited graphically in fig. 2, in which
the points +, 0, show the values of G as a function
of B in iron and steel respectively, while the straight line
represents the function B/ \/Sirg.
The points # are taken from the former experiments,
and represent the square root of the tractive force per sq.
centim., which in that case was theoretically equal to —
\/Sirg
All the observed points lie near the straight line ; the
greatest difference is about 3 per cent. Most of the points
lie under the line ; this is probably due to the spreading
action of the gap in the isthmus, which diminishes the tractive
164
Dr. E. Taylor Jones on
9 9 Jo
Magnetic Tractive Force. 165
force. The agreement is, however, on the whole such that
the equation (3)
B 2 
Sirg
can be regarded as verified.
According to Maxwell's theory (I. c. § 643) the first term
B 2
rr — is the total electromagnetic tension existing in the narrow
airgap between the ends of the two bars ; similarly, H 2 /S7rg
is the tension in the gap, depending only on the " external "
field H. The correctness of this interpretation of the term
H 2 /87r# follows from the wellknown experiments of Quincke
and others.
This latter part of the stress is not included in the tractive
force in the present arrangement : it merely serves to com
press the brass pillars S l5 S 2 , S 3 .
If the bar C 2 were firmly fixed in its polepiece (instead of
sliding in it), and if the two halves of the electromagnet were
pulled from each other, the pull in grammes weight in this
case, corrected for the tension due to all those tubes of in
duction which do not pass along the isthmus, would be equal to
B 2
■5 — X area of contact.
%Trg
This case corresponds theoretically to the arrangement of
my former experiments, but cannot of course be practically
carried out.
Measurement of High Inductions.
After Maxwell's law of tractive force, or electromagnetic
stress, had been verified as explained above for inductions up
to 40,000 C.Gr.S., its general truth could be assumed. And
just as various physicists have conversely used the tractive
force for the measurement of lower inductions, it now seemed
feasible to apply a similar process to the measurement of the
highest attainable inductions, all other known methods failing
in this case.
I therefore arranged a few more experiments with this
object. Two polepieces of 120° vertical angle were made of
the very best annealed Swedish iron. The point of one was
slightly flattened, ground plane, and polished, so that it pre
sented a small circular poleface whose diameter was only a
fraction of a millimetre. Through the opposite polepiece a
slightly conical hole was bored, through which a long wire of
166 Dr. E. Taylor Jones on
annealed soft iron (of diameter 0586 millim.) could be drawn,
which fitted accurately in front and more loosely behind.
The friction was thus only a fraction of 1 gramme weight.
The end of the wire was also carefully polished and, as well
as the opposite small poleface, examined and measured with
a Zeiss microscope.
With a magnetizing current of 25 amperes, the weight
supported in this case was 249 grms. weight ; this corresponds
to a pull of 92*39 kilogs. weight per sq. centim. (1314 lb. wt.
per sq. inch).
The maximum value of the magnetization was, moreover,
determined by means of an isthmus consisting of a bundle of
50 pieces of the same wire pushed through the former pole
pieces of 78° 28' vertical angle. This value was found to be
about 1800 C.G.S.
Taking also 500 grms. weight as the approximate value of
1 P 1
the integral  I TLdL we have from these data the values of
ffj*
B 2 — H 2 , and of B — H = 47rl ; from these were deduced the
values
B= 61,900 C.G.S.
H= 39,300 „
These experiments were extended still further by drawing out
the wire to a thickness of 0*2412 millim., and narrowing
down the hole in the polepiece correspondingly.
After the wire had been once more annealed in a spiritflame
and its end ground plane, the weight supported with a
current of 40 amperes was 52*5 grms. weight, corresponding
to a pull of 114*9 kilogs. weight per sq. centim. (1634 lb.
wt. per sq. inch).
Hence follow, as above, the values
B = 74,200 C.G.S.
H= 51,600 „
and the permeability is only
^=B/H = 1*44.
The calculated value of H from the StefanEwing loga
rithmic formula is 50,450 C.G.S., assuming that the conical
polepieces are uniformly magnetized with a maximum inten
sity of 1800 C.G.S. To this must be added about 750 G.G.S.
due to the direct action of the coils of the electromagnet with
Magnetic Tractive Force. 167
a current of 40 amperes. The total calculated field is thus
51,200 C.G.S., which agrees well with the value found above.
In this experiment the area of the isthmus was about
5 of the area of the base of the conical polepiece.
200>000
This induction (B = 74,200) exists also by continuity in the
narrow airgap at the contact, and here it is to be regarded
as " external " field. This will not be much altered at the
moment of separation as long as the gap is narrow in com
parison with its transverse dimension. It was then observed
with a magnifying glass that, unless the apparatus was very
carefully cleaned, small microscopical iron filings flew into the
gap when separation took place and formed new " isthmuses "
across it, whose thickness was very small in comparison with
that of the \ millim. iron wire, so that their selfdemagnetizing
effect could be neglected. The induction B' in these is thus
B'= 74200 +4^.1800=96,800 C.G.S.
We may therefore conclude that inductions of nearly
100,000 C.G.S. can exist in iron without any special occur
rence.
If the wire is pulled back so that its end is in the plane of
the edge of the opening, the field between the polefaces is
then about 60,000 C.G.S. ; it only extends, however, over a
few tenths of a millimetre*.
At the instant of separation a tension of (74200) 2 /8?r^ = 224
kilogs. weight per square centim. (3185 lb. wt. per sq. inch)
is transmitted across the airgap ; this is about the limit
of elasticity of lead. In this connexion it is interesting to
remember that the tension between the plates of an air
condenser at atmospheric pressure cannot far exceed 2 grms.
weight per square centim. without a spark appearing. An
absolute vacuum, however, or a body of high specific induc
tive capacity can transmit much greater tensions.
My best thanks are due to Dr. H. du Bois, in whose
laboratory the above experiments were carried out, for his
help and advice.
Berlin, Christmas, 1895.
* If the field extends over several millimetres an intensity of over
40,000 C.G.S. can scarcely he reached. See du Bois, Wied. Ann. 1.
p. 547 (1894) ; J. B. Henderson, Phil. Mag. Nov. 1894.
[ 168 ]
XXII. Researches in Acoustics, — No. X.
By Alfred M. Mayer*.
[Contents. — The Variation of the Modulus of Elasticity with Change
of Temperature, as determined by the Transverse Vibration of Bars at
Various Temperatures. The Acoustical Properties of Aluminium.]
Summary of the Research,
POISSON, in his Traite de Mecanique (Paris, 1833, t. ii.
pp. 368392) t discusses the laws of the transverse
vibrations of a bar free at its ends and supported under its two
nodes. He shows that the frequency of the vibrations of the
bar is given by an equation, which, reduced to its simplest ex
pression, is N=Vx r0279jj; in which N is the number of
vibrations per second of the bar, t its thickness, I its length,
and Y the velocity of sound in the direction of the length of
the bar.
To ascertain how nearly the frequency of the transverse
vibrations of a bar, computed by Poisson's formula, agrees
with the result obtained by experiment, the following method
of experimenting was used,
Rods of steel, aluminium, brass, glass, and of American
whitepine (Pinus Strobus) — substances differing greatly in
their moduli of elasticity, densities, and physical structures —
were carefully wrought so as to have the length of 1*5 + metre,
the thickness of 0*5 cm., the width of 2 cms., and a uniform
section throughout their lengths. The velocity of sound in
these rods was determined by vibrating them longitudinally
at a temperature of 20°, while held between the thumb and
forefinger, and their frequencies of vibration ascertained by
the standard forks of Dr. R. Koenig's tonometer.
Out of each of these long rods were cut three bars of the
length of 20 cms., and these bars, also at 20°, were supported
on threads at their nodes, vibrated transversely by striking
them at their centre with a rubber hammer, and their
frequencies of vibration determined by the forks of the tono
meter.
The mean departure of the observed from the computed
numbers of transverse vibrations (see Table I.) is g^g ; the
computed frequency being always in excess of the observed,
* From an advance proof from the American Journal of Science for
February 1896 communicated by the Author, having been read before
the British Association at Oxford, August 1894.
t See also < The Theory of Sound/ by Lord Rayleigh, 1894, vol. i.
chap. 8.
Dr. A. M. Mayer's ResearcJies in Acoustics. 169
except in the case of glass, where the computed is ^js below
the observed frequency.
In Table I. I — length and t — thickness of bar in centimetres
at 20°; V = velocity of sound in centimetres in bar at 20° ;
N = number of vibrations per second at 20°.
The close agreement of the computed and observed values
shows that, by vibrating a bar at various temperatures,
the variation of its modulus of elasticity with change in its
temperature can be obtained. We observe N at various
N
temperatures of the bar; then Y= is computed,
10279 %
Y 2 d
and the modulus M= — . As t, l> and d (the density of
the bar) vary with the temperature, the coefficient of
expansion of each bar and its density at 4° were determined,
so that the dimensions and density of the bar could be
computed for each of the temperatures at which it was
vibrated.
Experiments were made on five bars of different steels, on
two of aluminium, on one of St. Gobain glass, one of brass,
one of bellmetal, one of zinc, and one of silver. The results
of these experiments may be summed up as follows : —
The modulus of elasticity of St. Gobain glass is 1*16 per cent, less at 100° than at 0°.
the five steels „ 224309
,, „ brass ,, 3*73
„ „ bellmetal „ 4'3
,, „ aluminium „ 5"5
silver „ 247 „ „ 60
zinc „ 604 „ „ 62
The decrease of the modulus of elasticity of glass, aluminium,
and brass is proportional to the' increase of temperature;
straight lines referred to coordinates giving the results of
experiments on these substances. The five steels, silver, and
zinc give curves, convex upwards, showing that the modulus
decreases more rapidly than the increment of temperature ;
while bellmetal alone gives a curve which is concave
upwards, its modulus decreasing less than the increment of
temperature. (See Curves, fig. 5, p. 185.)
The more carbon a steel contains, the less is the fall of its
modulus of elasticity on elevating the temperature of the
steel. Thus, the modulus of the steel with 1*286 per cent, of
carbon is 2'24 per cent, less at 100° than at 0°, while the
steel containing 0'15 per cent, of carbon has a modulus at
100° which is 3*09 per cent, lower than its modulus at 0°.
So far as experiments on a single steel containing nickel
Phil. Mag. S. 5. Vol. 41. No. 250. March 1896. N
170 Dr. A. M. Mayer's Researches in Acoustics.
permit of any general deductions, it appears that the presence
of nickel in a low carbon steel lowers its modulus of elasticity.
Thus, steels No. 3 and 4, having respectively *47 and "51 per
cent, of carbon, have a modulus of 2130 xlO 6 ; while steel
No. 5, containing "27 per cent, of carbon and 3 per cent, of
nickel, has a modulus of 2080 x 10 6 , which is 2*35 per cent,
lower than that of steels Nos. 3 and 4.
The presence of nickel in a steel may, in a diminished
degree, have the effect of carbon in lessening the lowering of
the modulus when the temperature of the steel is increased.
Thus the percentage of the lowering of the modulus, by
heating from 0° to 100°, of steel [No. 5 containing 0'27 of
carbon and 3 per cent, of nickel, is the same as that of steel
No. 3 with 0*47 per cent, of carbon.
If a bar of any one of the substances experimented on is
struck with the same energy of blow, by letting fall on the
centre of the bar a rather hard rubberball from a fixed height,
the sound emitted by the bar diminishes in intensity and in
duration as the temperature of the bar is raised. Thus :
Brass at 0° vibrates during 75 sees. ; at 100° it vibrates during 45 sees
Bellmetal ,, „ 55 „ „ „ 15
Aluminium „ ,, 40 „ „ „ 12
J. & C. Cast Steel „ „ 80 „ „ „ 5
Bessemer Steel „ „ 45 ,, „ „ 1*5
St. Gobain Glass „ „ 6 „ ., „ 3'5
Zinc at 0° vibrated during 5 sees. ; at 20° only during
1*5 sec. At 62° it vibrated for so short a time that it only
gave three beats with forks of 1090 and 1082 v. s. At 80° it
was not possible to determine the pitch of the bar, and at 100°
the bar when struck gave the sound of a thud. The bar of
silver acted in a similar manner to the bar of zinc — it was
even less sonorous than zinc, — thus flatly denying the " silvery
tones " attributed to it.
These phenomena do not depend on the fall of modulus, but
on changes in the structure of the metal on heating it, which
cause the blow to heat the bar and not to vibrate it.
Bellmetal was found to be an alloy peculiarly well suited
for bells, as the intensity and duration of its vibrations were
the same at 50° as at 0°, all other substances showing a
marked diminution of intensity and duration of sound at 50°.
A bar of unannealed drawn brass, after it has been heated
to 100°, has its modulus at 20° increased j^ per cent.
(See Table III. and fig. 11, p. 188.)
In this research I had the good fortune to have the
assistance of Dr. Eudolph Koenig, of Paris. He not only
placed at my service the resources of his laboratory and
Dr. A. M. Mayer's Researches in Acoustics. 171
workshop, but generously gave me constant assistance during
the experiments — making the determinations of the numbers
of vibration of the rods and bars with the standard forks of
his tonometer. Without his aid this work could not have been
done. For instance, in the cases of the bars of silver and zinc
the beats they give with a fork are so few that they cannot
be compared with a chronometer ; but Dr. Koenig, from his
long experience in the estimation of beats, was enabled to
form an accurate judgment of their number per second from
the rhythm of the beats. The determination of pitches
extending through such a range of vibrations as occurs in this
research can only be made with Dr. Koenig's " grand tono
metre " — a unique apparatus of precision, giving the fre
quency of vibrations from 32 to 43690 v. s., and really
indispensable to the physicist who would engage in precise
quantitative work in Acoustics.
We now proceed to give accounts of the several operations
performed in the progress of this research.
Determination of the Velocity of Sound in Rods.
In the determinations of the velocity of sound in the rods of
1*5 m. in length, I used the method of Chladni*. Kundt's
method of obtaining nodal lines of fine powders in a tube, by
vibrating a rod whose end carries a cork which fits loosely the
end of the tube, is not accurate. The weight and friction of the
cork, the necessity of firmly clamping the rod at a node, and,
above all, the want of knowledge of the velocity of sound in the
air of glass tubes of different diameters, renders this method, so
beautiful and ingenious, worthless for accurate measures of the
velocity of sound in solids.
The curves in fig. 1 show the very diverse determinations
of the velocity of sound in the air in tubes of different dia
meters by the physicists Kundtf, Schneebeli J, Seebeck§,
and Kayser ] . The velocity of sound in metres is given on
the axis of Y ; the diameter of the tube in centimetres on the
axis of X. Ku stands for Knndt, Sch for Schneebeli, Se for
Seebeck, and Ke for Kayser. The most precise measures of
velocities are those of Kayser, who closed the end of the tube
with a cork attached to the end of a steel bar, while the other
end of the bar was securely clamped. The frequency of the
transverse vibrations of the bar was registered by a style
* Traite dAcoustique, Paris, 1809, p. 318 et seq.
t Bericht. der Akad. der Wiss. zu Berlin, 1867.
t Fogg. Ann. 1869, t. 136. § Yogg. Ann. 1870, t. 139.
 Pogg. Ann. 1877, t. 2. p. 218.
N2
172 Dr„ A. M. Mayer's Researches in Acoustics.
describing the sinusoids of the vibrating bar. Thus the
weight and friction of the cork introduced no error. In a
similar manner I obtained the velocity, marked M in fig, 1,
by vibrating a rod of aluminium. The frequency of the
Ffc. 1.
8 em.
longitudinal vibrations of the rod was measured while the
cork at the end of the rod was vibrating in the mouth of the
tube. The result agrees closely with Kayser's. It is needless
to discuss the curves of fig. 1.
Dr. A. M. Mayer's Researches in Acoustics. 173
The method of Chladni, used exactly as that eminent man
used it, remains the best we have. It is important, however,
to note that the rod must be held between the thumb and
forefinger when it is vibrated and not clamped when vibrated.
When clamped it always gives a higher frequency, as shown
by the following experiments : —
Steel rod clamped 3429 2
Steel rod held between fingers . . 3428*4
Aluminium rod clamped .... 3377*0
Aluminium rod held between fingers. 3376*4
The frequency of the vibrations of the rods of steel, brass,
aluminium, glass, and pine wood, when held at the middle of
their lengths and vibrated so as to give their fundamental
tones, gave exactly the octaves of these fundamental tones
when held at onequarter of their lengths and vibrated.
Determination of the Lengths of the Long Rods and of the
Lengths and Thicknesses of the Bars.
The lengths of the rods of 1*5 + metres were ascertained by
comparison with the rod of steel whose length was measured at
the Bureau International des Poids et Mesures. The lengths
and thicknesses of the bars which were vibrated transversely
were measured with micrometer calipers. The readings of
these calipers were tested by comparison at 20° with a series
of standards of various lengths of inches and fractions of
inches, made for me with great care by Mr. George M. Bond,
who has charge of the gauge department of the Pratt,
Whitney Co. In reducing the comparisons to centimetres,
I adopted the value of the inch as equal to 25*4 millimetres.
In obtaining the length of a bar, the mean of several measures
in the axis of the bar and in directions parallel to the axis and
at various distances from it was adopted. The thickness of a
bar was taken as the mean of measures taken throughout the
length of the bar at points designated by the intersections
of lines drawn parallel and at right angles to the axis of the
bar.
The dimensions of the bars were measured at 20 , except
those of steels Nos. 3, 4, 5, which were measured at 18°'25.
Determinations of the Coefficients of Expansion of the Bars.
To determine the coefficients of expansion of the bars,
I devised the apparatus shown in fig. 2. In a brass tube, T,
174 Dr. A. M. Mayer's Researches in Acoustics.
the bar, B, rests in slots in the supports, S, S'. The tube T
is slightly shorter than the bar B. Washers of rubber (shown
in black in the figure), of the same diameter as the outside
diameter of the tube, are placed in the screw caps, C, C.
These washers are perforated with holes of diameters smaller
than the thickness of the bar. When the caps are screwed up,
the rubber washers press against the ends of the bar. This
Fiar. 2
pressure is further increased by flat rings which surround the
holes in the washers, and are pressed against these washers by
means of the springs, D, D'. By this arrangement the sur
faces of the ends of the bars are exposed, while the contact
of the washers on the bars makes a water and steamtight
joint. Thus the bar may be surrounded with ice, or with
steam, or with a current of water of different temperatures,
and be cooled or heated up to its terminal planes, while the
holes in the washers allow the micrometerscrews, M, M', to
be brought to contact with the terminal planes of the bar.
Two helical springs are attached to the column A. The
other ends of these springs are fastened to rods projecting
from the tube T. Thus the same pressure of contact is always
made between the bar and the end of the micrometerscrew M.
The tube T is supported in Vs, V, V, and the greater part of
Dr. A. M. Mayer's Researches in Acoustics. 175
the weight of the tube is taken off the Vs by helical springs
fastened to a frame above the apparatus. The tension of these
springs can be so regulated that the tube rests on the Vs with
the same pressure when it has steam passing through it and
when it is filled with ice. The column, A, and the Vs, V, V,
are insulated from the base of the apparatus by thin plates of
ebonite, e. Between the bindingscrews, E and E f , and con
nected by wires, are the voltaic cell F, the galvanometer Gr,
and a box of resistancecoils, R. The micrometerscrew, M',
with which the variations in length of a bar are measured,
is mounted as follows : — The screw passes through its nut in
a massive brass plate which rotates around nicely fitted
centres at H. These centres are supported by two side plates
not shown in the figure. A spring, K, is fastened to the
lower part of the swinging nutplate and brings this plate
against the plate, L, firmly fastened to the base of the appa
ratus. When the swinging plate is vertical and the axis of
the screw horizontal, the swinging plate fits accurately the
surface of the fixed plate, L. By turning the rod, N, the
swinging plate and its screw can be rotated away from the
bar. This arrangement allows the screw to be swung out
of the way while the tube, T, is being placed in the Vs.
Also, it prevents any strain between the micrometerscrew, M',
and the column, A ; which would take place if M' were fixed
and it should be brought in contact with a hot bar in the
tube, T.
With careful manipulation, successive electriccontacts can
be made on a bar in the tube, T, surrounded by ice, so
that the variations in a series of measures will not exceed
Woo mm ? "with a resistance of about 200 ohms placed in
the circuit.
It may be reasonably objected to this apparatus that when
the micrometerscrew touches the bar at 0° it is cooled and
shortened, and that when it touches the bar at 100°, or at
temperatures higher than that of the screw, the latter is
heated and elongated. This error, however, is quite small,
and may be neglected in our work. If we assume that one
centimetre of the screw is heated 10°, which is a large
estimate, considering the duration of contact of screw and
bar during a measure, the shortening or elongation of 1 cm.
of the screw by cooling or heating it 10° amounts to only
•0012 mm., or jqqqqq of the length of the bar. This change
in the length of the screw will affect the coefficient of
expansion of the bars only 00000006.
176 Dr. A. M. Mayer's Researches in Acoustics.
Determination of the Densities of the Bars at 4°.
The bar whose density was to be determined was immersed
in water at 4° for a couple of hours. The bar was then sus
pended by a platinum wire in water at 4° and weighed. The
bar was then removed from the wire and a quantity of water
equal in volume to the volume of the bar was added to the
water in the vessel, and the platinum wire, now immersed
exactly as it was when the bar was attached to it, was weighed.
This weight, subtracted from the previous weighing, gave the
weight of the bar in water. Every precaution was taken to
prevent, by means of screens, the action on the balance of
the currents of cold air in the balancecase, which are pro
duced by the constant descent of air from the sides of the
cool vessel.
The Apparatus in which the Bars were Heated and Cooled. On
the precautions used so that one is sure of having the real
temperature of the bar when it is vibrated.
The apparatus used to heat and cool the bar is shown in
fig. 3. In a brass box, C, is inclosed a box, C, containing
the bar, B, supported on its nodes, N, N, by threads held by
upright rods. From this central box two tubes, T, P, pass
through the outer box C. The inner box is made watertight
and steamtight by a rubber washer which is pressed between
the top of the box and its cover by means of screws. Through
the tube, T, the bar is vibrated by letting fall upon its centre
a rubber ball fastened to a light wooden rod. On the blow of
the ball it rebounds, and the rod is caught by the fingers in its
upward motion. The cork is then at once replaced in the
tube, T. The sound from the bar is conveyed to the ear, at
E, by means of a tube (fig e 4). One branch of this bifurcated
tube leads through a rubber tube to the pipe, P, of the box,
fig. 3. The other branch leads to the fork, F, the number
of whose beats per second made with the vibrating bar is
measured by a chronometer. The pipe, S, allows the steam
to issue when water is boiled in the box, C, by a gas lamp.
The flow of gas through this lamp was neatly regulated by a
stopcock turned by a long lever. The box, C, is covered,
except at the bottom, with thick felt.
To determine the frequencies of vibration of a bar through
a range of temperature from 0° to 100°, the following method
was used. The box, C, was filled with ice, surrounding the
inner box, C. It thus remained for an hour so that the
boxes were cooled down to 0°, and the moisture in the inner
Dr. A. M. Mayer's Researches in Acoustics. 177
Fig. 3.
"S"
8
Fisr. 4.
D=aa^E
178 Dr. A. M. Mayer's Researches in Acoustics.
box had been condensed so far as it could be at 0°. The bar,
which had been in ice for two hours, was wiped dry and quickly
introduced into the inner box. A thermometer, T (made by
Baudin and corrected), which entered the boxes through
stuffingboxes, and whose bulb touched the under surface of
the bar, was read till it became stationary. The bar was now
vibrated, and its frequency of vibration determined for the
temperature given by the thermometer.
The lamp was now placed under the box, and the water in
it boiled till the thermometer reached its maximum reading
and the reading remained stationary during a halfhour.
The vibration frequency at this temperature was taken. The
flame of the lamp was now lowered and the box allowed to
cool very slowly, at the rate of 1° fall of temperature in about
eight minutes. When the thermometer read 80°, 60°, 40°,
the flame of the lamp was carefully adjusted, so that these
successive temperatures were maintained during 15 minutes.
We then took the frequency of vibration of the bar.
The numbers of vibrations of the forks used in the deter
minations of the pitches of the bars were corrected for tem
perature by the coefficient '0001118, determined by Dr. Koenig
in 1880 (Quelques Experiences d'Acoustique, Paris, 1882,
p. 172 et seq.).
The subsequent tables show the results of the experiments
and give the computations of velocities and moduli founded
on them. The curves express graphically the effect of change
of temperature on the modulus of elasticity of all the bars
experimented on. The circles, on or near the curves, give
the data as determined by the experiments.
In Table III., T = temperature of bars, Z = the length, £ = the
thickness, and V = the velocity of sound through the bars, in
centimetres. M = the modulus in grammes per square centi
metre section of the bar. g, at Paris, equals 980*96. D = the
density, and N=the number of vibrations of bar per second
at temperature, T.
All of the bars were annealed, except those of Jonas and
Colver steel, of the French aluminium, and of brass ; these
were experimented on just as they came from the drawbench.
For the analyses of the substances of the bars experi
mented on, I am indebted to my colleagues, Professors
Stillman and Leeds.
Dr. A. M. Mayer's Researches in Acoustics. 179
las ^h I sq
+ + +
I!
ii ii
,_
h
CO
on
,_l
,— i
o
IH
ri
, — i
r H
« II
§
OS
o
CI
so
so
CO
CO
CO
CM
O
CO
oT
§
oo
CO
9
CO
CO
s
CO
CO
o
o
CO
<*
o
CO
1
m
En
a
> B
? a
©
fc ©
CO ©
00
3
2
CO^
Xco
»o
s?
8»
II
6 II
s"
► a
r ©
00 2
?§
Xco
>
* a
cq ©
£co
22o
CO fc 00
no
CO
iH •* O
cq co oi
os os os
l— f T— 1 Jt— CO
o o o
os os cs
c
O O tH O
*P f
5 »p
ip IQ ip 1— O0
CO 1C
CO CC CO
Cq
»f
3 OS CO
CO
©
t—
CM
p
i— 1
512602
511908
510112
509389
508470
507065
N
i— 1 CO lO CO © ■— I
CC © Tf CM — i o
LO O iC iC i£3 O
00 00 © CM CO ©
CO O CO O CO rfi
(MW»OOh
b 1 CO CO © ©
CM © CM CC iQ O
IC I— lO r 1 CO ©
O CO X 1 © tH
h o o o o o
IC iQ iO iCJ O iO
OOCD^HH
CM »l CO GO t ib
CC CC iQ iO © o
© CC © © © ©
CO — i O O OO
CO CS iQ t t o
© CO i'~ tH CM ©
Cl ,— I ^H — I rH ,(
oo go oo on oo oo
Tfi GM i— I CO CC ^
CM CM CM rt . H
CO 00 00 CO 00 CO
CO tf CO i—l CM ii
CM CO CC CO © b
cb h co t ib cm
1— h © OO ©
oo oo co cooo oo
CO CO GO ^ © tH
CM Ol — i — ( O ©
GO CC CO CO GO 00
t *>■ fc J>» t ij
© ©
© cH © GM CC — i
t hh CO © CM CM
CO GO CO CO CO 00
t t i> b i> tr
ie
© — ' Tfrl il CO ©
Tf Thl CO CO CM rl
oo coco oo co co
CM l^ ^h vo © tH
© rfi ttfi CO CM CM
CO 00 00 00 GO 00
t l> t tr t tr fc fc>« i> fc fc fc
K
n
© ©
LC3 © lO iQ iC O
CM CM CM CM CM CM
rfi © tCO © ©
CO O © © ^f< tH
Tfl ^CH TH ■* "+ 1 ^
iC iO iC lO >o o
© O «T~tH CO ©
CM CM — < ^h o ©
iO »C O uo iO o
© © © © © o
CO f © © CM ©
© © © on co t
iC iC iO m iO iO
© t00 © O r4
CO CO CO CO * f
© © © © © ©
LG O C O O iO
© CO tH Th © rH
* cc tr Co © CM
©© © © il iH
^ f Tfl +l f !f
CO © CC © © ©
iO CM CO tr © CO
35h'MCOCW
CM CO CO CO CO CO
© © CO © CC ©
OO CM 00 CO 00 Tfl
CO © ri CO TtH CC
ih CM CM CM CM CM
*H"< TJH T^t Tfl "^1 Tfl
CO CC © CO CO CO
i© © co © ©
© rr cc cm cc ©
CM GM CM CO CO tJH
©© © © ©©
CO 00 f t © CM
— < © © © o ©
© lO © © t— t—
CO CO © CO CO CO
t CM tH © © ©
i— ' CC iH ^ — ' ©
© © cc © t r
© CO CO © © ©
©©©£©©
,— i iO © © © ©
»CiO?OCCNt>
© © © © CO ©
6 o © © o©
CM CM CM CM CM C J
CM CO
OO © © H © CO
CM T^ © GO ©
iO
CM ©
© cb ©© © ©
IH tf © 00 ©
«3 .
o
182 Dr. A. M. Mayer's Researches in Acoustics.
%
CO CO CO
O O O O O O ©'© O O O O CO
X X X XX X
i— I rH © CO i— I CO
IMhOQONiQ
i— * i— I i— i o o ©
oqoqoq oqoq oq
X X X XX X
os oq co coco os
o o as oo t~co
H rH O O O O
X X
oca
rH O
to co to to co to to
O O OO O OO
X X X X X X X
cb t oo rH © ib o
^
coi> oq
CO GO CO
rH cq o rH os o
1>Q0h
go oq co ii o os os
CD ^ 00 H O Tt* O
VO rH CO CO CO Ol rH
OS OS OS OS OS 0s OS
►MR*
OS
O rH Ol i— l OS 00
rl i— I rH rH O O
t~r~co oq t>. oo
COOOiQ^t
CO CO i—i O C CO
GO t^cO O CO t t~
OOONiOHcq
CO O O O O O
rH rH Tt* rH rH rH
rH lO
O CO
© GO
co o
rH rH
(NOSNCOHCOH
1> CO CD CO CD CD CO
iQ C iQ "O O iO >0
HTticDOOiMTH
oq cd o rH os co
HCOCONHI
CO GO GO GO 00 GO
b 1 l> t fc> !>■
00 00 00 00 00 GO
00 O
tco
tCO
rH rH
GO 00
O O CM CO 00 rH iO
© O O id O H 'M
HMCO(M(NhO
CO CO CO CO CO CO CO
ob oo oo oo oo oo do
O tTH OtM
00 t^ t~ l> CO CO
CO CO CO CO CO CO
oo
O o
cq l>
GO 00 rH oo oq rH
oq o t rM oq os
—  oq cq co rH rH
o o o o o o
t^GO
oq o
rH oq
oo
oq© goo o ooo
O OS CO rH O 00 rH
rH rH O CO CD t~ 00
oq oq oq oq oi cq oq
o
rj<
CO
CO
rH
H
O©
r?S
©
8
i—OC5G}
© ©
© rH © © © rH OS
oq
CXI
0q rH CD CO OS
oq
oq rH OCO fcOS
co co
GO
03 A
63
*Ss
* .
n 03
*u

© cq co oo oo rn t~©©cqio©
00 CO
CO © 00^ CO
©© © © 00
00 00 t^t^t
'£§222 222222
v v v v v XXXXXX
55££5 qooo^tcqco
tp CO i— i © t— rH © 00 © iO CO
i— I r— I i— 1 i— I O lO "*H "^P ^ ^p t}H
rH rH rH rH rH fc t> I> t> t t
52 co oo © co ©
cp oo b © oo t
Cq © cq rH SO rH
rH os »c © co Tti
C3 CN i— I OS SO I>
co as oi co oo os
© Cq CO 00 00 —l
CN ^h © OS 00 00
l> l> t CO CO CO
^_J U.J \*^ U'J ^
lC © t 00 T* CO
© Cq 0 0(M'0
iHinOOiOON
l>t £~SO © ©
©CJ © ©OS
© TP © 00 fr
CO © © iO ©
CO © CI © ©
H t^ C~ L~ CO
co © oo ^ti co
© © © © oo
QOQOttt
^ CO OS (M ©
hh ON iO CO !>•
»C SO © CO CO
hhhO
© ii tP © © lO
CO CO CO Cq i— i th
00^ ©© »0
t^"^r cq © ©
CO CO CO C4 ©
t^ b 1~ I> b
cq cqcq cq cq
CO 1— I TP Tfl H
SO © Tp OS OS
© io oo cq cq
© © © © ©
■* ^P ■* "tf CO
CO © CO © £ lO
p— I so © © © o
co t~ tp os tp ©
CO CO CO COCO CO
1Q iO iO O iQ O
NhihhOO
so © © © © ©
© © cq © o cq
1Q TfH TT CO 00 Ot
so so so © © ©
t~t~© IO CO
CO CO CO CO CO
tp hh so cq h
cp op tcq so
© b CO r< CO
© © © 00 b©
lQ Hi Hi HH HH HH
b b b b b b
b© CO iO tp ©
CM © Tp rH J^HH
© ©© os oo oo
t so © © © so
cq cq cqcq cq cq
© © cq Tf © ©
© r> co © ics cq
co cq cq iH i— 1 1— i
b b b b b b
cq cq cq cq cq cq
cq cq cq © oo © © © © oo
■*(Nsooj> co — © cq io
— ■ © © © b rHOoooor>
ip >p tp tp hh cp qo b b 1
© © © © © so © so © © cqcqcqcqcqcq
co b©© © io
ii © ©OS 00 CO
CO 00 GO b 1^ b
br I iO 00 © id
io o tp co co cq
"^H "^^ ^i ^^ Tf "^^
cq cq cq cq cq cq
© ©cq © oo ©
b © CO or©
i— i cq cq cq cq co
io H0 io io >o >o
IQlQlQlQlOlO
. _■ © CO ' _
© © © OS © ©
OS © © © © ©
<* ■* Th tp" t* 1 "3
tr^cq © ©
HCOIQCDN
iQiOiOiQiO
T^t ^H T^l T^l ^1
^^ ^^ ^^ *^ ^^
oo cq © cq rp io
© © ii cq co •*
oo © © © os ©
^ ^ "^ "^ ^i "^
t^» »>• t 1 1> t
© cq
cq cq <*< cq ^ cq
rH CI CO'* io ©
so © © © © ©
,1^ ,1^ tIh rH rH ri
cq cq cqcq cq cq
rH cq CO ^ IO©
©©©©©©
© © tH © ©
t^XO 00 rH 00
rH cq cq co co
cqcq cqcq cq
■* © 00 tH ©
os © © © cq
© CN CO CO ^P
cq cq cq cq cq
cb do do go do
© © io © cq ©
cq^n © oo©
©©© rH^>
Cq TP © 00 rH
co io io
© Th ©© © 6s
cq tp so oo ©
184: Dr. A. M. Mayer's Researches in Acoustics.
a .2
ih
EH
rd
CD
s
h3
43
■+3
bf
rH
pj
CD
»j
^3
■**
2^
' J
CD
H
o
bt
d
•3
^>
o
o
rH
5*
s
drd
1— 1 •+=
o
o
br
<*>
o
O
.3
"en
en
d
CD
s
cj
C3
HH
rl
v
O
rn
CD
rd
©*
i—l
0)
H
t>>
en
f<
C*
ctf
3
2
d
CD
O
en
13
CD
a
en
CD
o
d
C3
a
p
r d
rH
CD
P.
H
CD
cd
• to
^
CD
en
CD
r^
CO
en
"3 ^ .
^QrQ ^
o
73 bJD
CD CD mh
S
m
d 
MH
o
en
h3
O
rd
+3
CD
CD
•d d
CD rd
^2
o
0§ Ch
S
O
s
d
"d c3
S rH
•"§
a?
rd
^T3
CD
,?s
CD en
u
0)
d
rl en
rg cd
rO ^ :
(73
o
C+H &*
H
a
W CD
»
CD
m cd
r^ rC
d Sh
+3
*>
o a
n
CD
H CD
i— i
^ "c3
CD ^
O CO OS rH 00 O
O O p CM cp r
O OS CO t cb th
O OS OS OS OS OS
en
g
rH
OffiHtncoN
O CM ip t O CM
o os cb ^ i^ cb
O OS OS OS OS OS
rH
li
S CO.
O CO OS lO CO rH
O iQ OS CO ZD OS
6 Q 00 00 t i
O OS OS OS OS OS
Sot OtCCOONH
gco ONiOCOOQO
rK is o bs OS OS OS cb
^ oososososos
6q
id
6
cl>
TJl
OXOtHCOiQ
O lO rH lO OS CO
6 6 6 cs n i>
O OS OS OS OS OS
o
cq
to o *c oo co :
^ p qq t qs :
o bs t^cb
O OS OS OS
6
ft
'cd
CD
m
O O rH p^cp
go o6ot>cbib+ 
5 U OOSOSOSOSOS
4& "
CD
CD
go
Q
H5*
OHlONWCO
o 6s cts cb cb r~
O OS OS OS OS OS
'S rl O OI CO lO t^ OS
.22 O OS 00 t> CD iO
5© oob^cbib^i
g ^ oososososos
EH
00 c
CN
i
—
c
c
a
I—
Eh °c
c
c
t
c
c
CO
C
C
Dr. A. M. Mayer's Researches in Acoustics. 185
Phil. Mag. S. 5, Vol. 41. INo. 250. March 189(5.
186 Dr. A. M. Mayer's Researches in Acoustics
Fia. 6.
k :::i : ii
:.::::
v<; : !.;•*' ■■■ ■
.:._■
L _ : »i!:i:'
: ; . : . i.
: "
"■ir'i'rr'i '■'/, ',' r, n a ,t
laiiji
..
#j
.' :.::
: ; : :
.:
: ■.::■ ::;: ; .::i
• ■ .:
"o .:•:::
.. .
■■. : :.
x:
;^: : ;
':. r .'■' '.
ij^4
:: ':■■ ■:
: : . ~; ;:.■: ;;•.; ;:::; : :: . :; •::. ^i
'
~*^^
i§3p
Jlii''\~
"flTi.
^f^ :; ;^::.::i, : :;;:::
■ "" : : :;;; ~
yp Si 51:
:s&; : ;^<^: v
I § m m jjp =h p ;^!irr ^ t
IIS
**,!**>•
k &pi^=LnSpP ifejs:
^^ii;iL : ! : ! ■: : :ii;!:;^
"■■ ;;  ; ' ''■ ■ ; " ''■■ y  r  ~ ■':'■
: J '
" .
'.':":' :': .■':':
i::;:^r^
^^^p^mmmk
ggjlj
ii:
J " E (^h
\^;::W
" ■::; ;:: : : i
:: :
'■ :■.:: — irn^s^;!
:. ..: ....
.
■—
■■.
: :: J
 ::: : :; :••:;::::■.;■ i: 1 ::;;^::
"^,
■
?"';.■
ii
^\
. ■:.
ill
: : ;":t:.l!#
J
::lj:
:^ :: ; ■■■■■■
?p! 1
• : ■:• : : : . ! . ! : ::' "N.
.:■ ■■ T
'—::■■■■ 7: :: ho— : ■ "feT ■ .H " : ~ ■ ' ~hr" : " r ~ frf
\ ii ■:: :;i
i .:
?° 5 r4' :
;
. .. v
■
iii ; 
::
—',.
;"::;;.. :j;);:ii:
:.; : ;. ;:::£.
:::::;::!;:i;j::.: ;;::f::,. : ~/ ; ::; : : :
 :'■;;;
; : '■:■:■.■ ::;i.: : :
gi
:. ':■ ' ':
. ■::": ■■:■:■■. ::i:.i: ■ . .. ;:;; t :.:;. .. :
. : ': :. ; ;.pi;.';; : .
..;■. :
 :
■
Fig. 7.
10 10. }0 4 5
BO oo
Dr. A. M. Mayer's Researches in Acoustics. 187
Fie. 8.
P(#
^^ $~ ^j^. W4 cU M' f i w 
f :
jflE]
. 'i . : ■':■' ■[ " ; '1 : "'
i : i

. ■ , '
iEpppBEl
.
Sgffi
. ^ _ r — .
p  + +■■
: ...
 I I
• i '
i[ ;;;: ■
MiHw 1
mMk'n
l :
 
;
::(""  ■ . :i : .u:p~j~ifer

W &ffl
■■[M^yy
m
BHi
■:

H "
jjl.f{i; :
 j\a1 * kc/c
2
2
2
2
g" 10 2 9 30 40
/<* »« ;b :  ; li i: i;: j^;^
;■:;;!
iijSl
h:!
lili;
1
m
7:
•; ;..__J  ri" I ■'■ ; ; ..._.^J^..j. L J ^ "y j n'vb ^
Fisr. 13.
..
■
j_ ; :
.
: :
:::
L .:="
■.,
!";
•■':
.■i:
::::
■
:■'
;;;:
7 : . 9
vpf
**
%
■ ;;
:■:
5
^
~^
^_
tl!  } l^tr:r;: ,:t:;;;:^
; .. ... ;
::!.
: : .
4 *
!
j.H
7 I
^^^g^^0l^^g^$
•^ ;_
TIMMMMi^^^n^MmiiBB^Ji
*:
§
' :
....
.;■.:
j
~\7.
.....
(
m
ijeitei
_
: ; . :
:■: ..
_j_M
1
;;::
;?
______ :^ ____
:,.
'.:
:: ::::•
'■■
■ .
#W)lN^
J
■.:.
"'::
f.
:;•:.:
:: J"
ik : ?s
• OV:;!":
:vn:'t:; :  ;: j" : ^H:>:;;r;:: ::: ! r~>:}::.::: : i^
:i
■;
■ . :
1
.::„}.;._:
■
;•:— :; :: :! :  :; ':' : : •■.:. " ; :!: .;: : : : : ;
;_•

■■
__^
i. ,:
i .:
:::
:..
....
1

■ j
,.:
,.
'
..
::;;
190 Dr. A. M. Mayer's Researches in Acoustics.
Fi 2 \ 14.
Fig. lo.

x r
■ i"
■ f
j
#f 1
i
aftij
■
'^
•,';'.
: .r
; U
iii;
!>■!!
j p
"
ft
in
;■;!
"■j:
III
. ";. : ;
"...
: ~
' J ;
' ' i:lN;ifra%g&g$»^
§
:: !:;;:r::4
7 s<;
H
:1...:L.£
m Sag
.:.Jf
■Mil!';!'
1
lljllf
s.'..\\i. KF[FJFLF_L.j ^ ■ i"'F j ' '
■i : ii ; ipjjt3j
; u j
Jll
1
s
1
B
•£
•:,
i!ii
:: :;
!!!!
:' :
8
F
ifi
s
ill
pf
■
:<
iili
:!!;
;;;.
:f
: i' ; .: : :il
10 20
Dr. A. M. Mayer's Researches in Acoustics. 191
Fijr. 16.
 ■ / ■"■'
:;
iSte: r/ ; : ^ ' " ^ M?T ' i ; j "%' i: 4
;: : L"fi?
zimtjL
1
H
:i!
;■
i
: :;•;;. :
J::^: 7 ':
IjrMpp
H
:;::
5,3 f£
rrT
"' 1
.wWi

Lw 1
p 9 J a 3.0
192 Dr. A. M. Mayer's Researches in Acoustics.
Results obtained by other Experimenters on the change of the
Modulus of Elasticity tcith change of Temperature.
I have found five researches on this subject.
Wertheim, 1844. Ann. de Chim. et de Rhys.
Iron.
Modulus 5'2 per cent, greater at 100° than at 18°.
Modulus 191 per cent, less at 200° than at 100°.
Iron Wire.
Modulus 4*9 per cent, greater at f 10° than at —11° 6.
Modulus 742 per cent, greater at 100° than at 18°.
Wire of English Caststeel.
Modulus 2323 per cent, greater at 100° than at 18°.
Modulus 946 per cent, less at 200° than at 100°.
Modulus at 200° is 11*57 per cent, higher than modulus at
18°.
Steel Wire tempered to Blue.
Modulus 1*97 per cent, higher at +10° than at —10°.
Modulus 51 per cent, higher at 100° than at 18°.
Cast steel.
Modulus 28 per cent, less at 100° than at 18°.
Modulus 5'73 per cent, less at 200° than at 100°.
Silver.
Modulus 5 per cent, less at + 10° than at — 13°'8.
Modulus 187 per cent, greater at 100° than at 18°.
Modulus 1287 per cent, less at 200° than at 100°.
Copper.
Modulus 653 per cent, less at + 10° than at 15°.
Modulus 658 per cent, less at 100° than at 1 8°.
Modulus 20 per cent, less at 200° than at 100°.
Wire of Berlin Brass (Cu = 6755, Zn =3235).
Modulus 7*95 per cent, less at +11° than at —10°.
Kupffer, 1856. Mem. de V Acad, de St. Retersb.
Modulus of iron wire 5*5 per cent, less at 100° than at 0°.
Modulus of copper wire 8*2 per cent, less at 1 00° than at 0°.
Modulus of brass wire 3*9 per cent, less at 100° than at0°.
Dr. A. M. Mayer's Researches in Acoustics. 193
Kohlrausch and Loomis, 1870. Pogg. Ann.
Modulus of iron wire 5 per cent, less at 100° than at 0°.
Modulus of copper 6 per cent, less at 100° than at 0°.
Brass &2 per cent, less at 100° than at 0°.
H. Tomlinson, 1887. Phil. Mag. xxiii.
Says, "my own experiments show that both the torsional and
longitudinal elasticities of iron and steel are decreased by about
2 J per cent, when the temperature is raised from 0° to 100°."
M. C. Noyes, 1895. The Physical Review.
Modulus of a piano wire of ^ 4 q mm. diam. 5 per cent, less
at 100° than at 0°.
The results of Wertheim's experiments giving an increase to
the modulus, as the temperature rises, of iron, iron wire, wire
of English caststeel, steel wire drawn to blue, and silver,
have not been confirmed in any instance by subsequent
experiments ; only for caststeel rod and copper did he obtain
a diminution of modulus for a rise of temperature from lb°
to 100°. Yet he found that a wire of English caststeel had
a modulus 23 per cent, higher at 100° than at 18°.
On the Acoustical Properties of Aluminium.
The low density (2*7) of aluminium combined with a
modulus of elasticity of only 712 x 10 6 render this metal easy
to set in vibration ; a transverse blow given to a bar of this
metal causes it to vibrate with an amplitude of vibration
greater than that which the same energy of blow gives to a
similar bar of steel or of brass. This fact has given rise to
the popular opinion that aluminium has sonorous properties
greatly exceeding those of any other metal. This opinion is
erroneous. If a bar of aluminium and a bar of brass having
the same length and breadth and giving the same note, are
struck transversely so that the bars have the same amplitude
of vibration, the bars give equal initial intensity of sounds ;
but the bar of aluminium from its low density and because of
its internal friction will vibrate less than onethird as long as
the bar of brass. Thus, a bar of aluminium and a bar of brass
of the same length and width and of such thickness that they
gave the same note, SOL 4 of 768 v. d., were vibrated so that
the sounds at the moment of the blows were, as near as could
be judged, of the same intensity. The duration of the sound
of the brass bar was 100 seconds; the sound of the aluminium
bar lasted 30 seconds.
The readiness with which a bar of aluminium vibrates when
acted on by aerial vibrations of the same frequency as those
19 4 Dr. A. M. Mayer's Researches in Acoustics.
given by the bar, gives one the means of making many charm
ing experiments in which " sympathetic vibrations " come
into play.
I here describe an experiment which I devised to show the
interference of sound in a manner similar to analogous ex
periments in the case of light. The resonant box on which
Koenig mounts his UT 5 (1024 v. d.) fork is open at both ends
and has a length of nearly a half wave of the sound of the
fork. If this resonant box is held with its axis vertical, above
an aluminium bar in tune with the vibrating fork, the bar does
not enter into sympathetic vibration with the fork, because the
sonorous pulses, on reaching the aluminium bar from the two
openings of the resonant box, differ in phase by one half wave
length. But if the axis of the box is held parallel to the axis
of the bar, then the sonorous waves reaching the bar have
travelled over equal lengths from the openings at the ends of
the box, and these waves conspire in their action and the
aluminium bar enters into sympathetic vibration.
As this experiment is an interesting one I here give details
as to the manner of making it. The bar of aluminium has a
large surface, having a length of 17 cms. and a width of
5 cms. The two nodal lines, which are at a distance from the
ends of the bar equal to §ths of its length, are drawn on the bar.
The bar is supported under these nodal lines on threads
stretched on a frame. This frame is of such a height that the
under surface of the aluminium bar is 8'4 cms., or one quarter
wavelength, above the surface of the table, so that the vibra
tions of the bar and those of the waves reflected from the
table will act together. The upper surface of the bar is
covered with a piece of thick cardboard, in which is cut a
rectangular aperture, having for length the distance between
the nodal lines and a width equal to that of the bar. As this
piece of cardboard rests on supports which lift it a slight dis
tance above the surface of the bar, the latter, when it vibrates,
does not send to the ear the vibrations of the surfaces of
the bar included between its nodal lines and its ends, which
vibrations are opposed in phase to those given by the central
area of the bar. Thus the sound emitted by the bar is much
increased and the experiment rendered more delicate and im
proved in every way. I have found that the experiment
succeeds best when the centre of the resonant box is held
about 58 cms., or 7 7 above the surface of the aluminium bar.
4
This experiment works best in the open air, away from the
action of soundwaves reflected from the walls and ceiling of
a room.
Dr. A. M. Mayer's Researches in Acoustics. 195
The fact that aluminium gives, from a comparatively slight
blow, a great initial vibration, and that its vibrations last for
a short time, render this metal peculiarly well suited for the
construction of those musical instruments formed of bars
which are sounded by percussion and the duration of whose
sounds is not desirable.
1 had hopes that aluminium would prove to be a good
substance out of which to make plates on which to form the
acoustic figures of Chladni. Experiments have shown that
aluminium is not suited to this purpose. I had plates of
aluminium carefully cast, with 2 cms. of thickness. These
plates were turned down on the faceplate of a lathe to thick
nesses of 2 mm. and 3*8 mm. Three of these plates were
quite homogeneous in elasticity, for the Chladni figures when
obtained on them were symmetrical. Yet the Chladni figures
were difficult to produce, because it is difficult to obtain a pure
tone from an aluminium plate. The sound is generally more
or less composite ; therefore the plate iii its vibration tends to
form two or more figures at the same time, and the con
sequence is that either no figure is formed or one is given
that is not sharply defined. One square plate of 30*8 cms.
on the side and '38 cm. thick, gave quite clearly the three
following tones:— UT 2 (1), SOL 2 (2), and SOL 4 (3). Cor
responding respectively to the Chladni figures of (1) two
lines drawn between opposite points of the centre of sides of
plate; (2) figure formed of the two diagonals drawn between
the corners of plate ; (3) figure similar to (1) but with corners
of plate cut off' by curved lines. Figure 3 corresponded so
nearly to the sound of SOL 4 that a vibrating SOL 4 fork when
held near the plate set the latter into vigorous vibrat on.
Another difficulty met with in using plates of aluminium for
Chladni'' s figures is that sand, even when entirely free from salt
and from the globular grains of windblown sand, does not
move freely over a vibrating surface of aluminium, whether
this surface has been polished or has been slightly tarnished
and roughened by the action of alkali.
There is one serious objection to the use of aluminium in the
construction of musical and acoustical instruments, and that
is the great effect that change of temperature has upon
its elasticity. If a bar of aluminium and a bar of caststeel
be tuued at a certain temperature to exact unison, a change
from that temperature will affect the frequency of vibration
of the aluminium bar 2^ times as much as the same change of
temperature will affect the bar of caststeel.
[ 196 ]
XXIII. On the Freezingpoints of Dilute Solutions.
By W. Nernst and R. Abegg*.
rf 1HE lowering of the freezingpoint of dilute solutions has
JL been recently carefully investigated by two observers,
Mr. Jones t and Mr. LoomisJ, but they have found very
different values for the lowering in the case of nonelectrolytes.
No reader of the two researches could fail to see that
Mr. Loomis had worked with great care, and that Mr. Jones,
on the other hand, had neglected some very obvious pre
cautions.
Indeed we found that a very dangerous source of error,
viz. the influence of the external temperature, which Loomis
approximately avoided, made Mr. Jones's results in the case
of nonelectrolytes entirely worthless. At the same time we
showed how the influence of the external temperature is to be
computed, since we developed a mathematicalphysical theory
for its effect upon the freezingpoint.
To our surprise Mr. Jones sought to defend his results
by attacking our research in a most condemnatory manner.
How far he is justified in this can be seen from the following
brief observations.
Mr. Jones had nothing to say against the abovementioned
theory, except (page 386 of this Journal, 5th series, vol. xl.)
that "the introduction of a correctionterm when amounting
to more than 20 per cent, would not tend to increase our
confidence in the final results."" Ought such an argument
to be taken seriously when a very elementary acquaintance
with physical measurements shows that exact determinations
can be made in spite of large correctionterms, provided these
are correctly computed. So long as Mr. Jones has not
proved any error here he must discard his own results. In
reference to the same point Mr. Jones remarks, " Their
correctionterm appears to me to involve the assump
tion that K for solutions is the same as for water, which
assumption is gratuitous and unallowable" [p. 385]. Evi
dently Mr. Jones has not correctly understood our theory.
We have in no way assumed that K, the rate of dissolving
(Losungsgesch'windigkeit) solid substances in solutions and in
water is the same ; but, on the other hand, we have em
phasized the fact of the great difference in the values for
water and solutions, and we have computed them.
* Communicated by the Authors.
t Zeitschr. f. physikal. Chemie, xi. p. 529; and xii. p. 623.
X Wiedemann's Annalen, li. p. 500.
On the Freezingpoints of Dilute Solutions. 197
We had ourselves called attention to the fact that we had
at our disposal thermometers reading only to y^ ° direct, and
to yo 1 oo° by estimation, so that our determinations of the
lowering of the freezingpoints could for this reason be accu
rate to within only one to two thousandths of a degree. Mr.
Jones's numbers, which read to 7p^°, are, as we have
shown, untrustworthy to within at least hundredths of a degree.
Mr, Jones then remarked that it was not " apparent " to him
why our two series for NaCl in very dilute solutions should
vary by 5 per cent. He neglected to remark that in reality
such differences are limited to the most dilute solutions, and
on that account lie within the limits of experimental errors
given by us. Moreover in the second series for NaCl the
greater values for the lowerings may be accounted for by the
presence of larger quantities of the solid substance, since in
this method larger quantities of undercooled liquid are intro
duced. In any case it is selfevident to every one who knows
the elements of the computation of errors, that no conclusions
can be drawn presupposing a greater degree of accuracy in our
results than that given by us. Nevertheless in spite of this
Mr. Jones concluded, from our results with ethyl alcohol, that
a rise of the molecular lowering took place which was far
within the experimental errors.
That we used substances of sufficient purity for the purpose
we had in view, that is, substances whose possible impurities
were absolutely unessential, we certainly did not especially
mention. Every one knows that comparatively pure NaCI,
ethyl alcohol, and canesugar are easy to obtain.
We refrain from criticizing the few new experiments of
Mr. Jones, partly because we have not the least interest in
carryingon a further controversy, partly because the criticism
given in our earlier research would simply need to be repeated
word for word*. Moreover Mr. Jones admits indeed the
influence of the external temperature (page 389), and in that
the existence of a great source of error which he formerly
entirely neglected.
In conclusion a purely personal remark. Mr. Jones spoke
(page 385) of " the unusual lack of courtesy." That Mr. Jones
* For instance, Mr. Jones maintained that the influence of his " gentle "
stirring was imperceptible, i. e. determined from his feeling instead of
quantitatively. That the thermometerreading remained unchanged is
no proof for his assertion according to what our theory as well as experi
ments showed.
198 On the Freezingpoints of Dilute Solutions.
introduces no proof for this assertion no one could wonder
from the above. But that each one may judge for himself
we insert the following passage from a paper in which we refer
to the results of other authors, and to which alone Mr. Jones's
severe reproach could relate : —
" It is not without interest to test the earlier values for this
source of error through which partly they have been rendered
so considerably inaccurate : for instance, for the molecular
lowering of the freezingpoint (computed from Raoult) for
dilute (about 1 per cent.) canesugar solutions the following
values were found by
Arrhenius. Raoult. Jones. Loomis.
202 207 218 181
but we find, as was mentioned, 1*86 (uncorrected from 1*6 to
2*1). Arrhenius used the usual Beckmann apparatus with
quite an energetic coolingmixture : this explains why his
value is considerably too large. Raoult gives more experi
mental details, and from these one can conclude that he kept
the coolingbath about 3° below the freezingpoint of the
solution. This investigator seems to have appreciated the
essential importance of the coolingbath, for he says, ' If the
influence of the coolingbath upon the temperature of the
liquid at the moment of freezing is not nought, yet it is
indeed the same in the experiments to be compared and
vanishes from the differences, so that the lowering of the
freezingpoint is not influenced by it.' The assumption which
Raoult here makes is identical with the supposition that K
has the same value for pure water and for solutions, which
is certainly not the case with canesugar according to our
experience. His values must accordingly be considerably too
high. Still more erroneous are the values of Jones, who used
a coolingmixture of ice and salt, therefore an exceedingly
strong cooler. If Jones had used a single time in the ciise of
canesugar another cooling mixture, or even only a freezing
vessel of other dimensions, he would have observed with the
great accuracy with which he read the apparent freezing
points the influence of these factors, and would have refrained
from publishing his essentially accidental numbers. [Note: —
" Since the corrections for Jones's values amount to hun
dredths of a degree, if the accuracy is to be increased to
within O'OOOl with the same external temperature and rate
of stirring only by using a greater volume of the liquid, in
order to reduce K (cf. equations (3) and (7) to its hundredth
part, it is necessary to increase the linear dimensions of the
Prof. J. D. Everett on Resultant Tones. 1S9
freezing vessel ( — ~ , cf. page 683) an hundredfold. The
accuracy for which Jones strives would have been attained
therefore, cceteris paribus, by using a vessel not of a 1 litre's
capacity but of that of a million litres ! " Loomis un
doubtedly worked with precaution "
Gottingen, November 1895.
XXIV. On Resultant Tones.
By Professor J. D. Everett, F.R.S*
1. ri^HE received theory of the generation of resultant tones
JL in the ear may be summed up with rough accuracy
as follows f: —
The druinskin being pulled inwards by the end of the
handle of the " hammer/' which is attached to its centre,
offers unsymmetrical resistance to displacement in the inward
and outward, directions, so that the equation for the move
ment of its centre in free vibration would be
— x = co 2 x + ax 2 ,
or to a closer approximation
— x = co 2 x + ax 2 + (3x*,
co, a, ft being constants.
The value of co when the second is the unit of time is less
than 60, hence the frequency of free vibration, being coj2ir, is
less than 10.
When two harmonic forces of frequencies m and n act upon
the drumskin, they produce, in addition to their own tones,
certain " resultant tones," the one of largest amplitude being
the "first differencetone," of frequency n—m. The next
/ 71 — 17l\ 2
largest amplitude is about I ) of this, and belongs to
the "first summation tone/' of frequency n + m. Neither
of these tones will be audible unless the excursion x is so large
that ax 2, is sensible in comparison with co 2 oc. There will also
be differencetones of frequencies 2?n—n and 2n—m, but
neither of these will be audible unless fix' 6 is sensible com
pared with co 2 x.
2. This theory does not appear sufficient to account for the
loud resultant tones which are sometimes heard. When a
Helmholtz siren is driven rapidly, with the rows of holes 9,
* Communicated by the Physical Society : read January 24, 1896.
t See Tonempfindungen, Appendix xii. ; Rayleigh on Sound, art. 68 ;
Bosanquet, Proc. Phys. Soc. vol.iv. p. 240, arts. 5769.
200 Prof. J. D. Everett on Resultant Tones.
12, 15, and 18 open, the resultant tone of frequency 3 on the
same scale is the most prominent tone in the whole volume of
sound.
3. The view which I desire to put forward is closely con
nected with the wellknown theorem of Fourier, that eveiy
periodic variation can be resolved in one definite way into
harmonic constituents, whose periods must be included in the
list 1, J, J, J, &c, where 1 denotes the period of the given
variation itself. The corresponding frequencies will be as
1, 2, 3, 4, &c.
In the majority of cases, when this analysis is carried out,
the fundamental constituent, represented by 1 in the above
lists, is the largest or among the largest ; but in the case of
a variation compounded of two simple tones with frequencies
in the ratio of two integers, neither being a multiple of the
other, the fundamental will be absent, and the Fourier series
will consist of only two terms, which in the language of
acoustics are harmonics of the fundamental.
4. Clearness of thought is facilitated in these matters by
supposing a curve to be drawn, in which horizontal distance
represents time and vertical distance represents the quantity
whose variation is in question. Since the variation is periodic,
the curve will consist of repetitions of one and the same form,
in other words it will consist of a number of equal and similar
waves, and the wavelength stands for the complete period of
the variation.
The point on which I wish to insist is, that if such a curve
representing the superposition of two harmonics of the funda
mental is in the first instance very accurately drawn, and is
then inaccurately copied in such a way that all successive
waves are treated alike, the inaccuracy is morally certain to
introduce the fundamental.
Let y denote any ordinate, and 6 the time (or abscissa)
expressed in such a unit that 27r is the numerical value of
the wavelength or period ; then the amplitude of the funda
mental is the square root of A 2 { B 2 , where
1 f 2 * 1 f 2,r
A= ycosOdO, B= y S m0d0.
7tJ j ttJ j
In the original curve both A and B vanish. Let y' be the
altered value of y in the new and inaccurate curve, and let
z denote y' —y, then we have, between the above limits,
jV cos 0J0= jV cos Odd +jr cos 6d$=$z cos 6d0,
since (*?/ cos Odd is zero.
Prof. J. D. Everett on Resultant Tones. 201
But z may be regarded as a random magnitude, hence it is
infinitely improbable that its different values exactly fulfil the
condition J z cos 0dd—O. Therefore the new A is finite ; and
similar reasoning shows that the new B is finite.
If all the ordinates were changed in one uniform ratio, A
and B would remain zero, and no new constituent would be
introduced ; but any other change, unless specially planned
to avoid introducing A and B, is practically certain to give
A 2 + B 2 a finite value.
5. I maintain that such a change is effected in the form of
sonorous waves during their transmission from the external
air to the sensory fibres by which we distinguish pitch. The
waves are transmitted first from the air to the drumskin, then
through two successive levers, the hammer and anvil, to the
head of the stirrup, while the foot of the stirrup sits upon the
membrane of the oval window, and passes on the vibrations
through the membrane to the liquid on the other side in
which the sensory fibres are immersed. The levers turn upon
ligamentous fulcra, and have rubbing contact with each other.
The waveform cannot run the gauntlet of all these trans
missions without being to some extent knocked out of shape.
It is much as if a very accurately drawn curve, representing
the original waveform, were copied and recopied, five times
in succession, by five different pantagraphs not very firm in
their connexions. The final copy so obtained would be sure
to exhibit sensible departures from the original.
6. It appears likely that the chief seat of the disturbing
actions in the ear is the junction of the hammer and anvil.
" When the drumskin with the hammer is driven outwards,
the anvil is not obliged to follow it. The interlocking teeth
of the surfaces of the joint then separate, and the surfaces
glide over each other with very little friction "** Such action
is likely to introduce derangement, increasing generally with
the excursions of the drumskin, but not expressible as a defi
nite function of the ordinates of the wavecurve. For a
given pressure on the drumskin, the pressure communicated
to the liquid in the cochlea will vary according to the relative
position and relative motion of the two portions of this joint.
7. The principal resultant tone due to these actions is likely
to be that which corresponds to the complete period of the
actions, in other words the highest common fundamental of
the two primaries, or what old writers called the " grave har
monic" This will not be the same as the u first difference
tone " unless the ratio of the two primaries is of the form
* Ellis's * Helmholtz/ p. 133, 2nd edition.
Phil Mag. S. 5. Vol. 41. No. 250. March 1896. P
20*2 Prof. J. D. Everett on Resultant Tones.
m : m + 1 ; and I have satisfied myself, both by my own
trials and by a study of Koenig's experimental results, that
when the differencetone and the common fundamental are
not identical, the common fundamental is usually the pre
dominant, and often the only audible resultant tone. (See
Appendix.)
8. The common fundamental is, however, not the only
resultant tone that can be thus accounted for. Similar
reasoning to that employed in reference to A and B suffices
to explain the introduction of any or all of the harmonics of
the fundamental ; but it is to be expected, from the analogy
of ordinary experience in harmonic analysis, that the succes
sive constituents will usually be smaller and smaller as we
advance in the series. The octave is likely to be the largest
of them ; and Koenig found, in several experiments with
primaries in the ratio of 3 : 5, that both the fundamental 1
and its octave 2 were distinctly heard as resultant tones.
9. The following investigation bears on the relation be
tween beats and resultant tones. The expression
a cos mO + b cos nO
can be reduced to the form
where A and e are given by
A 2 = a 2 + b 2 + 2ab cos {n  m) 0,
, a — b. n—ni*
tan e = r tan — ~ — 0,
a + o 2 '
and the beating together of two tones not differing much in
pitch is explained by the fact, definitely expressed in these
formulas, that the whole effect may be regarded as a succession
of waves with gradually varying amplitude. The frequency
of the beats is the frequency of the maxima of A 2 , and is the
difference of m and n.
We have ascribed resultant tones to alterations made in
the waveform by the action of the ear, such alterations being
in general largest at those points at which the excursions of
the drumskin are largest. These excursions are measured by
±A, and the above investigation shows that their maxima
have a frequency corresponding to the differencetone. This
is true whether m and n are commensurable or incommen
surable. If they are commensurable, their greatest common
Prof. J. D. Everett on Resultant Tones. 203
measure will be the frequency of the complete cycle of change.
This cycle will not be conspicuous in the curve if the ratio of
n—m to ^(n + m) is very small, but will assert itself more and
more as this ratio increases ; and these remarks will apply to
the comparison of the fundamental with the first difference
tone.
10. If the ear is able so to alter the form of waves
impinging upon it as to generate resultant tones, it is natural
to seek for some instance of a similar action in external bodies.
A violin is very susceptible, like the ear, to vibrations of all
frequencies between wide limits, and the soundpost serves,
like the ossicles of the ear, to transmit vibrations from one
portion to another. It is easy to produce resultant tones by
bowing two strings of a violin together. For example, in
the ordinary process of tuning, when the fourth and third
strings with frequencies as 2 : 3 are combined, the resultant
tone 1 is very observable if attention be directed towards it.
But more striking effects are obtained when the resultant is
at a larger interval from the primaries. The major sixth 3 : 5,
the major second 8 : 9, and the minor seventh 5 : 9, are suitable
intervals for calling out the fundamental 1, the strings em
ployed being either the first and second or the second and
third. The deep resultant tone thus obtained can not only
be heard by the ear but felt as a tremor by the hand which
holds the instrument. This is clear evidence of its objective
existence, and I have succeeded in confirming the fact by
means of a Helmholtz resonanceglobe, the largest of the
ordinary set, responding to C of 128 vibrations. When held
with the edge of its mouth resting against the side of the
violin, it responds to the combination C of 256 and G of 384
on the 4th and 3rd strings, or to the combination C of 512
and E of 640 on the 2nd and 1st ; or, still better, to the 3rd
and 2nd open strings each flattened one note, so as to be C of
256 and G of 384. Here, then, we have distinct evidence
that the violin possesses the power which I have ascribed to
the ear — the power of manufacturing the fundamental when
the two primaries are supplied.
11. Sir John Herschel, in his treatise on Sound (Enc.
Met. arts. 238, 239), mentions the fact that the common
fundamental can be called out by sounding two or more of
its harmonics on very accurately tuned strings or pipes, and
says that the effect cannot be obtained from a pianoforte tuned
in the ordinary way, because the intervals are tempered. I
find, however, on trying the experiment with an upright
Broadwood of date about 1860, that C of 64 is easily called
out by simultaneously striking eight or ten of its harmonics ;
P2
204 Prof. J. D. Everett on Resultant Tones.
and trie effect is greatly enhanced if the key of C 64 is held
down. In the latter case its note continues to be heard for
a long time after the keys which were struck are released.
From these experiments it appears probable that the sounding
board of a piano possesses the same property which we have
proved to exist in the violin.
12. I now come to the explanation of the experiments of
Professor Eiicker and Mr. Edser (Proc. Phys. Soc. vol. xiii.
p. 412, Phil. Mag. 1895, xxxix. p. 341). They were made
with a Helmholtz siren, and in each instance the two primaries
were produced in the same box, sometimes the upper and
sometimes the lower box. The following explanation is a
development of suggestions contained in Appendix xvi. of the
Tonempfindungen.
The rate of escape of air from the box containing the two
rows of holes which are employed may as a first approxima
tion be assumed to be jointly proportional, at each instant,
to the aperture for escape and the differential pressure
which produces the escape. Again, this differential pressure
may be regarded as the algebraic sum of two terms, one of
them constant, and representing its average value, while the
other represents the difference from the average due to the
varying amount of the aperture from instant to instant. As
a first approximation, equal increments of aperture must be
regarded as producing equal decrements of pressure, so that
the variable term will be proportional (with reversed sign) to
the excess of the aperture above its mean value. This excess
(defect being counted negative) will be a periodic function of
the time, and if the ratio of the two primaries in lowest terms
be m : n, the frequency for the complete period will be repre
sented on the same scale by 1. In other words it will be the
period of their common fundamental.
Let the aperture at time t be expressed in a Fourier series,
6 being put for 27rt/T, where T is the complete period ; and let
the variable part of the expression be denoted by/(0), while
a denotes the mean aperture, so that the aperture at time t is
«o+/(0) We shall have
/(0) = Asin0+ ... +a 1 sm(md + € ] )+ . .. + \ sin (nO + e 2 ) +
The largest amplitudes will be a v and b x corresponding to the
two primaries ; but A, which corresponds to the fundamental,
is likely to be sensible.
The pressure at time t is proportional to
0/W,
C being a constant ; and the aperture is
«o +/(0).
Prof. J. D. Everett on Resultant Tones. 205
Hence the rate of escape is proportional to
Ca o +(Ca o )/(0){/(0)P.
a is comparable with the maximum value of /(#), and C is
much greater ; hence a may be neglected in comparison
with C.
Developing {(/(#)} 2 , we shall obtain a term
2a 1 6 1 sin (md + e^ sin (nQ + e 2 )
= a 1 o 1 [GO${(n — m)0 + e 2 — € 1 }—cos{(n + m)d + 6 2 + 6 1 }],
representing a differencetone and a summation tone. From
(C— CLo)f(0) we have the common fundamental
(C— a )A sin#, or GA sin 0,
and the two primaries
C«i sin (?n# + e x ) , Chi sin (w0 + e 2 ) .
Suppose for simplicity that fli=6i, then, taking the amplitude
of each of the primaries as 1, the amplitude of the common
fundamental will be A/%, and the amplitudes of the summa
tiontone and differencetone will each be %/C.
When n— ra = l, the differencetone coincides with the
fundamental, and their joint amplitude may be taken as the
square root of the sum of the squares of %/C and A/%.
13. Professor Pucker and Mr. Edser in experiments i. and
ii. obtained the differencetone 64 from five distinct com
binations of primaries,
256 & 320, 192 & 256, 320 & 384, 512 & 1152, 96 & 160,
their ratios being
4:5, 3:4, 5:6, 4:9, 3:5.
The second combination appears to have given a stronger
effect than either the first or the third ; whence it would
appear that low frequencies are favourable to strong effects.
Nevertheless the fourth combination is mentioned as giving a
rather feebler effect than any one of the first three. This
confirms our conclusion that the differencetone is weaker
when it is distinct from the common fundamental than when
it coincides with it.
Experiment iii. was directed to testing for the presence
of the resultant 64 when the primaries were 256 and 576.
which are as 4 : 9. Their common fundamental is 64, and it
could not be detected. This may have been because the
pitch 576 was too high to give a good effect. Or the failure
may be an indication that A/% is decidedly smaller than %/C.
It would be interesting to repeat the experiment, employing
192 and 320 as the primaries.
206 Prof. J. D. Everett on Resultant Tones.
14. Near the end of chapter vii. of the Tonempfindungen
Helmholtz makes prominent mention of the slipping of the
hammer on the anvil as an important cause of resultant tones,
and appears to regard it as exemplifying his mathematical
formula for the restoring force as a function of the displace
ment. But it is clear that if the hammer, which holds the
drumskin, is liable to shift in its supports, the restoring force
cannot be a mere function of the displacement, but must also
depend on the relative position and relative velocity of the
hammer and anvil at the moment considered. I accept all
the consequences which Helmholtz deduces in the passage
in question from this slipping, including its application to
explain first differencetones ; but I regard these consequences
as lying outside the range of his general mathematical for
mulae as given in Appendix xii.
15. To sum up my objections to the received mathematical
theory of resultant tones : —
First. It assumes that the reaction of the drumskin against
the air is a definite function of the displacement of the drum
skin from a certain fixed position, whereas this reaction
depends also on the position and motion of the further end of
the hammer at the time.
Secondly. Even if the vibrations of the drumskin were in
accordance w T ith the received formulae, there is plenty of scope
for the introduction of additional constituents on the road
from the drumskin to the liquid in the cochlea. The auditory
ossicles, with their ligamentous supports and attachments,
probably serve to protect the oval window of the cochlea
against shocks and jars, and to smooth down asperities in the
waveform, thus mitigating the harshness of sounds and
rendering them more musical. The changes thus introduced
are very unlikely to fulfil the special conditions required for
the vanishing of the common fundamental.
Thirdly. The received theory makes the common funda
mental, when not coincident with the first differencetone,
depend on a term involving the cube or some higher power
of the displacement. When the primaries are as 3:5, the
fundamental 1 comes in as 2m — n, and depends on the cube
of the displacement. When they are as 4 : 11, the tone 1,
which Koenig found to be the loudest resultant, is 3mw,
and depends on the fourth power. When they are in the
ratio 4 : 15, as in Kcenig's experiment with the simple tones
ut 5 and si 6 , the common fundamental ut z , which was the
only resultant tone heard, is 4m— m, and depends on the fifth
power of the displacement ; the first differencetone, which
depends on the second power and should in theory be the
The Compound Law of Error. 207
loudest, being inaudible. This is surely a reductio ad absurdum
of the received theory.
I do not wish to be understood as denying that the theory
has any basis of truth. My contention is that the actions to
which it is truly applicable play only a subordinate part in
the production of resultant tones.
Appendix.
Examples selected from Kcenig's Experiences d'Acoustique,
pp. 103 and 104, illustrating the production of the
common fundamental. The " single vibrations " of
the original are here reduced to double or complete
vibrations : —
ut 5 and s%, which are as 8 : 15, gave only ut%.
ut 5 and 2816, which are as 4 : 11, gave ut B corresponding
to 1 louder than any other tone.
ut 5 and si 6 , which are as 4 : 15, gave no audible tone but ut 3 .
ut 5 and 3968, which are as 8 : 31, gave no audible tone but
ut 2 .
ut 6 and 3584, which are as 4 : 7, gave ut± more distinct than
the differencetone sol 5 .
ut 6 and si 6 , which are as 8 : 15, gave ut B distinct, the diffe
rencetone 7 being inaudible.
ut 6 and 3968, which are 16 : 31, gave ut 2 only.
ut 6 and 4032, which are 32 : 63, gave ut^ only.
XXY. The Compound Law of Error.
By Professor F. Y. Edgeworth, M.A., D.C.L.*
THE compound law of error is an extension to the case of
several dimensions of the simple law for the frequency
with which a quantity of one dimension (x) tends to assume
each particular value. A first approximation to the com
pound law has been obtained by several writers independently,
—by Mr. De Forest, in the 'Analyst' for 1881; by the
present writer, in the Philosophical Magazine for December
1892 ; and by Mr. S. IL Burbury, in the same Journal for
January 1894. I propose here to employ tbe method of
partial differential equations explained in a preceding paper 
to verify the first approximation, and to discover a second
approximation, to the compound law.
To begin with the case of two dimensions : let Q be the
* Communicated by the Author.
t " On the Asymmetrical ProbabilityCurve," Phil, Mag. February 1896.
208 Prof. F. Y. Edgeworth on the
sum (or more generally an expansible function *) of a number
of elements £i f 2 , &c, each of which, being a function of two
variables x and y, assumes any particular system of values
according to any law of frequency £ =/*(#>#) ; the functions
/ being in general different for different elements. If each
of these functions is referred to its centre of gravity at origin,
and expanded in powers of x and y, it appears, by parity of
reasoning with that employed in the case of the simple law,
that for a first approximation we need take account only of
terms of the second order. Integrate between extreme limits
of x*f(xy)dx dy for each element ; and let the sum of all
these integrals be h. Also let
l=t^xyf i {xy)dxdy,
the integration extending between the extreme limits of each
element, and the summation over all the elements. Then
s, the sought function which is to express the frequency of
Q, will be of the form
z=®(x,y; A,i,m)t.
This expression may be simplified by transforming the
axes to new ones making an angle 6 with the old ones, such
that the new I vanishes. This will be effected if we put
tan20 = 2Zr(&— m)%. Thus we may write with sufficient
generality : —
z=<&(x,y ; Is,m),
By superposing a new element after the analogy of the
* Cf. Phil. Mag. 1892, xxxiv. p. 431 et seq.
t I use a semicolon to separate the variables (x and y) from the con
stants (k, I, m).
X Put x =X cos 6  Y sin 6,
*/=Xsin<9+Ycos0.
The new I = SjJX Yf.dX dY ;
where f x is what fi(xy) becomes when for x and y are substituted their
values in X and Y ; each element is integrated between extreme limits,
and all the integrals are summed. Transforming back to the old axes
we have for the new I
Sjj/K^) (* (3/ 2 ^ 2 ) sin 20 ± W cos 20)4r dy= i(m~Jc) sin 26+1 cos 26 ;
which becomes null when
tan20=2/r(Am).
Compound Law of Error, 209
simple case * we obtain the differential equations
dz__ \d& /.x
dk~~ 2dx 2 ' { }
dz__ld^_ /«n
dm~ 2dy 2 W
Other differential equations are obtained by supposing the
units of x and y altered; substituting for x and y, x{\\a)
and 2/(1 + /3) respectively. The expression for z thus trans
formed must be multiplied by (1 +a) (1+/3) ; since the
measure of the solid contents of the parallelopiped inter
cepted between the surface, the plane of x^, and any two fixed
adjacent points in that plane will be increased in that pro
portion. Thus
s=(l+a)(l+£)0&,y; k,m).
Regarding a and ft as infinitesimal, expanding and neglect
ing higher terms, we have
•+£+»a<* (3 >
^ +to £=° ^
From (3) and (4) we have
where <£> and ty are arbitrary functions. Whence
1 / x y \
m \ vm vm'
Z = '
where % is an arbitrary function. The form of % is restricted
by the condition that its value is the same for positive and
negative values of x and y, the surface being symmetrical
about a vertical plane through each axis. For as we take
account only of the second powers in the expansion of each
element, we might replace the given system of elements by a
new system of symmetrical functions having each the same
centre of gravity and mean square of error as the old one f ,
* See the preceding article, Phil. Mag. 1896, xli. p. 90.
t This does not mean that the given elements must be symmetrical, as
is sometimes carelessly said with reference to the simple law of error.
The given elements may have any degree of asymmetry, provided that
their number is correspondingly great.
210 Prof. F. Y. Edgeworth on the
And a compound of symmetrical elements must itself be
symmetrical. "We have, therefore,
1 (^ f\
z =V^
To the five equations which have been stated there is to be
added the condition that the integral of zdxdy between
extreme limits =1.
To solve this system : substitute in (3) and (4) the values
dz dz
of pr and j — given in (1) and (2) respectively. We have,
then,
, dz . d 2 z
z+ VTy +m ^ =0 W
Integrating (6) with respect to #, and (7) with respect to
y, we have
».+ *=*(y), (8)
dz
^+™^=f(>); ( 9 )
where and yjr are arbitrary functions.
Both these functions reduce to zero; as may thus be proved: —
dz
From (5) it appears that when #=0, 7 also =0, whatever
the value of y. If, then, we put # = 0, the left side of equation
(8) vanishes for all values of y. Therefore the right side of
the equation vanishes for all values of y. Therefore (y) is
identical with zero. By parity yjr(y) is null.
By equation (8) thus reduced we have
z=®(y)xe u , (10)
z=V(x)e 2m (11)
Identifying the righthand members of (10) and (11) we have
Z=Ce 2fc 2m y (12)
where C is a constant : which is found to be  — 7= from the
27T \km
Compound Law of Error. 211
condition that
zdx dy=l.
Li
Transforming back from the principal axes which we have
employed*, we find for the general expression
1 (mz2— 2Zzy+*y2)
Z— .. e 2(kml2)
27T sjkm — l 2.
By parity of reasoning we obtain as the general form for
the law of error relating to any number of variables x 1} x 2 , %&
Ac.,
1 ~ K
,z 1 2+2L, 2 z ] x 2 +2L 13 z 1 z 3 +K 22 :r 2 2 +&c.
2A
2 —
(2tt)2 ^
s
where A is the determinant
*i
t>\2
Z 13 . . .
^21
K 2
'23 • • •
'31
'32
k s . . .
•
•
•
•
•
•
•
•
•
# 2 =2 jj J . . . %i x i d%\ dx 2 dx 3 ....
k 2 = 2§§§ ._... (J ^ 2 2 d#l ^ 2 cfa? 3
Z 12 = 2 j Jj* . . . £ x 1 x 2 dx x dx% dx 3 . . . . = Z 21 ,
the limits of the integrals and extent of the summation being
as before ; K x is the first minor of the determinant formed
by omitting the row and column containing k x ; L 12 is the
first minor formed by omitting the row and column con
taining l 21 , or l 12 ; and so on.
* The values of the k and m which we have "been employing with
reference to principal axes are in terms of our original k } I, m referred to
any axes respectively :
k cos 2 6 — 21 cos 6 sin d\m sin 2 6,
and k sin 2 6 4 21 cos 6 sin 6 +m cos 2 6 ;
where tan 26= (k — m)72l,
See note on p. 208.
212 Prof. F. Y. Edgeworth on the
If the units of the variables be taken so that k l9 Jc 2 , k 3 , &c.
each = J, then l 12 , l ls , &c. will he replaced by ^p 12 , ^p^, &c,
the coefficients of correlation which have been discussed in a
former paper *■
To obtain a second approximation to the compound law of
error by this method : beginning with the case of two varia
bles, put as before
&=?%$$ £ifd#dy>
principal axes being employed. Also put
n — %^^dxdy,
P = « 6«V »
9 = » S*V »
?^ 8 11
"We have then, for z the law of the compound, the following
system of equations : —
dz
lie
dz_
dm
dz
dn '
dz
dp
dz
dq
dz_
dr '
1 d 2 z m
2 dx 2 l
ld&
2dy 2 ;
1 d 3 z
~6d^
1 d*
2 da?dy'
1 d z z
2d^ay ;
1 d&_
6 ^ 3; '
, dz ~i dz , o dz t ~ dz , dz A
^ + ^^+2^77 +372^ +2/? j+^^ =0,
. dz dz t Q dz dz dz
* Phil. Mag. 1892, xxxiv. p. 194 et seqq.
(i)
(2)
(3)
(4)
(5)
(6)
(?)
(8)
Compound Law of Error. 213
From (7) and (8) we obtain
„__J_ /_£_ V_ . JJ JL. JL JL\ (9)
' sjkm X \sjV sjin k V km?' kW mV * W
Put z 1 for the first approximation which has already ^been
found, viz. :
1 (? + ^\
zi = = e v 2 * 2 »»/.
2 sjkm
Put z=Zi(l + are the most prominent*, and these
absorptionbands belong most probably to the aqueous vapour,
That Paschen has not observed any emission by watervapour
in this interval may very well be accounted for by the fact
that his heatspectrum had a very small intensity for these
short waved rays. But it may be conceded that the absorption
coefficient for aqueous vapour at this angle in Table II. is
not very accurate (probably too great), in consequence of the
little importance that Langley attached to the corresponding
observations. After this occurs in Langley's spectrum the
great absorptionband ty at the angle 39*45 (\=1*4 /a), where
in Paschen 's curve the emission first becomes sensible
(log y=. — 0" 1105 in Table II.). At wavelengths of greater
value we find according to Paschen strong absorptionbands
at \ = 1*83^ (H in Langley's spectrum), i.e. in the neigh
bourhood of 39°*30 and at \ = 2'64/x (Langley's X) a little above
the angle 39°' 15. In accordance with this I have found
rather large absorptioncoefficients for aqueous vapour at
these angles (log?/ =  0'0952 and —00862 resp.). From
X=3*0 \a, to X = 4*7yLt thereafter, according to Paschen the
absorption is very small, in agreement with my calculation
(logy = —0*0068 at 39°, corresponding to X = 4*3/a). From
this point the absorption increases again and presents new
maxima at A,= 5*5/i>, X=6*6 /u,, and \=7'7 fi, i.e. in the
vicinity of the angles 38°*45 (\ = 5'G fi) and 38°*30 (X=7 1/x).
In this region the absorption of the watervapour is con
tinuous over the whole interval, in consequence of which the
great absorptioncoefficient in this part (log y— — 0'3114 and
— 0*2362) becomes intelligible. In consequence of the de
creasing intensity of the emissionspectrum of aqueous vapour
in Paschen's curve we cannot pursue the details of it closely,
but it seems as if the emission of the watervapour would also
be considerable at X = 8"7//, (39 c *15) ? which corresponds with
the great absorptioncoefficient (log y= —0*1933) at this
place. The observations of Paschen are not extended farther,
ending at \=9*5yu,, which corresponds to an angle of 39°*08.
For carbonic acid we find at first the value zero at 40°, in
agreement with the figures of Paschen and Angstrom f. The
absorption of carbonic acid first assumes a sensible value at
* Langley, Ann. Ch. et PJiys. ser. 6, t, xvii. pp. 323 and 326, 1889,
Prof. Papers, No. 15, plate 12. Lamansky attributed his absorptionbands,
which probably had this place, to the absorbing power of aqueous vapour
(Pogg. Ann. cxlvi. p. 200, 1872).
t It must be remembered that at this point the spectrum of Paschen
was very weak, so that the coincidence with his figure, may be accidental,
248 Prof. S. Arrhenius on the Influence of Carbonic Acid
X=1'5/a, after which it increases rapidly to a maximum at
X=2'6 fi, and attains a new extraordinarily strong maximum
at \ = 4*6 (Langley's Y). According to Angstrom the ab
sorption of carbonic acid is zero at X = 0'9 /ju, and very weak
at X=l*69 /jl, after which it increases continuously to \=4*6//.
and decreases again to \= 60//.. This behaviour is entirely
in agreement with the values of log x in Table II. From
the value zero at 40° (A, = 1*0 /a) it attains a sensible value
(0*0296) at 39°'45 (X=l*4/i), and thereafter greater and
greater values (0*0559 at 39°*30, and 01070 at 3b°15)
till it reaches a considerable maximum (—0*3412 at 39°,
X=4*3/Lt). After this point the absorption decreases (at
38°45 = 56^1og^=02035). According to Table II. the
absorption of carbonic acid at 38 o, 30 and 38°*15 (\=7*t> 6.
and 8*7 //,) has very great values (log x= —0*2438 and
—0*3730), whilst according to Angstrom it should be o insensible.
This behaviour may be connected with the fact that Angstrom's
spectrum had a very small intensity for the larger wave
lengths. In Paschen's curve there are traces of a continuous
absorption by the carbonic acid in this whole region with
weak maxima at X=5*2//., X=5*9 //,, \=6'6/jl (possibly due
to traces of watervapour), X = 8*4 fi, and X=8'9 /ju. In
consequence of the strong absorption of watervapour in this
region of the spectrum, the intensity of radiation was very
small in Langley's observations, so that the calculated ab
sorptioncoefficients are there not very exact (cf. above,
pp. 242243). Possibly the calculated absorption of the car
bonic acid may have come out too great, and that of the
watervapour too small in this part (between 38°*30 and 38°*0).
This can happen the more easily, as in Table I. K and W
in general increase together because they are both propor
tional to the ** airmass." It may be pointed out that this
also occurs in the problems that are treated below, so that the
error from this cause is not of so great importance as one
might think at the first view.
For angles greater than 38° (\>9*5/a) we possess no
direct observations of the emission or absorption of the two
gases. The sun's spectrum, according to Langley, exhibits
very great absorptionbands at about 37°*50, 37°*25, 37°, and
36 o, 40°. According to my calculations the aqueous vapour
has its greatest absorbing power in the spectrum from 38° to
35° at angles between 37°*15 and 37°*45 (the figures for
35°*45, 35 o, 30, and 35°* 15 are very uncertain, as they de
pend upon very few measurements) , and the carbonic acid
between 36°*30 and 37°*0. This seems to indicate that the
first two absorptionbands are due to the action of water
in the Air upon the Temperature of the Ground. 249
vapour, the last two to that of carbonic acid. It should be
emphasized that Langley has applied the greatest diligence
in the measurement of the intensity of the moon's radiation
at angles between 36° and 38°, where this radiation possesses
its maximum intensity. It may, therefore, be assumed that the
calculated absorptioncoefficients for this part of the spectrum
are the most exact. This is of great importance for the fol
lowing calculations, for the radiation from the earth * has by
far the greatest intensity (about two thirds, cf. p. 250) in this
portion of the spectrum.
II. The Total Absorption by Atmospheres of Varying
Composition.
As we have now determined, in the manner described, the
values of the absorptioncoefficients for all kinds of rays, it
will with the help of Langley's figures  be possible to cal
culate the fraction of the heat from a body at 15° 0. (the earth)
which is absorbed by an atmosphere that contains specified
quantities of carbonic acid and water vapour. To begin with,
we will execute this calculation with the values K=l and
W = 0*3. We take that kind of ray for which the best deter
minations have been made by Langley, and this lies in the midst
of the most important part of ths radiation (37°). For this
pencil of rays we find the intensity of radiation at K=l and
W = 0*3 equal to 62*9; and with the help of the absorption
coefficients we calculate the intensity for K = and W=0,
and find it equal to 105. Then we use Langley's experiments
on the spectral distribution of the radiation from a body of
15° C, and calculate the intensity for all other angles of devia
tion. These intensities are given under the heading M. After
this we have to calculate the values for K = l and W = 0'3.
For the angle 37° we know it to be 62*9. For any other
angle we could take the values A from Table II. if the moon
were a body of 15° C. But a calculation of the figures of
Very J shows that the full moon has a higher temperature,
about 100° 0. Now the spectral distribution is nearly, but
not quite, the same for the heat from a body of 15° C. and
for that from one of 100° C. With the help of Langley's
figures it is, however, easy to reduce the intensities for the
hot body at 100° (the moon) to be valid for a body at 15°
* After having been sifted through an atmosphere of K = l*l and
W=0 3.
t ' Temperature of the Moon/ plate 5.
j "The Distribution of the Moon's Heat," Utrecht Society of Arts and
Sc. The Hague, 1891.
250 Prof. 8". Airhenius on the Influence of Carbonic Acid
(the earth). The values of A reduced in this manner are
tabulated below under the heading N.
Angle... 40°. 3945. 3930. 3915. 39 0. 3845. 3830. 38'15. 38"0. 3745. 3730.
M 34 116 248 459 840 121 '7 161 189 210 210 188
N 31 101 113 137 180 181 112 196 444 59 70
Angle... 37°15. 370. 3645. 3630. 3615. 360. 3545. 3530. 3515. 35'0. Sum. P.c.
M 147 105 103 99 60 51 65 62 43 39 2023 100
N 755 629 564 514 391 379 392 376 360 287 7432 372
For angles less than 37° one finds, in the manner above
described, numbers that are a little inferior to the tabulated
ones, which are found by means of the absorptioncoefficients
of Table II. and the values of N. In this way the sum of the
M's is a little greater (6*8 per cent.) than it would be accord
ing to the calculation given above. This nonagreement
results probably from the circumstance that the spectrum in
the observations was not quite pure.
The value 37*2 may possibly be affected with a relatively
great error in consequence of the uncertainty of the Mvalues.
In the following calculations it is not so much the value 37*2
that plays the important part, but rather the diminution of
the value caused by increasing the quantities K and W. For
comparison, it may be mentioned that Langley has estimated
the quantity of heat from the moon that passed through the
atmosphere (of mean composition) in his researches to be 38
per cent.* As the mean atmosphere in Langley's observa
tions corresponded with higher values of K and W than K = l
and W = 0'3, it will be seen that he attributed to the atmo
sphere a greater transparence for opaque rays than I have
done. In accordance with Langley's estimation, we should
expect for K = l and W = 0*3 a value of about 44 instead of
37*2. How great an influence this difference may exert will
be investigated in what follows.
The absorptioncoefficients quoted in Table II. are valid for
an interval of K between about 1/1 and 2*25, and for W between
0*3 and 2*22. In this interval one may, with the help of those
coefficients and the values of N given above, calculate the value
of N for another value of K and W, and so in this way obtain
by means of summation the total heat that passes through an
atmosphere of given condition. For further calculations I
have also computed values of N for atmospheres that contain
greater quantities of carbonic acid and aqueous vapour. These
values must be considered as extrapolated. In the following
table (Table III.) I have given these values of N. The
numbers printed in italics are found directly in the manner
* Langley, ' Temperature of the Moon/ p. 197.
in the Air upon the Temperature of the Ground. 251
described, those in ordinary type are interpolated from them
with the help of Pouillet's exponential formula. The table has
two headings, one which runs horizontally and represents the
quantity of aqueous vapour (W), and another that runs verti
cally and represents the quantity of carbonic acid (K) in the
atmosphere.
Table III. — The Transparency of a given Atmosphere for
Heat from a body of 15° C.
^
1
ico 2 .
03.
05.
10:
15.
20.
30.
40.
60.
100. !
1
37'2
350
301
269
239
193
160
107
89
12
347
327
286
251
222
178
147
97
80
15
315
296
259
226
199
159
130
84
69
2
270
253
219
191
167
131
105
66
53
25
235
220
190
166
144
110
87
53
42 !
3
201
188
163
H'2
123
93
74
42
33
4
158
147
127
108
93
71
56
31
20
6
109
102
87
73
63
48
37
19
093
10
66
61
52
43
35
24
18
10
026
20
29
2o
22
18
15
10
075
039
007
40
088
081
067
056
046
032
024
012
002
Quite different from this dark heat is the behaviour of the
heat from the sun on passing through new parts of the earth's
atmosphere. The first parts of the atmosphere exert without
doubt a selective absorption of some ultrared rays, but as
soon as these are extinguished the heat seems not to diminish
as it traverses new quantities of the gases under discussion.
This can easily be shown for aqueous vapour with the help of
Langley's actinonietric observations from Mountain Camp
and Lone Pine in Colorado*. These observations were
executed at Lone Pine from the 18th of August to the 6th of
September 1882 at 7 h 15 m and 7 h 45 m a.m., at ll h 45 m a.m.
and 12 h 15 m p.m., and at 4 h 15 m and 4 h 45 m p.m. At Mountain
Camp the observations were carried out from the 22nd to the
25th of August at the same times of the day, except that only
one observation was performed in the morning (at 8 h m ). 1
have divided these observations into two groups for each
station according to the humidity of the air. In the following
little table are quoted, first the place of observation, and after
this under D the mean date of the observations (August 1882),
under W the quantity of water, under I the radiation observed
by means of the actinometer, under l x the second observation
of the same quantity.
* Langley, ' Researches on Solar Heat/ pp. 94, 98, and 177.
252 Prof. S. Arrhenius on the Influence of Carbonic Acid
Morning. Noon. Evening,
D. W. I. I r D. W. T. I x . D. W. I. I r
Lrae /293 061 1424 15541 f23'6 046 1692 1715 I \ 266 051 1417 1351 1
Pine. \211 084 1458 1583 J" \ 269 059 1699 1*721 / j 232 0"74 1428 1359/
Mountain f 235 0'088 1790 1 /22'5 0182 1904 18731 J 245 0205 1701 1641]
Camp. [235 0153 1749 f \245 0215 1890 1917/ 1 225 032 1*601 1527/
At a very low humidity (Mountain Camp) it is evident that
the absorbing power of the aqueous vapour has an influence,
for the figures for greater humidity are (with an insignificant
exception) inferior to those for less humidity. But for the
observations from Lone Pine the contrary seems to be true.
It is not permissible to assume that the radiation can be
strengthened by its passage through aqueous vapour, but the
observed effect must be caused by some secondary circum
stance. Probably the air is in general more pure if there
is more watervapour in it than if there is less. The
selective diffusion diminishes in consequence of this greater
purity, and this secondary effect more than counterbalances
the insignificant absorption that the radiation suffers from the
increase of the watervapour. It is noteworthy that Elster
and Geitel have proved that invisible actinic rays of very
high refrangibility traverse the air much more easily if it is
humid than if it is dry. Langley's figures demonstrate mean
while that the influence of aqueous vapour on the radiation
from the sun is insensible as soon as it has exceeded a value
of about 0*4.
Probably the same reasoning will hold good for car
bonic acid, for the absorption spectrum of both gases is of the
same general character. Moreover, the absorption by car
bonic acid occurs at considerably greater wavelengths, and
consequently for much less important parts of the sun's
spectrum than the absorption by watervapour*. It is,
therefore, justifiable to assume that the radiation from the
sun suffers no appreciable diminution if K and W increase
from a rather insignificant value (K = l, W = 0'4) to higher
ones.
Before we proceed further we need to examine another
question. Let the carbonic acid in the air be, for instance,
the same as now (K = l for vertical rays), and the quantity
of watervapour be 10 grammes per cubic metre (W = l for
* Cf. above, pages 246248, and Langley's curve for the solar spec
trum, Ann. d. Ch. et d. Phys. ser. 6, t. xvii. pp. 323 and 326 (1889) ;
' Prof. Papers/ No. 15, plate 12.
in the Air upon the Temperature of the Ground* 253
vertical rays). Then the vertical rays from the earth traverse
the quantities K = l and W = l; rays that escape under an
angle of 30° with the horizon (airmass = 2) traverse the
quantities K = 2, W = 2 ; and so forth. The different rays that
emanate from a point of the earth's surface suffer, therefore,
a different absorption — the greater, the more the path of the
ray declines from the vertical line. It may then be asked
how long a path must the total radiation make, that the
absorbed fraction of it is the same as the absorbed fraction of
the total mass of rays that emanate to space in different
directions. For the emitted rays we will suppose that the
cosine law of Lambert holds good. With the aid of Table III.
we may calculate the absorbed fraction of any ray, and then
sum up the total absorbed heat and determine how great a
fraction it is of the total radiation. In this way we find for
our example the path (airmass) 1*61. In other words, the
total absorbed part of the whole radiation is just as great as
if the total radiation traversed the quantities 1*61 of aqueous
vapour and of carbonic acid. This number depends upon the
composition of the atmosphere, so that it becomes less the
greater the quantity of aqueous vapour and carbonic acid
in the air. In the following table (IV.) we find this number
for different quantities of both gases.
Table IV. — Mean path of the Earttis rays.
H 2
(C0 2
03.
05.
1.
2.
3.
067
169
168
164
157
153
1
166
165
161
155
151
15
162
161
157
151
147
2
158
157
152
146
143
25
156
154
150
145
141
3
152
151
147
144
140
35
148
148
145
142
If the absorption of the atmosphere approaches zero, this
number approaches the value 2.
Phil. Mag. S. 5. Vol. 41. No, 251. April 1896.
254 Prof. S. Arrhenius on the Influence of Carbonic Acii
III. Thermal Equilibrium on the Surface arid in the
Atmosphere of the Earth.
As we now have a sufficient knowledge of the absorption
of heat by the atmosphere, it remains to examine how the
temperature of the ground depends on the absorptive power
of the air. Such an investigation has been already performed
by Pouillet*, but it must be made anew, for Pouillet used
hypotheses that are not in agreement with our present
knowledge.
In our deductions we will assume that the heat that is con
ducted from the interior of the earth to its surface may be
wholly neglected. If a change occurs in the temperature
of the earth's surface, the upper layers of the earth's crust will
evidently also change their temperature ; but this later pro
cess will pass away in a very short time in comparison with
the time that is necessary for the alteration of the surface
temperature, so that at any time the heat that is transported
from the interior to the surface (positive in the winter, nega
tive in the summer) must remain independent of the small
secular variations of the surface temperature, and in the
course of a year be very nearly equal to zero.
Likewise we will suppose that the heat that is conducted
to a given place on the earth's surface or in the atmosphere
in consequence of atmospheric or oceanic currents, horizontal
or vertical, remains the same in the course of the time con
sidered, and we will also suppose that the clouded part of the
sky remains unchanged. It is only the variation of the
temperature with the transparency of the air that we shall
examine.
All authors agree in the view that there prevails an equi
librium in the temperature of the earth and of its atmosphere.
The atmosphere must, therefore, radiate as much heat to
space as it gains partly through the absorption of the sun's
rays, partly through the radiation from the hotter surface of
the earth and by means of ascending currents of air heated
by contact with the ground. On the other hand, the earth
loses just as much heat by radiation to space and to the
atmosphere as it gains by absorption of the sun's rays. If
we consider a given place in the atmosphere or on the ground,
we must also take into consideration the quantities of heat
that are carried to this place by means of oceanic or atmo
spheric currents. For the radiation we will suppose that
* Pouillet, Comptes rendus, t. vii. p. 41 (1838).
in the Air upon the Temperature of the Ground. 255
Stefan's law of radiation, which is now generally accepted,
holds good, or in other words that the quantity of heat (W)
that radiates from a body of the albedo (1 — v) and tempera
ture T (absolute) to another body of the absorptioncoefficient
/3 and absolute temperature 6 is
where y is the socalled radiation constant (1*21 . 10~ 12 per
sec. and cm. 2 ) . Empty space may be regarded as having the
absolute temperature 0*.
Provisionally we regard the air as a uniform envelope of
the temperature 6 and the absorptioncoefficient a for solar
heat; so that if A calories arrive from the sun in a column of
1 cm. 2 crosssection, ak. are absorbed by the atmosphere and
(1— a) A reach the earth's surface. In the A calories there
is, therefore, not included that part of the sun's heat which
by means of selective reflexion in the atmosphere is thrown
out towards space. Further, let f3 designate the absorption
coefficient of the air for the heat that radiates from the earth's
surface ; /3 is also the emissioncoefficient of the air for radia
tion of low temperature — strictly 15° ; but as the spectral
distribution of the heat varies rather slowly with the tempe
rature, /3 may be looked on as the emissioncoefficient also at
the temperature of the air. Let the albedo of the earth's
crust be designated by (1— v), and the quantities of heat that
are conveyed to the air and to the earth's surface at the point
considered be M and !N" respectively. As unit of time we
may take any period : the best choice in the following calcu
lation is perhaps to take three months for this purpose. As
unit of surface we may take 1 cm. 2 , and for the heat in the
air that contained in a column of 1 cm. 2 crosssection and
the height of the atmosphere. The heat that is reflected
from the ground is not appreciably absorbed by the air
(see p. 252), for it has previously traversed great quantities
of water vapour and carbonic acid, but a part of it may be
returned to the ground by means of diffuse reflexion. Let
this part not be included in the albedo (1 — v) . 7, A, v, M, N,
and a are to be considered as constants, /3 as the independent,
and and T as the dependent variables.
Then we find for the column of air
0y0*=#yv(T*0*) + *A + M. ... (1)
The first member of this equation represents the heat
* Langley, 'Prof. Papers,' No. 15, p. 122. " The Temperature of the
Moon," p. 206.
T2
256 Prof. S. Arrhenius on the Influence of Carbonic Acid
radiated from the air (emissioncoefficient j3, temperature 6)
to space (temperature 0). The second one gives the heat
radiated from the soil (1 cm. 2 , temperature T, albedo 1— v) to
the air ; the third and fourth give the amount of the sun's
radiation absorbed by the air, and the quantity of heat ob
tained by conduction (aircurrents) from other parts of the
air or from the ground. In the same manner we find for the
earth's surface
/3 7 = 0*5. For the watercovered parts of the earth I
have calculated the mean value of v to be 0*1)25 by aid of the
figures of Zenker*. We have, also, in the following to make
use of the albedo of the clouds. I do not know if this has
ever been measured, but it probably does not differ very much
from that of fresh fallen snow, which Zollner has determined
to be 0*78, i. e. j>=0"22. For old snow the albedo is much
less or v much greater ; therefore we have assumed 0'5 as a
mean value.
The last formula shows that the temperature of the earth
augments with /5, and the more rapidly the greater v is. For
an increase of 1° if v=l we find the following increases for
the values of v= 0*925, 05, and 0*22 respectively : —
/3.
!/=0925.
i>=05.
^=022.
065
0944
0575
0275
0*75
0940
0556
0261
085
0934
535
0245
095
0928
0512
0228
100
0925
0500
0220
This reasoning holds good if the part of the earth's surface
* Zenker, Die Vertheilung der Wdrme auf der Erdoberjlache, p. 54
(Berlin, 1888).
in the Air upon the Temperature of the Ground. 257
considered does not alter its albedo as a consequence of the
altered temperature. In that case entirely different circum
stances enter. If, for instance, an element of the surface which
is not now snowcovered, in consequence of falling temperature
becomes clothed with snow, we must in the last formula not
only alter fi but also v. In this case we must remember that
a is very small compared to /9. For a we will choose the
value 0*40 in accordance with Langley's* estimate. Cer
tainly a great part of this value depends upon the diffusely
reflected part of the sun's heat, which is absorbed by the
earth's atmosphere, and therefore should not be included in a,
as we have defined it above. On the other hand, the sun
may in general stand a little lower than in Langley's measure
ments, which were executed with a relatively high sun, and
in consequence of this a may be a little greater, so that these
circumstances may compensate each other. For ft we will
choose the value 0*70, which corresponds when K = l and
W = 03 (a little below the freezingpoint) with the factor 1*66
(see p. 253). In this case we find the relation between T
(uncovered) and T ± (snowcovered surface) to be
T4 . T 4= A(l + l040) + M A(l + O50020)+M
• i r (l+l070) : 7(1 + 050035)
160 + 130+0
= 130 : 115 '
if M=0A. We have to bear in mind that the mean M for
the whole earth is zero, for the equatorial regions negative
and for the polar regions positive. For a mean latitude
M = 0, and in this case Tj becomes 267*3 if T = 273, that is
the temperature decreases in consequence of the snowcover
ing by 5°'7 C.f The decrease of temperature from this cause
will be valid until 0=1, i.e. till the heat delivered by con
vection to the air exceeds the whole radiation of the sun.
This can only occur in the winter and in polar regions.
But this is a secondary phenomenon. The chief effect that
we examine is the direct influence of an alteration of /8 upon
the temperature T of the earth's surface. If we start from a
value T = 273 and /3 = 0*70, we find the alteration (t) in the
* Langley, u Temperature of the Moon," p. 189. On p. 197 he estimates
a to he only 0*33.
t According to the correction introduced in the sequel for the different
heights of the absorhing and radiating layers of the atmosphere, the
number 5°*7 is reduced to 4 o, 0. But as about half the sty is cloud
covered, the effect will he only half as great as for cloudless sky, i. e, the
mean effect will be a lowering of about 2° C,
258 Prof. S. Arrhenius on the Influence of Carbonic Acid
temperature which is caused by the variation of /3 to the
following valnes to be
13=060 t= 5°C.
080 + 56
0*90 +117
100 +186.
These values are calculated for v=l, i.e. for the solid crust
of the earth's surface, except the snowfields. For surfaces
with another value of v, as for instance the ocean or the
snowfields, we have to multiply this value t by a fraction
given above.
We have now shortly to consider the influence of the
clouds. A great part of the earth's surface receives no heat
directly from the sun, because the sun's rays are stopped by
clouds. How great a part of the earth's surface is covered
by clouds we may find from Teisserenc de Bort's work* on
Nebulosity. From tab. 17 of this publication I have deter
mined the mean nebulosity for different latitudes, and found: —
Latitude. . 60. 45. 30. 15. 0. 15. 30. 45. 60.
Nebulosity. 0603 048 0*402 0511 0581 0463 053 0701
For the part of the earth between 60° S. and 60° N. we
find the mean value 0'525, i. e. 52'5 per cent, of the sky is
clouded. The heateffect of these clouds may be estimated in
the following manner. Suppose a cloud lies os 7 er a part of
the earth's surface and that no connexion exists between this
shadowed part and the neighbouring parts, then a thermal
equilibrium will exist between the temperature of the cloud
and of the underlying ground. They will radiate to each
other and the cloud also to the upper air and to space, and
the radiation between cloud and earth may, on account of the
slight difference of temperature, be taken as proportional to this
difference. Other exchanges of heat by means of aircurrents
are also, as a first approximation, proportional to this dif
ference. If we therefore suppose the temperature of the
cloud to alter (other circumstances, as its height and compo
sition, remaining unchanged), the temperature of the ground
under it must also alter in the same manner if the same supply
of heat to both subsists — if there were no supply to the
ground from neighbouring parts, the cloud and the ground
would finally assume the same mean temperature. If, therefore,
the temperature of the clouds varies in a determined manner
* Teisserenc de Bort, " Distribution moyence de la nebulosite," Ann.
du bureau central meteorologique de France, Ann6e 1884, t. iv. 2 de partie,
p. 27,
in the Air upon the Temperature of the Ground. 259
(without alteration of their other properties, as height, com
pactness, &c), the ground will undergo the same variations of
temperature. Now it will he shown in the sequel that a
variation of the carbonic acid of the atmosphere in the same
proportion produces nearly the same thermal effect indepen
dently of its absolute magnitude (see p. 265) . Therefore we
may calculate the temperaturevariation in this case as if the
clouds covered the ground with a thin film of the albedo 0*78
(v=0'22, see p. 256). As now on the average K = l and
W= 1 nearly, and in this case ft is about 0'79, the effect on the
clouded part will be only 0*25 of the effect on parts that have
v=l. If a like correction is introduced for the ocean
(v= 0*925) on the supposition that the unclouded part of the
earth consists of as much water as of solid ground (which is
approximately true, for the clouds are by preference stored
up over the ocean), we find a mean effect of, in round num
bers, 60 p. c. of that which would exist if the whole earth's
surface had v=l. The snowcovered parts are not considered,
for, on the one hand, these parts are mostly clouded to
about 65 p. c. ; further, they constitute only a very small
part of the earth (for the whole year on the average only
about 4 p. c), so that the correction for this case would not
exceed  5 p. c. in the last number 60. And further, on the
border countries between snowfields and free soil secondary
effects come into play (see p. 257) which compensate, and
perhaps overcome, the moderating effect of the snow.
In the foregoing we have supposed that the air is to be re
garded as an envelope of perfectly uniform temperature. This
is of course not true, and we now proceed to an examination
of the probable corrections that must be introduced for elimi
nating the errors caused by this inexactness. It is evident
that the parts of the air which radiate to space are chiefly
the external ones, and on the other hand the layers of air
which absorb the greatest part of the earth's radiation do not
lie very high. From this cause both the radiation from air
to space (/37# 4 in eq. 1) and also the radiation of the earth
to the air (/37v(T 4 — 4 ) in eq. 2), are greatly reduced, and
the air has a much greater effect as protecting against the
loss of heat to space than is assumed in these equations, and
consequently also in eq. (3). If we knew the difference of
temperature between the two layers of the air that radiate to
space and absorb the earth's radiation, it would be easy to
introduce the necessary correction in formulae (1), (2), and
3). For this purpose I have adduced the following con
sideration .
As at the mean composition of the atmosphere (K = l,
260 Prof. S. Arrhenius on the Influence of Carbonic Acid
W = l) about 80 p. c. of the earth's radiation is absorbed in
the air, we may as mean temperature of the absorbing layer
choose the temperature at the height where 40 p. c. of the
heat is absorbed. Since emission and absorption follow
the same quantitative laws, we may as mean temperature of
the emitting layer choose the temperature at the height where
radiation entering from space in the opposite direction to the
actual emission is absorbed to the extent of 40 p. c.
Langley has made four measurements of the absorptive
power of watervapour for radiation from a hot Leslie cube
of 100° C* These give nearly the same absorptioncoeffi
cient if Pouillet's formula is used for the calculation. From
these numbers we calculate that for the absorption of 40 p. c.
of the radiation it would be necessary to intercalate so much
watervapour between radiator and bolometer that, when
condensed, it would form a layer of water 3*05 millimetres
thick. If we now suppose as mean for the whole earth K = l
andW = l (see Table VI.), we find that vertical rays from the
earth, if it were at 100°, must traverse 305 metres of air to
lose 40 p. c. Now the earth is only at 15° C, but this cannot
make any great difference. Since the radiation emanates in all
directions, we have to divide 305 by 1*61 and get in this way
209 metres. In consequence of the lowering of the quantity
of watervapour with the height f we must apply a slight
correction, so that the final result is 233 metres. Of course
this number is a mean value, and higher values will hold
good for colder, lower for warmer parts of the earth. In so
small a distance from the earth, then, 40 p. c. of the earth's
radiation should be stopped. Now it is not wholly correct to
calculate with Pouillet's formula (it is rather strange that
Langley's figures agree so well w 7 ith it), which gives neces
sarily too low values. But, on the other hand, we have not
at all considered the absorption by the carbonic acid in this
part, and this may compensate for the error mentioned. In
the highest layers of the atmosphere there is very little water
vapour, so that we must calculate with carbonic acid as
the chief absorbent. From a measurement b}^ Angstrom J,
we learn that the absorptioncoefficients of water vapour and
of carbonic acid in equal quantities (equal number of molecules)
are in the proportion 81 : 62. This ratio is valid for the
least hot radiator that Angstrom used, and there is no doubt
* Langley, "Temperature of the Moon," p. 186.
t Hann, Meteor oloyische Zeitschrift, xi. p. 196 (1894).
% Angstrom, Bihang till K. Vet. Ah. Handl. Bd, xv. Afd. 1, No. 9,
pp. 11 and 18 (1889).
in the Air upon the Temperature of the Ground. 261
that the radiation of the earth is much less refrangible. Put
in the absence of a more appropriate determination we may
use this for our purpose ; it is probable that for a less hot
radiator the absorptive power of the carbonic acid would
come out a little greater compared with that of watervapour,
for the absorptionbands of C0 2 are, on the whole, less
refrangible than those of H 2 (see pp. 246248). Using the
number 0*03 vol. p. c. for the quantity of carbonic acid in
the atmosphere, we find that rays which emanate from the
upper part of the air are derived to the extent of 40 p. c. from
a layer that constitutes 0*145 part of the atmospbere. This
corresponds to a height of about 15,000 metres. Concerning
this value we may make the same remark as on the foregoing
value. In this case we have neglected the absorption by the
small quantities of watervapour in the higher atmosphere.
The temperaturedifference of these two layers — the one ab
sorbing, the other radiating — is, according to Glaisher's
measurements* (with a little extrapolation), about 42° C.
For the clouds we get naturally slightly modified numbers.
We ought to take the mean height of the clouds that are
illuminated by the sun. As such clouds I have chosen the
summits of the cumuli that lie at an average height of
1855 metres, with a maximum height of 3611 metres and a
minimum of 900 metres j. I have made calculations for
mean values of 2000 and 4000 metres (corresponding to dif
ferences of temperature of 30° C. and 20° C. instead of 42° 0.
for the earth's surface).
If we now wish to adjust our formulae (l)'to (3), we have
in (1) and (2) to introduce 6 as the mean temperature of the
radiating layer and (0 + 42), ((9+30), or (<9 + 20) respectively
for the mean temperature of the absorbing layer. In the
first case we should use v=l and v = 0*925 respectively, in
the second and the third case v = 0"22.
We then find instead of the formula (3)
K
lrv(l/3)'
another very similar formula
T4= l + CI
£
CO
b
OS
co
b
CO
o
b
CO
b
CM
CO
CM
00
o
CO
o
CO
8
S3
1
lH
cc
b
CO
b
00
CO
00
CO
CO
b
CO
b
1—1
CO
CO
OS
b
o
CO
1—1
CO
OS
b
•oarr
CO
GO
CO
00
00
b
CO
b
b
fe
b
b
CO
OS
iO
b
IO
b
1—1
00
co
CO
§3
a
H
< e3
a;
JO U129JH
o
b
1
CM
+
b
00
+
co
7— 1
+
OS
CM
+
rn
ib
CM
+
IO
ib
CM
+
T— 1
ib
CM
+
CM
+
b
os
+
TH
+
b
cb
1—1
CM
+
•aojs[
o
cb
1
CM
CM
+
b
OS
+
CO
cb
+
O
+
OS
ib
CM
+
iO
ib
CM
+
o
ib
CM
+
1H
cb
CM
+
CO
OS
+
CM
J— 1
CM
CO
4
tp
+
•Sny
iO
b
+
iO
cb
+
b
cb
7—1
4
op
An
CM
+
o
cb
CM
+
CO
cb
CM
+
+
o
CM
+
oo
+
cb
I— 1
+
os
ib
+
6
1
co
CO
1
CM
6
+
op
b
+
iO
i—i
+
ip
CM
+
iO
»b
CM
+
00
ib
CM
+
lO
ib
CM
+
o
+
1Q
+
CM
ib
+
IO
OS
o
CM
+
•09(1
A)
CM
1
CM
i— 1
1
1
CO
+
o
b
I— 1
+
CM
CO
CM
+
IO
ib
CM
+
b
ib
CM
4
OS
CM
+
CM
CM
+
ip
b
+
I— 1
+
CO
ib
+
•9pn}t^i
OOOQOOOOOC
b co *o rp co cm th iic5
1 1
o c
CO ^
1 1
1
o
CO
1
in the Air upon the Temperature of the Ground. 265
By means of these values, I have calculated the mean
alteration of temperature that would follow if the quantity of
carbonic acid varied from its present mean value (K=l) to
another, viz. to K = 067, 1*5, 2, 2'5, and 3 respectively. This
calculation is made for every tenth parallel, and separately
~for the four seasons of the year. The variation is given in
Table VII.
A glance at this Table shows that the influence is nearly
the same over the whole earth. The influence has a minimum
near the equator, and increases from this to a flat maximum
that lies the further from the equator the higher the quantity
of carbonic acid in the air. For K = 0'67 the maximum
effect lies about the 40th parallel, for K = 1'5 on the 50th,
for K = 2 on the 60th, and for higher K values above the
70th parallel. The influence is in general greater in the
winter than in the summer, except in the case of the parts
that lie between the maximum and the pole. The influence
will also be greater the higher the value of v, that is in
general somewhat greater for land than for ocean. On account
of the nebulosity of the Southern hemisphere, the effect will
be less there than in the Northern hemisphere. An increase
in the quantity of carbonic acid will of course diminish the
difference in temperature between day and night. A very
important secondary elevation of the effect will be produced
in those places that alter their albedo by the extension or
regression of the snowcovering (see p. 257), and this secondary
effect will probably remove the maximum effect from lower
parallels to the neighbourhood of the poles *.
It must be remembered that the above calculations are
found by interpolation from Langley's numbers for the values
K = 067 and K=1'5, and that the other numbers must be
regarded as extrapolated. The use of. Pouillet's formula
makes the values for K = 0*67 probably a little too small,
those for K = 1'5 a little too great. This is also without
doubt the case for the extrapolated values, which correspond
to higher values of K.
We may now inquire how great must the variation of the
carbonic acid in the atmosphere be to cause a given change of
the temperature. The answer may be found by interpola
tion in Table VII. To facilitate such an inquiry, we may
make a simple observation. If the quantity of carbonic acid
decreases from 1 to 067, the fall of temperature is nearly the
same as the increase of temperature if this quantity augments
to 15. And to get a new increase of this order of magnitude
(3° # 4), it will be necessary to alter the quantity of carbonic
acid till it reaches a value nearly midway between 2 and 2*5.
* See Addendum, p. 275.
266 Prof. S. Arrhenius on the Influence of Carbonic Acid
JO WSdJft
t cq
os do
co g
■AO^J
•■jdag
OS
OS
Oq
OS
00
do
OS
1Q
co
b
cp
CO
do
00
do
CO
OS
i
•Sny
9unp
ds
OS
00
CO
do
CM
do
lp
cq
tr
Cq
^1
o
do
CO
do
6s
rH
OS
1
co
OS
OS
OS
o
OS
co
do
ip
cp
CO
ip
do
1>
do
cq
OS
io
OS
•q ••■•■•■ w
MM <*'
Of the quantities involved in these equations, n u n 2 , »/, vj
are known, and p may be determined by density measure
ments before and after mixture. The form of the functions
in (3) and (4) maybe determined if measurements of the
conductivities of sufficiently extended series of simple solu
tions of the constituent electrolytes are made. We have thus
four equations with but four unknown quantities.
If we employ the symbol V to represent the dilution (v/nv 7 ),
we may write the above equations as follows : —
rrv ■ ■ ■ ■ • • < l >
280 Prof. J. G. MacGregor on the Calculation of
which, in the case of mixtures of equal volumes, becomes
f=MVi), (3)
^ =fc(V f ). '..... (4)
I determined ot l and a 2 from these equations by the fol
lowing graphical process: — Equation (3) was employed by
drawing, from experimental data, for simple solutions of elec
trolyte 1, a curve with values of the concentration of the ions
(a/V) as abscissas and corresponding values of the dilution (V)
as ordinates. This curve was drawn once for all and was used
in all determinations. The curve embodying equation (4)
had to be drawn anew for each mixture examined. If this
mixture was formed of solutions containing n Y and n 2 gramme
molecules pei unit volume of electrolytes 1 and 2 respectively,
the curve had as abscissas the concentrations of ions of a series
of simple solutions of electrolyte 2, and as ordinates, since
Bender's mixtures were mixtures of equal volumes, n 2 /n v
times the corresponding values of the dilutions. Equations
(1) and (2) were applied by finding, by inspection, two points,
one in each curve, having a common abscissa (a!i/V 1 = a 2 /V 2 ),
and having ordinates (Y"{~and — V 2 respectively) of such
Hl
magnitude as to have a sum equal to p times the sum of the
ordinates of the points on the curves determined by the dilu
tions (V/ and V 2 respectively) before mixing. The value of
the abscissa common to the two points thus determined gives
the concentration of ions of both constituents in the mixture.
The corresponding ordinate of the first curve, and that of the
second curve multiplied by njn^ give the dilutions (Vi and
V 2 ) of the constituents in the mixture. The products of the
common value of a/V into V 1 and Y 2 are the required values
of a 1 and a 2 respectively.
It will be obvious that the values of a x and a 2 for a solution
containing two electrolytes with a common ion may be deter
mined in this way, whether it has been formed by the mixing
of two simple solutions or not. It may always be imagined
to have been formed in this way ; and if data are not available
for the determination of p } special density measurements may
be made.
the Conductivity of Mixtures of Electrolytes. 281
Data for the Calculations.
Bender's paper contains all the data required for the cal
culation of the conductivities of mixtures of solutions of
Potassium and Sodium Chlorides, with the single exception
of the specific molecular conductivity of the simple solutions
at infinite dilution. Owing to the want of this datum, I have
drawn the curves «/V = 0(V) by means of data based on
Kohlrausch and Grrotrian's and Kohlrausch's observations*
of the conductivity of solutions of KC1 and NaCl. They are
are as follows : —
...",'.'. NaCl Solutions.
Grrm.molecules
per litre.
Specific
Molecular
Conductivity.
Litres per
grm.molecule.
Concentration
of Ions.
05
757
2
03682
0884
71042
11312
06109
1
695
1
06761
1830
61859
05465
11012
2843
53993
03517
14932
3
528
03333
15418
3924
46635
02548
17802 .
o
398
02
1936
5085 
39253
01967
19416
5325
37765
01878
19562
5421
37195
01845
19611
KC1 Solutions.
Grm. molecules
Specific
Molecular
Conductivity.
Litres per
Concentration
per litre.
grm.molecule.
of Ions.
05
958
2
3939
0691
93313
14472
05304
1
919
1
07558
1427
89070
07008
10452
2208
85552
04529
15535
3
827
03333
20409
3039
82395
03291
20592
3213
81794
03112
21612
These data are quite sufficient for drawing the curves repre
senting a/V as #(V) in the parts corresponding to small
dilutions, but they are few for the parts corresponding to the
* Wiedemann's Annalen, vi. p. 37 (1879), and xxvi. p. 195 (1885).
282 Prof. J. Gr. MacGregor on the Calculation of
greater dilutions, where the curvature is most rapid. I there
fore obtained interpolation formulae, by means of which I
drew in the latter parts of the curves, expressing «/V in the
case of each salt in terms of the reciprocals of powers of V.
These formulae, having no permanent value, need not be given
here. The table of results below shows that they were accu
rate enough for the purpose in hand.
As Bender measured the specific gravities of both his
simple solutions and his mixtures, his paper affords the neces
sary data for determining the change of volume on mixing.
Such change will have a double effect on the calculated con
ductivity : (1) it will affect the value of a as determined from
the curves, and (2) it introduces the factor p in the final com
putation. In the case of Bender's solutions, though in some
cases they were nearly or quite saturated, the first effect was
so small as to be much less than the error incidental to the
graphical process, and I did not therefore take it into account.
The second effect was also very small ; but as in some cases
it was nearly as great as Bender's estimated error, I took it
into account in all cases.
While Kohlrausch's solutions had at 18° C. both the con
stitution and the conductivity specified in his tables, Bender's
solutions had at 15° the constitution and at 18° the conduc
tivity ascribed to them. I found that it did not appreciably
affect the values found for a x and « 2 to take the concentra
tions at 15° as being the concentrations at 18° ; but that this
approximation was inadmissible in calculating the conduc
tivity, as in some cases it made a difference of about the
same magnitude as Bender's estimated error. Hence in the
calculation I took the values of n ± and ?i 2 to be Bender's
values multiplied by the ratio of the volume of the solution
at 15° to its volume at 18°. As Bender measured the thermal
expansion of his solutions, his paper affords the necessary data
for this correction.
The conductivities given by Bender as the results of his
observations are the actual results of measurements, and are
thus affected by accidental errors, which in some cases are
considerable. In order that his measurements may be
rendered comparable with the results of calculations, these
accidental errors must, as far as possible, be removed. I
therefore plotted all his series of observations on coordinate
paper, drew smooth curves through them, and estimated as
well as I could in this way the accidental errors of the single
measurements. The correction thus determined is referred
to in the table on p. 285 as correction a.
Bender himself draws attention to certain differences be
tween his observations of the conductivity of simple solutions
the Conductivity of Mixtures of Electrolytes. 283
of KC1 and NaOl and those for solutions of the same strength
contained in Kohlrausch's tables of interpolated values, ascri
bing them (1) to his own observations being the results of
actual measurement, and (2) to the different temperatures at
which their respective solutions had the specified strengths.
These differences are shown in the following table : —
Salt in
Solution.
Conductivity.
Difference.
Bender.
Koblrausch.
NaCl
KOI
388 380
478 471
702 698
916 911
977 974
1217 1209
1362 1328
1425 1412
1594 1584
1741 1728
1745 1728
1845 1846
2106 2112
2484 2480
2820 2822
+ 8
+ 7
+ 4
+ 5
+ 3
4 8
+34
413
+ 10
+ 13
+17
 1
 6
+ 4
 2
NaCl
KOI
NaCl
NaCl
KOI
NaCl
NaCl
KC1
NaCl
NaCl
KC1
KC1
KC1
Again, it will be noticed that the differences are all of the
same sign up to conductivities of about 1800, and nearly all of
the opposite sign for higher conductivities: also that for any
given conductivity the difference is of the same sign and
order of magnitude for solutions of both salts. If they were
due to the first of the above assigned causes, since Kohl
rausch's interpolated values agree well with his observations,
we should expect much more alternation of sign : if to the
second, there should be no change of sign : if to both, there
should be greater and more irregular variation in magnitude.
The fact that the differences are practically the same for both
electrolytes at any given value of the conductivity, would
seem to show that the cause of the differences — a defect in the
apparatus possibly or in the distilled water — was operative in
the measurements of both sets of simple solutions, and there
fore probably in the measurements of the mixtures. Hence,
to render the results of calculations based on Kohlrausch's
data for the simple solutions comparable with Bender's results
for mixtures, we must determine what the conductivities of
Bender's mixtures would have been found to be if Kohlrausch
had prepared and measured them. To find this out as nearly
as possible, I have plotted the data of the above table with
Bender's conductivities as abscissae and the differences between
284 Prof. J. G. MacGregor on the Calculation of
them and Kohlrausch's corresponding values as ordinates,
and drawn a smooth curve through the points. By the aid of
this curve I determined the correction b of the table given
below. The correction is, of course, a more or less doubtful
one ; for it is not certain that the observations on mixtures
suffered from the same unknown source of error as the obser
vations on simple solutions. It seems probable, however,
that they did ; and the results of the table given below would
appear to render it almost certain. __ . .; *
It may bo well in one case to give an example of the mode
of calculation. We may take for this purpose the mixture of
solutions containing 1 grm .molecule of salt each. It is
found by the graphical process that the value of a/V for this
mixture is 0*718 grm.molecule per litre, and that the
dilutions of the mixture are 0*937 and 1*063 litre per grm.
molecule for the NaCl and the KC1 respectively. The den
sities of the constituent solutions were l  0444 and 1*0401
respectively, and that of the mixture 1*0422. The expansions
per unit volume between 15° and 20° C. were 0*0013569 and
0*0012489 respectively. The values of the conductivity at
infinite dilution I took to be 1028 and 1216, according to
Kohlrausch's observations. Hence the conductivity of the
mixture,
,_1 2 x 10422 /lx 0718x0937x1028 1 x 0718 x 1063 xl216 \_ 809 , 2
2* 20845 \ 1 +06 x 000136 + 1+0 6x000125 /
Bender's observed value (he used the same standard as
Kohlrausch) was 814. To this a correction of about —3
must be applied to make the observation agree with the
others of the same series (correction a), and a correction of
about —3 to make it comparable with a calculated value
based on Kohlrausch's data (correction b). Bender's reduced
result is thus 808, which differs from the calculated value by
1*2 or 0*15 per cent.
Results of the Calculations.
The following Table gives the results of the calculations ;
the second and third columns containing the numbers of
grm .molecules per litre in the simple solutions at 15° C; the
fourth column, Bender's observed values of the conductivities
of the mixtures; the fifth and sixth, corrections a and b referred
to above ; the seventh, Bender's reduced values ; the eighth,
the calculated values; and the ninth, the excess of the calcu
lated values over those observed, expressed as percentages
of the latter.
the Conductivity of Mixtures of Electrolytes. 285
bOWlOM
I—* t— « t — '
(_l _ _l 1— '
1— « )— I
3
bo^pjo
O0 CO tO w 1
o
0^
3
1?
«.   rf*.
  00
  ^ to
. <. > .. w 1— '
„»„,.©
gs
si
©
©
©
©
toi
Q
to S'
cTP
Q CD
& S
sfs
►o 2
CD £
4^ co to o
co tot"
00 ton 1 ©
tOHHOOO
h©©© ©
^ ff.
©on © co
666
d© 6h
© oil d^icb h
On 4 to! ob iJ
p; o
 © 3
tO © X ~~I Oi 4>
00 tor 4> CO to
os 4>oo 3 ©
I; p
©^ 00
00 toi 00 ■ 4*
< C2.
co4> to
tO 00 tor ©
4*. 4 4*
©l
©co© oo
CO tor 00 4^0T« ©
00 07! ©~4l
g
ei
•"■N
CD
co td
tO tO ''!'
tCHM
Q CD
4» CO 00 00
© © CO
oo 4 ©a
to^^^t
to © 00 "
OmOO^Ot
00 tot 4* CO to
C B
00 O tor 4>
© OS to
os 4> co©©© 4» OS
00 to! 4^ CO to
to! CO to 4 00
o"
oo os to
00 ©4
© to! oo a
00 00 OS 00
c
© O"! © • © H
00 4 © 00 CO
p
co toco4»
I
ds ob d
© to to h os 4>
ob © H i tot
1 1 1 +
1 1 +
I+++
'l++l 1 +
I 1 l + l
1 ~
3
co too©
ooo
© ©o ©
©©©©©©
© ©©©©
CD
SSS©"
dtp to
tO
. CD
2
l—4> CO
CO© oo to
 © to tor4j to
! p
p
cs
a
1
1
Phil Map. 8. 5. Vol. 41. No. 251. ^4pn/ 1896. X
286 Prof. J. G. MacGregor on the Calculation of
It will be seen that in the case of the more dilute solutions
Nos. 117 and 19, the differences, which are in all cases less
than 1 per cent, and for the most part considerably less, are
one half positive and one half negative ; and that whether the
solutions are arranged in the order of conductivity or in the
order of mean concentration, they exhibit quite a sufficient
alternation of sign to warrant the conclusion that they are
due, chiefly at least, to errors in the observations and the
graphical portion of the calculations.
In the case of the stronger solutions, Nos. 1618 and 1922,
the alternation of sign has disappeared. In the weakest solu
tions of these two series the differences are positive and small;
but as the concentration increases, the differences become
negative and take increasing negative values, the negative
difference having its greatest value in No. 22, which is a
mixture of a strong solution of NaCl with a saturated solution
of KC1. The tendency towards a negative difference as the
concentration increases may be recognized also in Nos. 11
and 15 ; and it is perhaps worth noting that, while the mean
value of the positive differences is slightly greater than that of
the negative differences up to a concentration of 1 gramme
molecule of salt per litre, the mean negative difference is the
greater for higher concentrations.
It is manifest from these results that for solutions of these
chlorides containing less than, say, 2 grammemolecules per
litre, it is possible to calculate the conductivity very exactly,
but that for stronger solutions the calculated value is less than
the observed.
This excess of the observed over the calculated conduc
tivities shows one or more of the assumptions implied in the
mode of calculation to be erroneous. It would seem to be
probable that the error is at any rate largely due to the
assumption that the molecular conductivity of an electrolyte
at infinite dilution is the same whether it exists in a simple
solution or in a mixture, and that the discrepancy is thus due
to the effect of mixing on the velocities of the ions. The
mode of calculation assumes that in the mixture the con
stituents are not really mixed, but lie side by side so that the
ions of each electrolyte in their passage from electrode to
electrode travel through the solution to which they belong
only. They must rather be regarded, however, as passing in
rapid alternation, now through a region occupied by one
solution and now through a region occupied by the other*
The actual mean velocities of the ions in the mixture will
therefore probably differ from their values in a solution of
the Conductivity of Mixtures of Electrolytes. 287
their own electrolyte only. In the case of dilute solutions
the difference will be small, in sufficiently dilute solutions
inappreciable ; but in the case of the stronger solutions it
may account in large part for the discrepancy observed above #
We have, however, so far as I am aware, no data for calcu
lating the effect of mixture on the ionic velocities, or the
extent to which the discrepancy is due to this effect.
To obtain some rough conception of its magnitude, I have
calculated the conductivity of the mixture No. 18 on two
assumptions, which seemed more or less probable, — viz. (1)
that the velocities of the ions of each electrolyte in the mix
ture were the same as they would be in a simple solution of
their own electrolyte of a concentration (in grammemolecules
per litre) equal to the mean concentration of the mixture ;
and (2) that the velocities of the ions of each electrolyte,
when passing through a region occupied by the other electro
lyte, were the same as they would be in a simple solution of
the former of a dilution equal to that of the latter. The
expression used for the conductivity was
where w, and u 2 are the sums of the velocities of the ions of
electrolytes 1 and 2 respectively in simple solutions of the
dilutions which they have in the mixture, while u/ and u 2 f
are the values these ionic velocities would have according to
the particular assumption employed, the velocities in all cases
being those corresponding to the same potential gradient.
As the graphical process above gave the dilution of each elec
trolyte in the mixture, the values of u and u' were readily
determined by the aid of Kohirausch's table of ionic velo
cities*. I found that according to assumption (1) the con
ductivity would be greater than Bender's reduced value by
1*6 per cent., and that according to assumption (2) it would
be greater by 1*3 per cent. Similar calculations could not be
carried out in the case of solutions stronger than No. 18
owing to lack of data. Such calculations are of course of
little value ; but they strengthen the suspicion that the excess
of the observed values of the conductivity of mixtures over
the calculated values is due to the impossibility of taking into
account the effect of mixing on the velocities of the ions.
* Wiedemann's Annalen, 1. p. 385 (1893).
X2
[ 288 ]
XXXIII. Thermodynamic Properties of Air.
By A. W. WlTKOWSKI*.
[Plates I. & H.]
Pakt I.
A. Thermal Expansion of Compressed Air.
§ 1 AIM of the Work. — It was an important advance in the
theory of gaseous matter, when the experimental
investigations of Despretz, Pouillet, Rudberg, Regnault, and
several others demonstrated the approximate character of
the fundamental laws of Boyle and Charles. Instead of a
common and single law of compressibility and thermal
expansion for different gases, there arose the necessity for
determining specific corrections of these laws for every one
of them.
But as soon as the range of temperatures and pressures
had been enlarged, new analogies between the physical pro
perties of these bodies became once more evident. First of
all, the theory of the critical state may be mentioned here,
supported by the discoveries of Andrews, Cailletet, Wrob
lewski, Olszewski, and many others. On the other hand,
these analogies found their expression in the laws of compres
sibility, thanks to the investigations of Natterer, Cailletet,
Amagat, and Wroblewski.
At the present time the theory of gases seems to be entering
a third phase of its development, initiated by Van der Waals,
and supported by Wroblewski, L. Natanson, Ramsay, Young,
and many other investigators. There are many facts which
seem to show that possibly there may be found, not merely
an analogy, but even an identity of properties of different
gases, provided that for every one of them special specific
units of measure (of temperature, pressure, and density) be
employed : all matter seems to be built on the same plan ;
but the scale is different for various bodies.
It is difficult to judge nowadays whether this grand law
is an exact truth, or only an approximation : whether it is
really universal, or confined only to certain classes of bodies.
To confirm or disprove it, numerous and very exact experi
mental data are greatly needed.
Investigations of the compressibility and thermal expansion
in very extended limits prove to be the best means of com
paring the thermodynamic properties of gases : in every case
* Translated from tlie xxiii. Vol. (1891) of the ' Rozprawy'' of the
Cracow Academy of Science (Math. Class), and communicated by the
Author.
Thermodynamic Properties of Air. 289
they give fuller and more exact information than observations
of critical points, or other singular states. Amongst recent
work in this direction the very important investigations of
Amagat, extending to very high pressures, must be placed
in the first rank. They contain exceedingly valuable data,
relating to the compressibility and expansion of gases at
ordinary temperatures and at higher ones. Amagat's results
give a very clear and extensive idea of the behaviour of
several gases, chiefly of those the critical point of which is
not far from ordinary temperatures.
Until quite recently the socalled permanent gases had not
been investigated at very low temperatures. So far as I
know the important paper by Wroblewski, u On the Com
pressibility of Hydrogen/' published after the author's death
in the Sitzungsberichte of the Vienna Academy, is the only
one relating to this range of temperatures. This want of
information as regards the compressibility and expansion of
permanent gases in the vicinity of the critical state has
induced me to undertake the experimental investigation of
which the present forms an account. The properties of
atmospheric air are described here, at temperatures ranging
from + 100° to — 145° Cent., and for pressures from 1 to 130
atmospheres.
§ 2. Outline of Method. — In order to determine the com
pressibility at different temperatures, and through it also
the expansion of gases, two experimental arrangements were
chiefly used : in one of them the quantity of gas remains
constant, in the other its volume. We might call them the
manometric and the volumetric method. In the first of
these methods (Andrews, Amagat, &c.) a long calibrated
capillary glass tube, enlarged at the open end into a bulb of
known volume, is employed. A certain quantity of gas is
shut up in the tube by mercury. By means of a compressing
arrangement the volume of the gas may be varied at will.
The experiment consists in determining the volume and
temperature of the gas and the amount of the applied pressure.
The method of constant volume was invented, so far as I
know, by Natterer. A vessel of known volume was filled
with gas under a known pressure. The experiment con
sisted in measuring the quantity of gas contained in the
vessel at a certain temperature. This was done in the so
called pneumatic trough, under atmospheric pressure. This
method has been used also by Wroblewski in his experiments
on the compressibility of hydrogen at low temperatures (the
first method being useless here on account of the freezing
of the mercury).
290 A. W. Witkowski on the
The determination of the compressibility of a gas at a set
of temperatures, together with its thermal expansion under a
single pressure (say the atmospheric), gives at once the ex
pansion under any of the pressures employed. And vice versa,
if the compressibility at one chosen temperature be known,
it is sufficient to determine the thermal expansion of the gas
under different pressures, in order to obtain directly the
compressibility at any of the temperatures employed.
The first of these two ways was followed by Wroblewski
in his abovementioned work on hydrogen. In the present
investigation I have used experimental appliances modelled
on those of Wroblewski, and my theoretical aim was a
similar one; but I preferred to apply the second way of
experimenting, viz., the thermal expansion, instead of the
compressibility, has been considered as the principal subject
of the experiments.
The following considerations have induced me to make
this change. First, the difficulty of measuring pressures
exactly. Since absolute manometers, suitable for laboratory
use and sufficiently trustworthy, are still to be invented, we
are compelled to use gasmanometers, founded on the com
pressibility of gases (air or nitrogen). The pressure calculated
according to the indications of an instrument of this kind,
as well as the law of compressibility of the gas under investi
gation, depend, in this case, directly on the assumed law of
compressibility of the gas used in the manometer. Suppose
we take this law as known, say through the experiments of
Amagat, then our results will be inextricably mingled with
these results. The method to be described presently, de
pending on determinations of expansion under constant
pressure, on the other hand, furnishes values of expansion
entirely independent of any accepted law of compressibility :
a dependence of this kind remains only in the values of the
pressures applied.
Another reason which induced me to depart from Wrob
lewski's combination of the gasmanometer with the constant
volume method of batterer, was the wish to invent a method
of constant sensibility as regards pressure meaurements,
and the determinations of expansion and compressibility
as well. The gasmanometer is an instrument of variable
sensibility ; the higher the pressure measured, the less is the
exactness of measurement. On the other hand, the volu
metric method is one of constant sensibility ; i. e. like incre
ments of pressure yield approximately equal increments
of the quantity of gas.
Instead of combining two methods of such opposite cha
Thermodynamic Properties of Air. 291
racter, I preferred to measure both pressures and expansions
by the volumetric method. By this device, as 1 hope, the
exactness and homogeneity of the results were materially in
creased.
§ 3. The Coefficient.— hi order to determine the thermal
expansion of air at constant pressure (1 to 130 atmospheres),
I have applied the following arrangement : — Two vessels of
known capacity are filled simultaneously with gas, under any
desired pressure, by connecting them with a reservoir con
taining a sufficient quantity of condensed gas. One of these
vessels is cooled or heated to any temperature 6; the other is
kept at the constant temperature of melting ice. Letp atmo
spheres be the common ' pressure in both vessels ; s { and s 2
their capacities, at the respective temperatures 6° and 0°, under
the common pressure p.
The quantities M, and M 2 of gas contained in the vessels
are then brought under atmospheric pressure and temperature :
having measured their volumes, we calculate M t and M 2 in
the usual way. As unit quantity of gas I take here, and in
the following pages, the mass contained in unit volume (cub.
mm.) at 0° Centigr. under the pressure of one atmosphere.
The densities of the two masses, when compressed in the
vessels s 1 and s 2 , are unequal; the colder one is also the
denser. Let their densities be p 1 and p 2 , then we have
But
therefore
Mi
Pi= — > P2 =
_M 2
*2
p2
Pi ~'l + « P ,e.
,6'
1M 2 5i
1
e
(1)
This formula may be employed to calculate a p>e ; i. e., the
mean coefficient of expansion of the gas between 0° and 6°,
when under the constant pressure of p atmospheres. It will
be remarked that the value of a p>e is made here to depend on two
ratios,^ and — , and on the temperature 6.
Mi s 2
In most of the experiments to be described the temperature
of the vessel s% was not 0°, but t° (usually +16°). In this
case we have
= Mi = __Po_. M 2= Po
292
therefore
A. W. Witkowski on the
M 2 5 X 1
a P)6 = (l + cip,t.t)
e
(2)
§ 4. Description of Apparatus. — A general representation
of the apparatus I have used to obtain the values of the co
efficient a. will be found in fig. 1.
Fig. l.
The vessels denoted above by s x and s 2 are two thickwalled
glass bulbs, melted on to capillary stems (T X and a 2 , the diameter
of bore of which is less than £ mm. On each of the capillary
tubes a mark m is drawn at a distance of 34 cm. from the
bulb, to limit the capacity s. The capacities of bulbs and
stems were measured repeatedly, before and during the
experiments, by mercury weighings. The capacities s x and
s 2 were nearly 2000 c.mm. (in some experiments at the lowest
temperatures, only 1000 c.mm.) ; the capacities of the capillary
stems (o  ! and
provided that t equals that temperature for which the values
of e were determined, or at any rate that the difference be
not too great. For this reason one of the bulbs of my appa
ratus was always immersed in a waterbath at + 16°, together
with a stirrer and a mercurythermometer divided in fc of a
degree. Since the temperature r of the space a differed but
slightly from + 16°, we may write with sufficient accuracy :
A(l+7*) ■" ,k\
r= 6 STV (5)
The method chiefly employed till now for measurements of
high pressures consists in the use of manometers charged
with a constant quantity of gas. I thought it interesting to
compare this method with the constantvolume method used
by myself. For this purpose I connected my apparatus with
a gasmanometer, charged at first with dry air, afterwards
with pure dry nitrogen. From a large number of comparisons
at various pressures, there resulted a slight but systematic
difference of results : the values of pressures, as determined
by my method, were constantly less, by several tenths per
' Phil Mag. S. 5. Vol. 41. No. 251. April 1896. Y
302 A. W. Witkowski on the .
cent., than the values recorded by the gasmanometer ; this
applies as well to the air as to the nitrogenmanometer. The
sign of the difference led me to suspect a leakage of gas in
some part of my apparatus. Therefore it was indispensable
to submit the apparatus to a severe test, the more so as any
leakage in it would vitiate the results, as regards expansion,
to an incalculable extent.
To test the apparatus I charged it (both bulbs at +16°)
with compressed air, under a sufficiently high pressure, the
value of which was determined simultaneously with the aid of
the gasmanometer. The bulbs were then immediately dis
charged into the eudiometers. Next I repeated this same
experiment, using the same pressure (as indicated by the gas
manometer), but instead of discharging the bulbs immediately
I left them charged for a relatively long time (an hour or
two) . I was satisfied to find that the quantity of air collected
after long imprisonment was not less than in the first case.
As an instance of this sort of testing, the following numbers
may be given : —
First experiment. — Nitrogenmanometer— 89'06 atm.
The bulbs discharged immediately after charging.
Bulb.
No.l...
No. 2...
Temp, of
bulb. s c. mm. a c.mm. A c. mm.
0=16 90190 87 77723
£=16 191947 150 164903
P
(calc. by 5)
8860
8849
Second experiment. — Nitrogenmanometer =
Bulbs kept charged 1 hour 15 min.
8910.
Bulb.
Temp, of
bulb. s. cr. A.
P
No. 1.,
No. 2 .,
,. +16 90190 87 77783
.. +16 191947 150 165096
8867
8860
It will be remarked that the pressure indicated by the gas
manometer exceeds in both cases that calculated by (5) by J
per cent, nearly. At the same time it is apparent that there
was no sensible leakage, since in the longcharge experiment
the difference of pressure is even less than in the first. I am
not able to give a sufficient explanation of the observed
difference, but considering that the constantvolume method
employed in the present work deals with larger quantities of
gas, and enables us to measure them with greater accuracy,
I am inclined to think that the results obtained with my
apparatus are at least not less trustworthy than those recorded
by an ordinary gasmanometer.
Thermodynamic Properties of Air. 303
§ 12. Determination of Temperatures. — All temperatures
referred to in this paper are reduced to the scale of the con
stantvolume hydrogenthermometer. In all experiments at
temperatures below zero there was placed in contact with the
cooled bulb s, in the same freezingmixture, the bulb of a
hydrogenthermometer. Yet the temperatures of the bulb
during the experiments on expansion were not read directly
on the hydrogen thermometer — because of the waste of time
unavoidable in such readings, and because of the slowness
of indications, which renders the hydrogenthermometer un
suitable to follow rapid variations of temperature. For this
purpose I constructed a small working thermometer based on
the variations of electric resistance of a fine platinum wire.
A description of this instrument will be found in the appendix
to the present paper. Here it will be sufficient to say that its
sensibility was about ^q degr. Centigr. and its quickness very
considerable.
This electric thermometer has been compared very often
with the hydrogenthermometer, and a table has been drawn
interpreting its indications in terms of the hydrogenscale.
Nevertheless I never used the electric thermometer otherwise
than under control of the hydrogenthermometer, because
slight secular changes of its resistance manifested themselves.
Comparisons of the working and the hydrogen thermometer
were made in the intervals between two consecutive experi
ments on expansion.
During the experiments themselves an assistant read the
electric thermometer a first time simultaneously with the
charging of the bulbs, a second time immediately after dis
charging them (it will be understood that the freezingmixtures
used to obtain very low temperatures do not keep their tem
perature quite steady) ; the mean of these two readings has
been accepted as the temperature (0) of the bulb. Finally,
the temperature of the bulb was determined a third time (0')
during the measurement of the gas quantity in the eudio
meters ; this temperature has been used to calculate the exact
value of the gasquantity remaining in the bulb.
§ 13. The Low Temperatures. — In the manner explained
above I executed some hundred and twenty determinations of
the coefficient of expansion a Pt0 , using different pressures up
to 130 atmospheres. One of the bulbs of the apparatus being
kept at + 16°, the other was heated or cooled to the following
temperatures: 100 c (steam), 0° (ice), —35° (a freezing
mixture of pounded ice and crystallized chloride of calcium),
— 78°' 5 (solid carbonic acid and ether), — 103°' 5 (liquid
ethylene, boiling under atmospheric pressure) , —130°, —135°,
Y 2
304
A. W. Witkowski on the
— 140°, —145° (liquid ethylene boiling under reduced
pressure).
The arrangement of the thermostat in which liquid ethy
lene has been boiled under reduced pressure is represented in
tig. 3. A tali cylindrical glass vessel A, standing on a horizontal
Tbimp
iiUkiiiiiiiuiiiiiiiiiiiiiiiiiiimimiihiTTi
brass plate, is covered with another brass plate PP, the dia
meter of which is somewhat larger than the diameter of the
upper flat edge of the glass vessel. By means of three brass
pillars, fastened to the lower plate, and screwheads pressing
Thermodynamic Properties of Air. 305
on the upper one, the cover can be made airtight. Near the
centre of the upper brass cover there are four circular open
ings. A fifth opening, with a short brass tube soldered in it,
is placed near the circumference ; it is connected by means of
a lead tube with the pneumatic pumps.
The just mentioned four openings are intended to introduce
into the apparatus (1) the bulb s, which is to be cooled ;
(2) the bulbT of the hydrogenthermometer ; (3) the electric
thermometer T'. The stems of these three pieces are held by
indiarubber stoppers cut into two halves and tightened by a
beeswax cement. Through the fourth opening enters in like
manner the delivering tube of the wellknown apparatus of
Wroblewski and Olszewski for liquefying ethylene.
The cold liquid ethylene flows down into the thin walled
tall glass beaker C, surrounded by a wide glass tube B.
This tube rests on three elastic pieces of indiarubber w 7 hich
press its upper edge firmly against the brass plate P. This
double walling is intended to cool the surroundings of the
beaker C by cold ethylene vapour, which is forced to circu
late as indicated by the arrows, until finally it is drawn out
by the pumps.
Liquid ethylene is very liable to get superheated and to
evaporate in an explosive manner, especially when boiling
under diminished pressure. This property is a great incon
venience when it is to be used to obtain constant tempera
tures. After trying different contrivances to avoid it (air
currents, Gernez's airbubbles, &c), I contented myself with
the use of a large flask D inserted just before the pumps, in
order to diminish the fluctuations of pressure accompanying
the strokes of pistons.
With the aid of this arrangement, it is possible to vary the
boiling temperature of ethylene through a range of some
forty degrees. Any desired temperature maybe obtained by
varying the number of airpumps (I had three large Biancni
at my disposal), by inserting Babinet's stopcocks, or by vary
ing the speed of the gasmotor driving the pumps. With
some practice it is possible to limit the range of fluctuations
of temperature in one series of experiments to no more than
2 or 3 degrees.
§ 14. Construction of Isothermal Curves. — The temperature
6 of the bulb s in different experiments belonging to one
series was not exactly the same. To obtain strictly isothermal
values of the coefficient «, the following graphical method of
interpolation was used. All series of experiments having
been completed I constructed, on a large scale, a diagram
similar to that represented on PL I., using uncorrected values
306 A. W. Witkowski on the
of a. By means of graphical interpolation isothermal s were
then drawn, differing but little from the final ones given on
PL I. With the aid of these isothermals I constructed
another large diagram on which the values of a were col
lected on lines of equal pressure; i. e. temperatures were
drawn on the axis of abscissae, the values of a belonging to
like pressures formed the ordinates.
The inclination of these lines to the axis of abscissae could
now be used to find the correction which was to be added to
the coefficient 0^,33 say, in order to obtain the value of
a p# _35, belonging to another isothermal, but to the same
pressure p ( — 35° in this example being the mean value of
temperatures in the chlorideof calcium series). This I did
by drawing through the point a Pt9 (experimental value) a
short piece of the corresponding isopiestic line, to its inter
section with the ordinate —35°, or any other desired tempe
rature. In most cases these lines were nearly straight ;
instead of drawing them I found it convenient to use a glass
plate on which a straight line had been drawn with a diamond
point. In some cases, however, the corrections were con
siderably larger, so that it was no longer possible to disregard
the curvature of these lines ; they were then drawn with
reference to the inclination and curvature of the neighbouring
isopiestic lines.
§ 15. Results. — The whole of the results obtained are repre
sented in a graphical form on PL I. The isothermal lines
for the nine temperatures experimented on have been drawn
by hand along the dots, representing experimental results*.
Although some of these dots fall off rather considerably from
their respective mean curves, yet, considering the whole of
the diagram, I suppose the final results may be considered
accurate, at least to four decimals. It is scarcely possible
at present to aim at a greater accuracy : this will be admitted
on considering the discrepant values of the coefficient of ex
pansion of air given by different experimenters in the relatively
simple case of atmospheric pressure and temperature of
boiling water.
From the diagram PL I., drawn on a large scale, I took
the mean values of the coefficient a p ^, reproduced in the
following table : —
* In the original memoir extensive tables are reproduced giving full
information on the particular data belonging to every experiment.
Thermodynamic Properties of Air.
307
Table of the mean Coefficients of Expansion of Atmospheric
Air. (Values of 100000 . a p>e .)
,_, Pressures
P atm.
Temperatures 9.
+1000
4160
350
785
1035
1300
1350
1400
1450
S75
376
15...
379
382
...
...
420
427
20...
383
387
401
4ib
427
440
450
25...
388
392
411
422
443
463
479
30...
392
398
420
434
462
477
492
519*
35...
397
403
429
448
483
506
538
40...
402
408
438
461
508
544
632
45...
406
414
448
474
536
594
50...
410
419
430
457
487
569
619
55...
414
424
436
467
500
598
623
60...
418
429
442
476
512
610
622
65...
421
434
448
485
525
612
621
70...
425
438
454
494
536
612
75...
428
442
461
503
547
610
80...
431
446
467
512
557
607
85...
434
449
473
520
566
90...
437
452
479
527
572
95...
439
455
485
532
577
100...
441
458
489
537
579
' 105..
443
460
493
542
580
• 110...
445
462
497
545
580
i 115...
447
463
499
548
579
120...
449
465
501
550
577
125...
466
503
551
574
130...
...
468
551
571
* Corresponds to 29 atmos.
This table, or, better still, the diagram on PI. L, shows very
clearly the kind of variation of the coefficient of expansion in
its dependence on pressure and temperature. For increasing
pressures the coefficient increases, at every temperature, to a
maximum, and decreases subsequently. The higher the
temperature, the less pronounced is this maximum.
The pressure corresponding to the maximum of expansion
diminishes with the temperature. From this it follows, that
the rate of increase of expansion is the higher, the less the
temperature. In the vicinity of the critical state (temp. =
— 141°, pressure 39 atm. nearly) the rate of increase of ex
pansion is extremely great, the corresponding isothermal runs
here nearly vertically. The same holds good for temperatures
below the critical, at points where liquefaction occurs, for
instance at —145° and 30 atm.
The isothermal lines of the coefficient of expansion form a
fanshaped bundle converging nearly to one point, namely,
to the value 0*00367 at 1 atm. of pressure. If this were
308 A. W. Witkowski on the
strictly true, the expansion of air, subject to a constant pres
sure of 1 a tin., would be independent of temperature. Now
this is known to be approximately the case ; but on the other
hand it is certain that the isothermals cannot converge strictly
to this point, since the pressure at one atmosphere has nothing
in common with the internal constitution of air. From some
experiments it would appear that the isothermals do not
intersect at all low pressures ; they come very near one
another, descending to a minimum at certain low pressures,
and probably return afterwards in a steep course towards
higher values.
B. Compressibility at Low Temperatures,
§ 16. Definitions. — It is now a simple matter to calculate
the compressibility of atmospheric air at any one of the nine
temperatures investigated above ; the compressibility at +16°
being assumed as known through the work of Amagat (§ 11).
Let v denote the normal volume (at 0° and 1 atm.) of a
certain quantity of air. When heated or cooled to any
temperature 6, and submitted to a pressure of p atmospheres,
the air assumes a volume v. Admitting Boyle's and Charles's
law, we should write
r
Now, in reality, the product pv is not independent of the
pressure p, and its dependence on 6 is not a linear one.
Instead of the above, we must write :
pv = V . v , . . (6)
y(p0) denoting a certain function of p and 6, the values of
which are to be calculated.
This we may do as follows : — Consider a volume v of air
in the normal state. Heat it, at constant pressure ( = 1 atm.)
to + 16° ; the volume increases to
v (l±y .16).
At this temperature submit it to a pressure of p atmospheres,
then according to the notation of § 11 we obtain the volume
t? (l + y.l6)
p ;
finally heat it at the constant pressure p to 6 degrees.
Denoting by a 16 and a e the coefficients of expansion corre
sponding to the pressure p and to the ranges 016 and
00, we get
tt (l + y.l6) l + 0.a Pt9
p l^lQ ,a Pt iQ
Thermodynamic Properties of Air.
Now this is identical with
309
therefore
r} = e
V —
P
1 + 16 .7
(1 +
%e)
(?)
' 1 + 16 . a pt iQ
§ 17. Results. — The values of the function r)(p,6) might
he also obtained directly from experimental data. This will
be evident when we remark, that the quantity of air =M
(expressed in normal volumeunits) contained, at a tempe
rature 6 under p atmospheres, in a vessel of capacity s is
M.=P S .
v
I preferred to calculate the values of rj according to (7)
because it is easier to obtain values of the coefficient a free
from accidental errors than those of rj. The following table
contains values of the function r)(p, 6) calculated in the
manner indicated.
Table of Compressibility of Air.
00
S s
Temperatures.
+ 100
+ 16
10587
35
785
1035
130
135
1
140 145
1...
1367
10000
08716
07119
06202
05229
05046
04862 04679
10...
13678
] 0550
09951
... I ...
15...
13685
10529
9923
04095 03786
' 20...
13691
10509
9897
06778
05697
04410
3808; 3447
25...
13698110488
9869
6689
5559
4183
3476 3015
30...
13704
10468
9842
6599
5417
3936
03502
3063 2444*
35...
13713
10449
9816
6510
5270
3650
3115
2419
; 40...
13725
10433
9793
6423
5125
3329
2598
1128
45 ..
13738
10419
9772
:::
6335
4980
2963
1942
50...
13754
10408
9754
08288
6252
4839
2544
1605
55... 13770
10399
9738
8253
6170
4701
2171
1553
60...
13784
10390
9723
8219
6089
4567
2013
1556
65...
13802
10384
9710
8187
6011
4439
1985
1576
70...
13821 10381
9701
8158
5937
4318
1989
75...
13842 10379
9694
8132
5863
4206
2013
80...
13866
10379
9688
8105
5796
4103
2043
85...
13887
10380
9684
8081
5734
4014
90...
13908
10382
9681
8058
5680
3948
95...
13929
10386
9680
8038
5634
3903
100...
13951
10390
9681
8023
5600
3881
105...
13977 j 10397
9685
8013
5568
3874
110...
14004
10406
9690
8006
5544
3877
115...
14034
10418
9699
8004
5530
3892
120...
14065
10432
9710
8006
5520
3914
125...
10448
9722
8012
5520
3944
130...
10467
9738.
5528
3981
* Corresponds to 29 atmos.
310 A. W. Witkowski on the
These numbers are represented in a graphical form on
PL II. by means of socalled curves of compressibility
(abscissas = pressures, ordinates = isothermal values of 7j=pv ;
the axis of abscissas through the point 0*1). It may be well
to remark that, on assuming Boyle's law to hold for all
pressures and temperatures, these curves would be straight
lines parallel to the axis of abscissas.
Every one of these curves shows a minimum of the pro
duct pv for a certain value of p (depending on the tempera
ture of the corresponding isothermal). This expresses the
fact, verified for many gases at higher temperatures, that with
increasing pressure the compressibility exceeds at first that
given by Boyle's law until a maximum is reached, afterwards
it diminishes indefinitely. In the vicinity of the critical point
the curves of compressibility run downwards very steeply ; at
points where liquefaction occurs their course is vertical.
§ 18. Comparison with other Gases. — The general proper
ties of atmospheric air as regards expansion and compressi
bility are quite analogous — apart from the large difference of
critical temperatures — with those of other gases which have
been investigated hitherto in this respect. It is interesting to
inquire whether this resemblance of properties is merely a
qualitative one, or whether it is more deeply rooted ; in other
words, is it possible to calculate beforehand the properties of a
gas, assuming the properties of another to be known ? — this is
the thesis of van der Waals. Wroblewski, in his memoir on
the Compressibility of Hydrogen, asserts the possibility of such
predictions. Take for any one gas the critical pressure for
the unit of pressure ; for the unit of temperature its critical
temperature (absolute) : then Wroblewski's theorem asserts
that the dependence between the temperature and that value
of the pressure for which the product pv is a minimum, is the
same for all gases. A more general theorem is due to
L. Natanson, namely, that all gases have a common character
istic equation, i. e., a common relation (not necessarily that of
van der Waals) between pressure, temperature, and volume,
provided that these elements be measured by means of units
specially adapted to the nature of every gas; the critical
elements form one of the infinitely numerous groups of such
units.
The critical elements of air are given by Olszewski as
follows : — crit. pressure = 39 atm. ; crit. temperature = — 140°
Cent. These data are also confirmed by my own experiments ;
I should only consider — 141° as a nearer approximation to
the true value of the critical temperature. A comparison of
atmospheric air with other gases may be best effected on the
Thermodynamic Properties of Air. 311
basis of Wroblewski's theorem. With the aid of the table of
compressibility given in the preceding paragraph, we calculate
the following values of the pressures for which the product
pv is a minimum : —
Temperature. Pressure. Min. pv (approx.).
+ 100 Less than 10 atm. 1'367
+ 16 79 10379
95 09680
 35 115 08004
 785 123 05519
1035 106 03873
130 66 01985
135 57 01551
The points corresponding to these values of p and pv are
connected on PL II. by a dotted curve.
Now, changing units, take the values 132° (abs. crit.
temperature) and 39 atm. as new units of temperature and
pressure (which will be denoted on this convention by t and
7r), then the foregoing table turns into the following : —
t=282 tt<025
„ 219 ,, = 203
„ 207 . „ 244
„ 180 „ 294
„ 148 „ 315
„ 129 „ 272
„ 1*08 „ 169
„ 1*05 „ 146
Draw a curve taking the t's and 7r's for abscissas and
ordinates, then according to Wroblewski this should be one
curve for all gases. Now it will be found that the curve
plotted in accordance with the above numbers runs really very
near that drawn by Wroblewski*, and based chiefly on experi
ments on carbon dioxide and methane. It is difficult to tell
if the remaining differences are real or else depend merely
on experimental errors. At all events it may be taken for
granted that these coincidences, as they stand' now, are most
remarkable.
I have limited my experiments to the gaseous states of air
but at the same time I tried to approach to the limits of lique
faction as near as possible. It is a very difficult . matter to
experiment near those limits ; very constant temperatures are
of great importance. It happened frequently that, in con
sequence of a slight variation of temperature, the glass bulb
of my apparatus was found full of liquid air, instead of com
pressed gas. For the present time I abstained from an ex
* PI. iv, of the paper on Hydrogen,
312
A. W. Witkowski on the
ploration of the region of liquefaction ; 1 intend to go through
it on another occasion, when investigating the properties of
simple gases.
In concluding I wish to express my thanks to Dr. J. Za
krzewski, to whom I owe many valuable suggestions, and who
undertook a great part of the very considerable labour which
was necessary to obtain the results given in the present paper.
Cracow, Physical Laboratory of the
Yagellonian University. «==
May 1891.
[To he continued.]
Fig. 4.
Appendix.
Electric Thermometer for Low Temperatures*.
Variation of the electric resistance of wires depending on
variation of temperature has been often
employed to construct thermometric appa
ratus. Fig. 4 shows (nearly true size) a
disposition of electric thermometer which
the author has found very useful in low
temperature work, on account of its sensi
bility and quickness. A short cylinder r
of thin sheetcopper is soldered at one end
of a narrow brass tube c through which
passes a rather thick silkcovered copper
wire d cemented with a mixture of resin
and indiarubber. The outer end of the
brass tube ought to be carefully covered
with this mastic in order to prevent con
densation of moisture on the thermometric
wire. The copper wire and the brass tube
are furnished with bindingscrews a, b to
introduce the current. On the outer side
of the cylinder r there are wound 2 or 3
metres of a very fine silkcovered platinum
wire (diameter about T ^ millim.) ; one of
its ends is soldered to the cylinder r, the
other to the end of the copper wire d.
To protect the coiled wire, another sheet
copper cylinder r' of somewhat greater
diameter is pushed over it. Both cylinders
r and r' are joined by a small quantity of
solder applied round the circumference of
their bases.
* Bulletin internat. de VAwd.de Sc.de Cracovie, May 1891,
Thermodynamic Properties of Air.
313
In this manner a platinum resistance is obtained of some
220 ohms at 0°. It forms one branch T (fig. 5) of a Wheat
stone bridge. A second branch is formed by a resistance of
Fisr. 5.
germansilver wire contained in a case K of exactly the same
construction as shown on fig. 4, but on a larger scale ; it is
kept at the constant temperature 0° by melting ice. The
third and fourth branches of the Wheatstoiie bridge are resist
ancecoils in one and the same box R, namely, a coil of
1000 ohms and a variable resistance R. B is a battery of
two elements of Leclanche, M a commutator, Gr a sensitive
galvanoscope.
To every temperature of T there corresponds a determinate
resistance R in the box. By comparison with a normal
hydrogenthermometer, it is possible to construct a diagram,
or a table, with the aid of which one may find by inspection
the temperature T corresponding to any observed resistance R.
Such a table does not cease to be true when the apparatus,
after having been dismounted, is to be used again several
days or months later ; the relation of T and R is of course
independent of the electromotive force of the battery or the
sensitiveness of the galvanoscope.
A table of this kind is given below; at the same time it
shows the relation of resistance and temperature (hydrogen
scale) for a certain platinum wire.
314
Thermodynamic Properties of Air.
T.
o
+ 50
10
20
30
40
50
60
70
80
The variation of resistance is about 2 ohms per degree ;
therefore it is easy to obtain a sensibility of ^° Cent.
Experience has shown that the relation between resistance
and temperature undergoes slight changes when the thermo
meter is employed at widely different temperatures. For
this reason it is better to avoid heating a thermometer destined
for low temperatures ; this might cause a variation of resist
ance which would not disappear until after several months.
E.
T.
K.
11059
o
 90
8018
10000
100
7789
9785
110
7558
9569
120
7324
9352
130
7089
9134
140
6853
8914
150
661*5
8693
160
6373
8470
170
6127
8245
180
5880
Note added by the Author.
It has been found since that fine iron wire answers the
purposes of lowtemperature thermometry still better than
platinum. Both constancy and sensitiveness are greater.
The following data apply to an iron thermometer which has
been now four years in use. Diameter of wire = 0'035 millim.,
resistance at 0° about 576 ohms.
Displacement of Zeropoint
January 1892
March 1892
June 1892
February 1893
December 1893
March 1894
February 1896
R =100180 ohms.
100150
100123
100097
100110
100117
100112
The sensitiveness is also nearly double that of platinum
thermometers, as will be seen by the following table : —
Analytical Study of tlie Alternating Current Arc. 315
Temp.
E.
10012
 55
7619
 65
7204
 79
6636
100
5810
130
4673
182
2910
£?R (ohms per
dT degree).
465
420
412
402
387
366
The author prefers now a slightly different type of electric
thermometer, differing from that shown on fig. 4 in this par
ticular, that the brass tube c joins the outer cylinder i J .
Although rather more difficult to construct, it allows the
resistancecoil to embrace the bulb of a hydrogenthermo
meter or any other piece of apparatus the temperature of
which is to be determined.
XXXIY. An Analytical Study of the Alternating Current Arc.
By J. A. Fleming, M.A., D.Sc, F.R.S., Professor of
Electrical Engineering in University College, London, and
J. E. Petavel*.
ALTHOUGH the physical phenomena of the Electric
Arc have received of late years very considerable
attention, there are many points in connexion with the study
of the alternating current arc which seem to us to have
been insufficiently explored. The investigation which is here
described is a contribution to the further examination of the
mode in which the variation of the luminous and electric
effects takes place in the alternating current arc.
The first portion of the research is concerned with an
analytical study of the variation of the light thrown out from
different radiating regions in the arc, and the delineation of the
periodic value of this illuminating power by graphical methods.
The object of the first set of experiments was to record and
represent by an appropriate graphical method the periodic
value of the light thrown out from the carbons and from the
true electric arc region when the arc is supplied with electric
power of known constant amount and varying magnitude,
and at the same time to record the periodic variation of
current through the arc, and potentialdifference of the
carbons. The instrumental appliances involved in the first
place the employment of means for keeping constant and
* Communicated by the Physical Society : read February 28th, 1896.
316 Prof. J. A. Fleming and Mr. J. E. Petavel
o
measuring trie mean value of the power supplied to the arc.
This was accomplished by the use of a suitable noninductive
resistance and a bifilar reflecting wattmeter, the series coil
of which was in the circuit of the arc, and the shunt coil
of which was connected across the carbons of the arc. The
following is a description of this wattmeter : —
The series coil was wound with 10 turns of thick copper
wire (No. 10 S.W.Gr.) and had a resistance of *01 ohm. The
shunt coil was wound with 23 turns of thin wire, and had a
resistance of 3" 76 ohms, and was placed in series with a
noninductive platinoid resistance of 1200 ohms The shunt
coil was suspended by two fine silver wires, which also served
to conduct the current in and ont of the coil. The con
trolling and deflecting forces so balanced each other at small
displacements as to make the angular displacement of the
movable coil very nearly proportional to the power passing
through the wattmeter. The movable coil was provided with
a mirror by means of which the image of a wire illuminated
by an auxiliary arc lamp was reflected on to a fixed scale.
This scale was carefully graduated, so as to read directly in
watts. It was found that the earth's magnetic field caused a
small deflexion of the movable coil when using continuous
currents and when the shunt current was passing through it,
when at the same time no current was flowing in the series
coil. This terrestrial field was neutralized by magnets, or
else the scale was shifted so that the part of the deflexion of
the shunt coil due to the terrestrial field was eliminated. The
best way would have been to have turned the wattmeter
round through a certain angle, but as it had to be screwed
up against the wall in a fixed position for steadiness, it was
found that the above method was the simplest plan for obviating
this source of error. The vibrations of the movable coil were
damped by means of a mica vane dipping into a dashpot
filled with oil. When all the adjustments were made, it was
found that this wattmeter produced a deflexion of the spot of
light on the scale almost exactly proportional to the power
passing through the instrument. This wattmeter was then
connected up to the arc lamp, so that the whole current
actuating the arc lamp passed through the series coil, the
terminals of the shunt coil being connected to the carbons
of the arc.
A series of preliminary experiments were then made for
the purpose of enabling us to eliminate from the wattmeter
readings, firstly, the power taken up in the wattmeter itself ;
secondly, the power taken up in the shunt coil of the arc
lamp and other shunt, resistances ; and, thirdly, the power
Analytical Study of the Alternating Current Arc. 317
taken up in the carbons themselves, right up to the point
where the arc is being formed. The observations, when
reduced by applying these corrections, gave us at once the
true mean power being taken up in the arc. The wattmeter
was constructed with all the precautions necessary in making
a wattmeter for measuring alternating current power, so that
it served to measure the power taken up either in alternating
or continuous current arcs. The current supplied to the
arc passed through a series of noninductive resistances, con
sisting of carbon plates, in such a manner that the power
given to the arc could be regulated with the greatest exact
ness. In addition to the wattmeter, an ammeter was placed
in series with the arc lamp so as to measure the current
passing through the arc, and a voltmeter connected to the
carbons so as to measure the potentialdifference of the
carbons, these instruments being suitable both for continuous
and alternating currents. In all the experiments the mean
power was kept constant in the arc, and this was done by
adjusting the current so that the wattmeter took a certain
deflexion corresponding to the power desired, and the watt
meter was kept at a constant deflexion by regulating the
carbon resistances in series with the arc. In addition to the
instruments described above, a lens was fixed so as to enable
the length of the arc to be measured in the usual manner.
In the course of the experiments three arc lamps were
employed — a handregulated arc lamp in which the distance
of the carbons was adjustable by a screw with great accuracy ;
a continuous current arc lamp (the Waterhouse Arc Lamp);
and an alternatingcurrent arc lamp (the Helios Arc Lamp) ;
these last two being selected as excellent arc lamps of their
respective types, our object being to select an arc lamp, the
mechanism of which enabled it to be worked with electric
powers varying over wide limits, and yet to yield a perfectly
steady arc. In addition to these instruments there was set
up the apparatus for delineating the curves of current of
electromotive force, which have been described by one of us
in the ' Electrician,' vol. xxxiv. p. 460, and which consists of
a synchronizing alternatingcurrent motor, having its fields
separately excited and its armature circuit traversed by a
shunt current from the circuit operating the arc lamp. This
alternatingcurrent motor was set up on the photometer bench,
and it was used to drive an aluminium disk pierced by four
openings, in such a manner that the disk revolved synchro
nously with the alternating current operating the arc. In
addition to this duty the alternatingcurrent motor carried a
Phil. Mag. S. 5. Vol. 41. No. 251. April 1896. Z
318 Prof. J. A. Fleming and Mr. J. E. Petavel :
contactmaker on its shaft for the purpose of enabling the
curves of current of electromotive force to be delineated. A
brief description of the motor is as follows : —
It consists of two sets of field magnets M, M (see fig. 1),
having eight poles on each side which are secured to two
castiron disks. Between these field magnets revolves a small
armature A, the iron core of which is formed of a very thin
strip of transformer iron wound up into a ring, the armature
coils being wound on this ring. The armature coils are joined
up in series with one another, so as to give a series of
alternating magnetic poles round the ring when a current
flows through the armature circuit. The diameter of this
armature is about six inches. The field magnets have eight
poles, and the armature eight coils. The field magnetic coils
are bobbins about 2 inches long and 1^ inch in diameter, and
when joined up in series in the proper manner the field
magnets take a current of about eight amperes to give them
a proper amount of saturation. The armature is carried
upon a hard wooden boss fixed to a steel shaft ; and the steel
shaft is carried through small ball bearings made like bicycle
bearings. In order to prevent any side shake of the armature,
there are at opposite ends of the base castiron pillars with a
gunmetal screw through each, against which the rounded
end of the shaft bears ; the position of the shaft can thus be
adjusted with great nicety, and runs with great freedom from
friction. The ends of the armature circuit are brought to
two small insulated collars fixed on the shaft, against which
press two light brass brushes marked B, B kept against the
shaft by means of an expanding steel wire W. The armature
shaft carries on one side an ebonite disk with a steel slip let
into it. Two insulated springs S, S are carried on a rocking
arm H; this rocking arm can be traversed through half a
circumference, and is centred upon the gunmetal screw
which prevents side shake in the shaft, and a pointer and
graduated scale enables the exact position of the contact
springs to be determined. One of the stopscrews keeping
the shaft from side shake is pierced with a longitudinal hole,
and through this hole passes a stiff steel wire ; this serves to
drive an aluminium disk 27 centims. in diameter and 4 milli
metres thick. This disk is carried on a shaft which runs in
a castiron bearing, and the disk is therefore driven syn
chronously by the motor. This aluminium disk has four slits
in it separated by angular intervals of 90° ; the slits are 0*5
centimetre wide and 4*5 centimetres long. If the field
magnets of the motor are excited by a continuous current of
Analytical Study of the Alternating Current Arc. 319
about eight amperes, and if an alternating current of about
two amperes is passed through the armature of the motor,
then, on turning the motor rapidly round by hand, which can
best be done by passing a strap round the shaft and pulling
at the strap so as to spin the motor like a top, the motor will,
Z2
320 Prof. J. A. Fleming and Mr. J. E. Petavel :
if sufficient speed be gathered, drop into step with the alter
nating current driving it. Since the motor has eight mag
netic poles, it makes one complete revolution in four complete
periodic times, so that if the motor is being driven from an
alternatingcurrent circuit having a frequency of 100, then
the motor has to run at 1500 revolutions per minute before
it will drop into step, but at that speed it will fall into step
with the current passing through its armature, and will be
driven as a synchronous motor. Under these circumstances,
if a ray of light is passed transversely to the disk in such a
manner as to pass through the slits of the aluminium disk
during the progress of rotation of the disk, the beam of light
will be interrupted, but will obtain passage four times during
each revolution through the slits in the disk as it goes round.
If the motor is being driven by the same alternating
current circuit which supplies the alternating current to an arc
lamp, it is evident that, on looking through the slits in the
revolving disk at the alternating current arc, it will be seen in
one constant condition during its periodic variation, such
instant being determined by the position of the slits with
reference to the phase of the current. Without entering
into a longer description, it will be evident that this synchro
nizing motor driving a disk and a contactbreaker enabled
two things to be done — first, to delineate all the current and
electromotiveforce curves of the arc taken in the usual way;
and, secondly, a ray to be taken from the arc selected at one
particular instant during the complete period through which
the variation of illumination passes. These arrangements
were completed by the construction of a photometer of a
particular kind. Owing to the slow variation of position of
the electric discharge in the alternating current arc, it would
have been useless to photometer the instantaneous value of
the light coming from the alternating current arc against any
fixed standard of light ; but it was found possible to make a
very exact comparison between the intensity of the light
coming from any part of the arc, and selected at any one
constant instant during the complete phase, with the mean
value of the light coming from that same part of the arc
during the complete period ; in other words, it was found
possible to photometer the arc against itself, and so eliminate
to a large extent the difficulties arising from slow periodic
variations of the light sent out from the arc in any one
direction. It is well known that the light of an alternating
current arc, taken in any one direction, undergoes a slow
periodic variation quite independently of the variation of
Analytical Study of the Alternating Current Arc. 321
current during the phase, neither is it dependent upon any
variation of the mean square value of the current, because it
takes place even when that current is perfectly constant. It
appears to be due to slow changes of position of the points on
the carbons between which the discharge takes place. The
discharge is as it were seeking out new points between which
to take place, and it continually changes these positions as the
arc burns. The photometric arrangements finally adopted
were as follows, and are shown in outline in fig. 2 : —
Arrangement of the Photometer and Revolving Disk.
A represents an alternating current arc. This arc was
enclosed in a metal lantern in which were three openings.
The light from this arc passed through a lens L x in a hori
zontal direction and fell upon a mirror M x placed at an angle
of 45°, capable of being rotated at this constant inclination
round a horizontal line colinear with the axis of the revolving
disk, and the motor placed in front of the lens L lt The ray
was then reflected upwards into another mirror M 2 , and by
this mirror reflected at an angle very nearly equal to 45° in
such a manner as to pass through the slits in the revolving
disk D, when any one of the slits was in a position to allow
the ray to pass. The two mirrors M x and M 3 were rigidly
connected to a rockingarm so centered that the line M. x M 2
could be rotated round into any required position, always
moving parallel to a radial line of the disk D. The disk D
was the disk carried on the shaft of the synchronizing motor
above described ; the motor, together with its associated disk,
was placed on the photometer bench in the required position
opposite to the arclight lantern. In fig. 2 the motor itself
is not shown, but its position is indicated by the letter C.
The angular position which the rockingarm carrying the two
mirrors Mj and M 2 occupied with respect to the vertical line
passing through the centre of the revolving disk could be
322 Prof. J. A. Fleming and Mr. J. E. Petavel :
observed on a graduated scale. It will be clear, then, that if
the disk driven by the motor was in synchronism with the
alternating current producing the alternating arc at A, an
observer, looking through holes in the rapidly revolving disk,
would see by reflexion in the mirrors M x and M 2 the alterna
ting current arc at A ; but he would see it, not as it is seen
when looked at directly, but in some constant condition taken
at one definite instant during the phase, which instant would
depend upon the position of the line ~M X M 2 with regard to the
vertical line through the centre of the disk. Thus by rocking
over the arm MjMg i n various positions the observer coald
see, through the window W as the disk revolved, the arc at
A, either in the condition when the electric discharge is
taking place, or when the true arc is extinguished, according
to the position in which the arm M : M 2 was set. If, instead
of observing with the eye, a disk of paper with a photometric
grease spot upon it was placed at P, the lens L could be so ad
justed as to throw an enlarged and welldefined image of the
arc upon the disk at P; and by rocking over the arm carrying
the mirror into successive positions, the observer woald see
the image of the arc pass slowly through all those successive
phases which in the arc itself actually take place during one
periodic time. As the image of the arc is much larger than
the photometer disk, it was possible, by slightly shifting one
of the mirrors, to bring any desired part of the image of the
true arc or of the craters of either of the carbons to cover the
grease spot. In addition to this interrupted ray, another ray
was gathered from the same part of the arc by a lens L 2 ,
placed on the same level with the lens L 1 but slightly to one
side of it. This lens gathered a beam which was reflected by
a mirror M 3 placed at an angle of 45°, and which reflected
the ray upwards to another mirror M 4 . In fig. 2, for the
sake of clearness the lens L 2 is shown beneath the lens L 1}
but it must be understood that in the real apparatus the
lenses L x and L 2 w 7 ere on the same level and placed side by
side. The ray reflected from the mirrors M 3 and M 4 was set
horizontally, so as to be received on a lens L 3 , and by this lens
L 8 was gathered to a focus at a point I. The lens L 2 was so
adjusted as to form a large image of the arc on the screen
which carried the lens L 3 , and by slightly moving the mirror
M 4 any part of this image could be made to cover the lens L 3 .
It will be seen, therefore, that the light gathered together at
a focus at the point I could be made to be light coming
from any assigned area in the arc or from the craters. A
movable stand carried two other mirrors M 5 and M 6 fixed at
Analytical Study of the Alternating Current Arc. 323
angles of 45° in such a way as to reflect the ray coming from
focus at 1, and reverse its direction so as to bring it round
and make it fall on the lefthand side of the photometerdisk P.
It will thus be seen that, by moving the mirrors M 5 and M 6 ,
the light falling on the lefthand side of the photometerdisk
P could be made to have any desired intensity within certain
limits, and could be gathered from any desired part of the
arc or craters ; and, moreover, this illumination was the mean
illumination, or proportional to the mean illuminative power
of any part of the arc selected for examination. It will thus
be clear that the arrangement enabled us to project on to the
righthand side of the photometer disk P the rapidly inter
mittent ray taken from any part of the arc, and always
gathered at one constant phase condition during the complete
period ; whilst on the lefthand side of the photometer disk
we could project a ray gathered from the same part of the arc,
but not interrupted. We could therefore compare the mean
value of the light proceeding from any part of the arc with
the instantaneous value of the light taken from the same part
of the arc and selected at any assigned instant during the
period. Thus the arc itself became its own standard, and
difficulties due to slow fluctuation of the mean light of the arc
disappeared. At the same time the contactmaker on the
motor enabled us to delineate in the usual manner the curves
of current and potentialdifference of the arc, and thus to
record the variation in the arc of the arc current, the carbon
potentialdifference, the power expended in the arc, the
resistance of the arc, and the luminous intensity of any part
of the arc. A long series of experiments was then made
with alternating current arcs of different lengths and powers,
the periodic electric quantities being delineated and the light
being taken, either from the centre of the true arc halfway
between the carbons or from one of the craters of the carbon
terminals, — generally the bottom carbon. The process of
taking measurements was as follows : —
After setting the lenses and the mirrors so that the lens L x
and the mirrors M 1 and M 2 gave a sharp image of the arc on
the righthand side of the photometer disk with the selected
area of the image covering the grease spot, the minors M 5
and M 6 were moved backwards and forwards until the balance
was obtained between the illumination falling on the right
and on the lefthand side of the photometer disk. The right
hand side of the photometer disk being illuminated by an
intermittent stream of light always selected in the same phase,
when considered as belonging to a periodically varying illu
324 Prof. J. A. Fleming and Mr. J. E. Petavel :
minating beam, whilst the light falling on the lefthand side
of the photometer disk is a uniformly illuminating beam of
the same quality and colour coming from the same part of the
arc but not interrupted, and representing therefore the mean
value of the light emitted from that selected area of the arc.
Observers were deputed to measure all the various quan
tities by the different instruments, and a power of constant
definite amount was supplied to the electric arc from an
alternatingcurrent machine driven by a continuouscurrent
motor. By means of the carbon resistance and the reflecting
wattmeter this power was kept constant at a selected value
for a sufficient time to enable all the various periodic quan
tities to be observed at sufficiently frequent intervals during
the phase. In all cases the arc was allowed to burn quietly
for half an hour to get the carbons into a constant position
before any observations were taken. It is hardly necessary
to go into the details of delineating the current and electro
motiveforce curves, as the process of doing this is now well
understood. The Kelvin vertical multicellular voltmeter,
having a half microfarad condenser placed across its ter
minals, was employed for the measurement of the potential
difference of the carbons in the following manner : —
The voltmeter, with its associated condenser, was connected,
through the contactmaker driven by the shaft of the alter
nating motor, to the carbons of the arc ; the contactmaker
thus closed the circuit at a certain instant during the phase,
and the voltmeter gave the instantaneous value of the potential
difference of the carbons. In series with the arc was placed
a noninductive resistance of suitable magnitude. A switch
was arranged so that the voltmeter with the condenser in
parallel with it, both being in series with the revolving
contactmaker, could be put across either the terminals of
this noninductive resistance, or else between the carbons of
the arc. By rocking over the arm carrying the spring
brushes of the contactbreaker, the voltmeter circuit was
closed at a particular instant during the phase, and the volt
meter reading gave therefore, when corrected, the instanta
neous value either of the potentialdifference of the carbons
or the instantaneous value of the current through the arc.
As the Kelvin multicellular voltmeter used by us only begins
to read at 60 volts, in order to get readings for lower values
than 60 volts, it is necessary to add a known electromotive
force to the voltmeter circuit in order to block up the needle
of the voltmeter to a false zero. This was done by connecting
a known number of small Lithanode secondary batteries, the
Analytical Study of the Alternating Current Arc. 325
potential of which was determined by the same voltmeter, in
series with the voltmeter. In this way the series of obser
vations were successively taken of the following quantities :—
First, the instantaneous value of the potential difference of
the carbons taken at equidistant intervals throughout a com
plete period ; secondly, the instantaneous values of the
current through the arc taken throughout the complete
period ; and, thirdly, the instantaneous values of the luminous
intensity of a certain selected portion of the arc taken at
intervals throughout the complete period and expressed in
terms of the true mean luminous intensity of the same portion
of the arc at that time. These quantities having been obtained,
it was then possible to plot them down in a series of curves,
and to deduce therefrom curves representing the periodic
variation of power through the arc, and the periodic variation
of the resistance of the arc. Five sets of experiments were
made, taken for different frequencies and different lengths
of arc, each set comprising an observation taken with the
light proceeding from the centre of the true arc, and also
an observation taken with the light proceeding from the
centre of the crater of the lower carbon. The frequencies
employed were 83, 50, and 26. In all cases the current
was kept at 14 amperes (mean square value) ; the results
of these observations are embodied in the following 10 tables
arranged in 5 pairs. Table I. A, for instance, gives the
results of the observations taken with an arc having a fre
quency of 83*3, the light being taken from the centre of
the true arc. Table I. B gives similar results of observations
for the same arc, the light being taken from the centre of the
lower crater. By holding a magnet at the back of the arc,
noticing which way the instantaneous image of the true
arc was projected by the magnet, and noting the pole of the
magnet presented to the arc, it was possible to determine
when the lower carbon was positive and when it was negative ;
and the diagrams corresponding to the above 10 tables are
marked so as to show the half of the wave when the lower
carbon is positive and when it is negative, in all those
diagrams which refer to the light coming from the crater.
The results of the observations given in the 10 tables are
delineated graphically in diagram in figs. 3 to 12 ; and on
referring to these diagrams a periodic line will be seen in
each, delineating the variation of the potential difference
of the carbons, and another periodic line indicating the varia
tion of the current through the arc, whilst a third line indicates
the periodic variation of the luminous intensity of the selected
326 Prof. J. A. Fleming and Mr. J. E.. Petavel :
portion of the arc. In the diagrams figs. 13 to 17 are given
curves representing the periodic variation of the light from
the crater in the various cases, and extending over several
periods. It will be noticed that those diagrams represent
ing the periodic variation of the light from the arc show
that this light undergoes a regular fluctuation between a
maximum and a minimum, the maxima having equal values.
The light in the centre of the true arc never falls quite to
zero. This seems to be due to a little luminosity which
hangs in the interspace between the carbons, but at the
present moment it is difficult to say whether this persistence
is due to a very small admixture of stray light (although
every effort was made to keep this out), or to a persistence
of the illuminatingpower of the incandescent vapour in which
the arc has been formed. On examining the true arc during
its complete periodic variation, it is found that the blue
or purple strip of light forming the true arc undergoes a
periodic variation in intensity. As far as the eye can judge,
the blue or purple light completely vanishes at a certain
instant during the phase ; but there is, outside the true arc, a
dim halo of golden light which is persistent; and it is therefore
probably on account of this persistent aureole of faint light
round the true arc that the ordinates of the carve representing
the periodic variation of the luminous intensity of the arc
never become zero, but always indicate the outstanding
constant amount of light. On the other hand, in the diagrams
which represent the periodic variation of light coming from
the centre of the crater of the lower carbon, we find the
luminous intensity of the crater varies between a minimum
value and two maximum values of different magnitude.
During the time when the crater is positive it reaches a higher
maximum intensity of illuminating power than during the time
when it is negative, and, moreover, the curve representing
the periodic variation of light rises more steeply than it
comes down, which indicates a slow cooling of the carbons
after they have been heated ; in other words, they heat more
quickly than they cool. This is particularly noticeable in that
part of curve corresponding to the crater being negative; and it
is only what would be expected, because after the carbon has
reached its negative maximum and is beginning to cool, the
opposite carbon is cooling from a condition in which it has
been positive, and as it has been heated to a higher tempera
ture than the negative carbon, it must assist by its radiation
in keeping up the temperature and retard the cooling of the
negative carbon, These diagrams will also show many other
Analytical Study of the Alternating Current Arc. 327
interesting facts : they show, for instance, that in the case
of the long arc the selfinduction of the arc is more marked
than when the short arc is employed, and in that case there
is a very distinct lag of current behind the potential 
difference.
It may also he noted in comparing the diagrams V. B and
IV. B in figs. 12 and 10, which represent the carves for arcs of
the same frequency but of lengths 1*2 centim. and '32 centim.
respectively, that in the case of the long arc there is no lag of
light behind power as far as regards the points of maximum
when the carbon is positive, but in the case of ihe short arc
there is a sensible lag. This might be expected to be the
case, because for a long arc the influence of the opposite
carbon in keeping uptho temperature of its neighbour as ^his
last is cooling is less felt than in the case of a short arc.
Broadly speaking, the facts may be summed up as follows : —
The purple light of the true arc undergoes a periodic varia
tion, and, as far as the eye can judge, is completely extin
guished during a certain interval during the phase ; it has
equal maxima values during the period, at instants slightly
lagging behind the instants of maximum powerexpenditure
in the arc. On the other hand, the illuminatingpower of the
carbon crater varies between a minimum value and two maxima
of unequal values ; the greatest maximum occurring when
the carbon is positive and at an instant slightly lagging behind
the instant of maximum powerexpenditure in the arc. A
series of curves are also given (see figs. 18, 19, and 20)
in which the periodic variation of the current, potential
difTerence, power, and apparent resistance of the arc for
various powers and frequencies are represented ; and it will
be seen from these curves that the resistance of the arc,
including in this any counter electromotive force which may
exist, varies periodically, the resistance being a minimum
when the current is a maximum, and vice versa.
328 Prof. J. A. Fleming and Mr. J. E. Petavel :
Table I. A.
Observations on the Periodic Variation of the Intensity of the
Light of an Alternating Current Arc.
Light from centre of Arc. Short Arc.
Frequency = 83*3 ~
Length of arc = 0*55 cm.
Potentialdifference (P.D.) of carbons ") _ qq i,
(mean square value) J ~
Current (mean square value) .....= 14 amperes.
Power expended in arc =546 watts.
Pressure at alternator terminals . . . = 104*5 volts.
Angle of
Phase.
Current
through Arc
(instantaneous
value).
P.D.
of Carbons
(instantaneous
value).
Angle of
Phase.
Intensity of
Light from
Arc.
26
38
20
38
23
40
4 56
4202
58
16
60
4 07
4 02
76
6
80
 37
242
80
16
100
 97
345
102
24
120
162
380
132
47
140
203
430
146
60
160
210
448
156
56
180
184
435
168
68
200
130
325
176
57
220
 62
192
196
48
240
 04
4 19
202
31
260
4 42
4255
242
11
280
+ 102
4381
252
19
300
4163
4408
274
12
320
4200
4465
304
50
340
+204
448 3
320
58
360
4184
4505
326
340
346
356
55
65
63
68
Analytical Study of the Alternating Current Arc, 329
Table I. B.
Observations on the Periodic Variation of the Intensity of the
Light of an Alternating Current Arc.
Light from Crater. Short Arc.
Frequency .....= 83*3^.
Length of arc = 0*42 cm.
Potentialdifference (P.D.) of carbons ~) _ _ on il
(mean square value) J "
Current (mean square value) . . . . = 14 amperes.
Power expended in arc = 504 watts.
Pressure at alternator terminals . . . = 98 volts.
Angle of
Phase.
Current
through Arc
(instantaneous
value).
P.D.
of Carbons
(instantaneous
value).
Angle of
Phase.
Intensity of
Light from
Crater of
tower Carbon.
+165
+42
12
60
20
+102
+253
33
345
40
+ 31
+102
53
28
60
 11
142
74
31
80
 54
277
95
24
100
134
324
116
44
120
187
384
137
71
140
204
489
157
101
160
198
523
178
148
180
15
484
200
91
200
 88
31
205
93
220
 33
11
226
61
240
+ 15
+114
247
42
260
+ 63
+264
268
3L5
280
+130
+349
289
41
300
+187
+394
309
56
320
+215
+444
330
65
340
+212
+458
351
55
360
+165
119
163
160
198
49
140
203
48
330 Prof. J. A. Fleming and Mr. J. E. Petavel
Table II. A.
Obser cations on the Periodic Variation of the Intensity of the
Light of an Alternating Current Arc.
Light from centre of Arc. Medium Arc.
Frequency = 50 ^.
Length of arc ... . . . ... = 0*63 cm.
Potentialdifference (P.D.) of carbons V__ qq ij.
(mean square value) . . . . . . J~
Current (mean square value) . . . . = 14 amperes.
 Power expended in arc . ' =546 watts.
Pressure at alternator terminals . . . =99 volts.
Angle
of
Phase.
Current
through Arc
(instanta
neous value.)
P.D.
of Carbons
(instanta
neous value).
Angle
of
Phase.
Intensity of
Light from
centre of Arc.
+ 198
444
10
44
20
+ 141
4385
32
13
Arc out.
40
+ 63
+27
55
11
„ „
60
4 27
4105
77
18
80
 22
16
100
17
100
 63
36
122
45
120
135
375
144
74
140
183
47
165
68
160
198
54
187
525
180
195
54
208
32
200
148
50
230
85
220
 79
34
252
9
240
 19
112
273
11
Arc out.
260
+ 19
+13
294
26
»> >>
280
+ 7'2
+31
315
(71)?
300
+126
+36
336
68
320
+195
+40
356
425
340
4217
+473
348
68
360
+198
+463
327
45
N.B.— Owing to an accidental shift of the brushes, in this table the angle
corresponding to the instantaneous light is not the same as that corresponding
to the current. The position of the curve of instantaneous light with regard
to the E.M.F., current, and power curves cannot therefore in this case be
determined.
Analytical Study of the Alternating Current Arc. 331
Table II. B.
Observations on the Periodic Variation of the Intensity of the
Light of an Alternating Current Arc,
Light from Crater. Medium Are.
Frequency = 50 ~.
Length of arc = 063 cm.
Potentialdifference of carbons (mean
square value) =38 volts.
Current (mean square value) . . = 14 amperes.
Power expended in arc =532 watts.
Pressure at alternator terminals . . . =96 volts.
Angle of
Phase.
Current
through Arc
(instantaneous
value).
P.D.
of Carbons
(instantaneous
value).
Angle of
Phase.
Intensity of
Light from
Crater of
lower Carbon.
20
56
43
30
+ 63
+17
40
45
50
 05
 25
60
32
70
 39
22
80
100
120
140
160
180
39
52
137
277
277
227
190
113
200
165
210
 55
22
220
63
230
+ 01
 1
240
64
250
+ 42
+ 16
260
61
270
+11
+284
280
300
320
340
360
340
320
78
85
72
87
(37)?
' 69
75
332
Prof. J. A. Fleming and Mr. J. E. Petavel
Table III. A.
Observations on the Periodic Variation of the Intensity of the
Light of an Alternating Current Arc.
Light from centre of Arc. Medium Arc.
Frequency . . . « = 26 ~.
Length of arc = 0*63 cm.
Potentialdifference (P.D.) of carbons ") _ q 7 ii.
(mean square value) J ~
Current (mean square value) .... =14 amperes.
Power expended in arc =518 watts.
Pressure at alternator terminals . . . = 62 volts.
Angle
of
Phase.
Current
through Arc
(instanta
neous value).
P.D.
of Carbons
(instanta
neous value).
Angle
of
Phase.
Intensity of
Light from
Centre of
Arc.
+ 19
+ 48
7
45
20
+ 125
+395
28
26
40
4 58
+215
49
6
Arc out.
60
+ 05
+ 25
71
12
>> »>
80
 22
225
92
7
JJ JJ
100
 97
24
114
14
Arc just starting.
120
168
345
136
67
140
195
455
158
71
160
218
475
180
66
180
193
50
202
24
200
115
42
224
195
Arc out.
220
 48
22
246
4
»> >>
240
 05
_ 2
268
8
» u
260
+ 20
+ 17
290
22
280
+ 108
+26
312
40
300
+ 158
+36
333
57
320
4207
+475
354
75
340
4213
+50
15
53
360
419
448
340
421
+51
345
62
320
420
+48
323
70
Analytical Study of the Alternating Current Arc. 333
Table III. B.
Observations on the Periodic Variation of the Intensity of the
Light of an A Iternating Current Arc.
Light from Crater. Medium Arc.
Frequency = 26 ~.
Length of arc = 0*63 cm.
Potential difference (P.D.) of carbons) t «a i,
(mean square value) )
Current (mean square value) ... = 14 amperes.
Power expended in arc . . . . . = 546 watts.
Pressure at alternator terminals . . =62 volts.
Angle of
Phase.
Current
through Arc
(instantaneous
value).
P.D.
of Carbons
(instantaneous
value.)
Intensity of
Light from
Crater of
lower Carbon.
+ 18
+52
64
20
+143
+55
77
40
4 67
+245
46
60
4 1
+ 45
42
80
 12
21
43
100
 5
27
52
120
175
37
64
140
182
47
145
160
206
52
285
180
20
54
310
200
11
47
260
220
 5
255
200
240
 1
 55
119
260
+ 2
+15
87
280
+ 85
+30
50
300
+14
+40
104
320
+ 19
+45
110
340
+22
+53
130
360
+21
120
Phil. Mag. S. 5. Yol. 41. No. 251. April 1896, 2 A
334
Prof. J. A. Fleming and Mr. J. E. Petavel :
Table IV. A.
Observations on the Periodic Variation of the Intensity of the
Light of an Alternating Current Arc.
Light from centre of Arc. Short Arc.
Frequency . = 50 ~
Length of arc = 0'32 cm.
Potentialdifference (P.D.) of carbons \ . . a* u
(mean square value) .... J
Current (mean square value) . . . = 14 amperes.
Power expended in arc = 504 watts.
Pressure at alternator terminals . . =; 85 volts.
Angle of
Phase.
Current
through Arc
(instantaneous
value).
P.D.
of Carbons
(instantaneous
value).
Angle of
Phase.
Intensity
of Light
from
Centre
of Arc.
+184
44L4
12
28
20
4138
+384
34
15
40
+ 57
421
56
24
Arc going out.
60
4 06
 86
78
13
Arc out.
80
 6
22
100
14
Arc out.
100
131
25
122
18
120
19
43
144
28
140
19
465
166
20
160
183
49
188
26
180
14*4
46
210
17
200
102
44*6
232
15
xlrc going out.
220
 50
20
253
6
Arc out.
240
4 11
4 68
274
4
Arc out.
260
4 86
4226
296
16
280
4145
429
317
26
300
4189
4376
338
23
320
4221
4406
360
30
■
340
4203
444
350
29
1
360
4181
4426
328
23
340
+215
4454
320
4211
4404
Analytical Study of the Alternating Current Arc. 335
Table IV. B.
Observations on the Periodic Variation of the Intensity of the
Light of an Alternating Current Arc.
Light from Crater. Short Arc.
Frequency . . = 50 ~
Length of arc = 0*32 cm.
Potentialdifference of carbons (mean") oc > ,
t v v 5 = 3b volts,
square value) j
Current (mean square value) . . . = 14 amperes.
Power expended in arc = 532 watts.
Pressure at alternator terminals . . =93 volts.
Angle of
Phase.
Current
through Arc
(instantaneous
value).
Angle of
Phase.
Intensity of
Light from
Crater of
lower Carbon.
20
4131
20
61
40
+ 72
41
56
60
+ 08
63
45
80
 50
84 .
58
100
105
51
120
126
101
140
160
180
200
148
170
191
212
220
227
216
111
220
 63
232
64
240
 04
253
73
260
4 46
273
66
280
+ 126
294
63
300
+ 176
315
77
320
+ 190
335
60
340
360
359
338
317
36
63
78
88
i
79
2 A 2
336
Prof. J. A. Fleming and Mr. J. E. Petavel
Table Y. A.
Observations on the Periodic Variation of the Intensity of the
Light of an Alternating Current Arc.
Light from centre of Arc. Long Arc.
Frequency = 50 ~ .
Length of arc = 1*2 cm.
Potential difference (P.D.) of carbons ) _ r o i,
(mean square value) J
Current (mean square value) ... =14 amperes.
Power expended in arc ..... = 742 watts.
Pressure at alternator terminals ... = 88 volts.
Angle
of
Phase.
Current
through Arc
(instan
taneous
value.)
P.D.
of Carbons
(instan
taneous
value.)
Angle
of
Phase.
Intensity of
Light from
Centre
of Arc.
+ 204
+586
10
40
20
+159
4486
31
44
40
+ 84
4264
52
9
60
+ 07
 76
76
12
Arc out.
80
 36
46
98
19
)> »
100
 81
536
120
24
120
151
636
142
72
140
186
67'6
164
50
160
196
68
187
63
180
176
67
; 209
44
200
126
536
232
12
Arc out.
220
 56
21
253
7
n >>
240
 06
+24
275
7
»> »»
260
+ 44
+47
297
34
280
+ 84
+53
320
62
300
+ 174
+55
343
87
320
+2L4
+596
348
63
340
+ 234
+ 59 6
326
78
360
+204
+556
340
+ 224
+596
320
+214
+ 586
Analytical Study of the Alternating Current Arc. 337
Table V. B.
Observations on the Periodic Variation of the Intensity of the
Light of an Alternating Current Arc.
Light from Crater. Long Arc.
Frequency = 50 ^.
Length of arc = 1*2 cm.
Potentialdifference (P.E.) of carbons ") _ ^q i,
(mean square value) j ~
Current (mean square value) . . . = 14 amperes.
Power expended in arc ..... 742 watts.
Pressure at alternator terminals . . = 93 volts.
Angle of
Phase.
Current
through Arc
(instantaneous
value).
Angle of
Phase.
Intensity of
Light from
Crater of
lower Carbon.
35.9
92
20
+ 159
18
92
40
4117
36
98
60
+ 16
53
84
80
 44
70
70
100
87
76
120
105
74
140
123
207
160
142
175
180
160
280
200
179
265
220
 65
199
193
240
00
218
76
260
+ 44
238
84
280
+ 104
257
71
300
277
66
320
296
101
340
315
98
360
334
107
340
120
45
90
Fig. 3.
r ^"^W
Angle of Phase. \
/ 's
/ //
1 if
/ //
1 / /
135
180 225J
276 315 SOtf
50
40
XD
30 k
20
10
The ordinates of the firm line marked ll Light Curve " represent the peri
odic variation of illuminatingpower of the centre of the true arc of an
alternatingcurrent arc. The other curves marked E.M.F., Current, and
Power represent the variation of potential difference of the carbons,
current, and power in the arc. Carbons, 15 mm. cored ; — = 833 ;
amperes = 14; volts =39; length of arc, 0*55 cm.; volts at alternator
terminals=104. (See Table I. A.)
Fig. 4. — Intensity of Light from Crater.
25

/ v «
/ %
r^^T^
20
r\
/ * \<^
2^Tv
\ \
/ //'
7 >
\
rt 15
!
v\\\
. f Carbon 'Positive \
/ .'/''
*\
a.
// W N
Lj/i'
g 10
<
m 'J » o y
~T \^ /
if/
/ / Carbon
5
/ "\% /• //
,gative
/ Angle of Phase % \f ///
/
L\ SO ' 180 //
270
\ \ <*'/
i
\ \ T.me ^/7
\ " Mi ^ **/ /
\ \ ») > ,*>/*/■
\ *\ * / *7
\ • .*v */
\\ / U//
\*S. / /
Frequency =833 ~. Length of arc=042 cm. (See Table I. B.)
Analytical Study of the Alternating Current Arc. 339
Fig. 5. — Intensity of Light from Arc.
60
50
40
<30
Length = 63cf
Frequency 50'
See table HA
20
• 10
J L
Ang/e of Phase
j I i
_t 1 1 /• f
20 40 60 \\80 100 120 140 160 ISO 200 220 240/ 260 289 .500 S2Q 340 3
Time
\ „
^ssr#
7a
60
50
40
30
60
340 Prof. J. A. Fleming and Mr. J. E. Petavel
Fig. 6. — Intensity of Light from Crater.
Frequency =50 ~. Length of arc= 063 cm. (See Table II. B.)
Analytical Study of the Alternating Current Arc, 341
Fig. 7. — Intensity of Light from Centre of Arc.
70
20
Freqaencij =» 26^^
Length of arc = 6.63cm
,/*v
/ \
60
342 5 Pro£ J, A. Fleming and Mr. J. E. Petavel
Fig. 8. — Intensity of Light from Lower Carbon.
20
Frequency* 26>^>
Length of Arc = 063 cm
Sec ta&le MB
I o
Angle of Phase
20.40 60 \^0 100 120 HO »60 ISO 200 220 240/7 260 280 300 320 340 3
)» »•
Time
s /
70
$0
50
40
>
30 ^
o
20 i/>
10
60
Analytical. Study of the Alternating Current Arc. 343
Fig. 9. — Intensity of Light from Centre of Arc.
Frequency * 20 nw
Length of Arc = 32cm
20
SeetadlelTA / \
/ V
Nt
0)
I
o^
Angle of Phase
20 40
\h 80 100 120 140 160 180 200 220 //240 260 280 300 320 340 3
\
m *
Time
70
60
50
10
60
344 Prof. J. A. Fleming and Mr. J. E. Petavel t
Fig. 10. — Intensity of Light from Centre of Arc.
\ \ / Angle of Phase \ / ;
180 / 270
360
Time
>» *■
Frequency =50 ~. Length of arc=032 cm. (See Table IV. B.)
Fig. 11. — Intensity of Light from Centre of Arc.
Frequency=50 ~. Length of arc=l'2 cm. (See Table V. A.)
Analytical Study of the Alternating Current Arc, 345
Fig. 12. — Intensity of Light from Crater.
Carbon \\ , . f  CarW ne  alive
negatively Angle of Phase \ /
/ 270
36'
Time.
Frequency =50 ~. Length of arc=l*2 cm. (See Table V. B.)
346
Prof. J. A. Fleming. am* .Mr.. J. E. Petavel
..^>'..
^\.+
o
^^N
rt
<
J
^J
_ _ _ _ _ _ _ .y _ _ _ _
e
y
03
Sl
o
Vir; +
2
^*"^>^^
o
^\
^3
.  . l
,£
•
# OC
i
^
/
«H
/ a)
o
^^ >
£3
X^ IS
.2
+I
.5
V z

) / o
CO
i—i
fee
'■ "7^^" t r ~ " " " ?
&4
X >
v ■"■"
^V
^^ P
>< s v 0
^ s " s ««»_ c
^^**>^ °
^S. o
\
C
o
^o
*+3
■a
rt
H
S3
co
^
CD
«4
o
o
II
a;
o
«s
Sh
<
D
O
g
,
rn
o
2 bD
E a
^3
i
! H
a.
: CT<
1
t
> P^
CD
r—i
ir
j
i)
?>r;
E
<
K
348 Prof. J. A. Fleming and Mr. J. E. Petavel
o
to
c
fo
bo
o
be
a
\
/ »
■ JO 6 AvX
J? H\ \
1000 / \
" / o \ \ \
« \
■ »/ !f \ , \
1 \
y t U\
800 i »
' / \ \ 
/ ; _ • o
Mean Wafts / > 6°
/ t /
^ \\
600 / ^ /
'! 'k \
' \ 1
1 \ \
400 / \ J j
1 ° \
* * r" ^
l \ X
i » / (J
1 \ \
' » 1 /
\ Q \
' S 6
200 / \ /
\A
/
Angle of Phase V^
15 30 45 60 75 90 105 120 2
Time
\ 150 165 180 195 2)0 225 240 255 270 285 300 // 330 345 3
\ V if
/ \ 3
— : *
i% \ f V A?
\ 50.
80
70 •
60
50
40
— /
/^v /
/ ">.
/ ^X/
\ >^
400
/ \
/ / X.
%IZ
\ \
350
/**«•»
's^ /
\ Vg \
^ ."* / '
y\u
\
300
y \
\
A
y / /' ' N
N \/*
/\
•7 \
\ \
250
i * /•*"
/Sc ' /
\ /\
//
\ S^\
\ \ ^
i.
/ /•"' *"*
/ / /\
\v
\ \^*"^
CI
£ 200
My
\
<£
(Tv
\
*^\ \
£ ISO
/ / '/ '
T" ~ar / / ' /
^A^J^
\^\
\ \
i \ \
o wo
A4J/]r
v /■ ^^
v/^
\ \
\ \
\ X ^\—— —
I/// 9 / s\s y '
\^^\/
%3^
\
\ J 1
50
4W/A
^ \*e&fcf~'
\
\
\
yw/x'^^x
i
1
I
1
^ ,0( Jw/
S\ AU
i 3
00
1
*
i
00
500 1 6f)0
^^L \
i
Hor/z\>ntaJ
f/ani
past
! nq thbough\ATC
"'^T^?
1
\
1
so
^xVx^aT^
"~^^ V n/
/
1
,\\ \ \ \( ^s.
s/ ^ > 7\ '
/ /
200
y \. y
"NM
i
/
250
\ V \ \
/ \\/
/ /
/ y\
/ T^s
\\ ^
4v
300
\\>"
/
\ "<
"*" ^^
\.X
\
y\. /
/
350
^\ — "^ ^ \
7\
' 1
400
\ ^
\ '"""^v^
/ /
/ /
•50
20
+ 10
+ 20
+ 50
+ 60 +70
■60
+50
358
pq
Prof. J. A. Fleming and Mr. J. E. Petavel :
d
CO
o3
CO
CO
s
Jh
00
S3
3
O
iH
bO
05
fl
^
03
d
^H
£
. o ..a
OS
CM
^H
CO
tH
C5
CM
CO
to
I— 1
• 9
o
CD
lO t— I CD
UO 9 9
rH O O
rH O
^1 CD
rH Oi
6 ©•
o o
oo •.
CM
O0
CM
CD
rH
1^
o
o
CM
CO
"'*'
00
o
ri
O
GO
t>
O CM
CO
*
00
rH
o
9 2o
LQ CD O 9 •
rH , — .
P
©
'"P oo
S3 P
© ■§
© _
o3
o
: £
P
©
. rrj
rQ
Pm
,__!
t/J
P c3
CD ■■
rt
F P
o3
05
O CD
pl
3
03
02
^^
O 60
r* P
pi 2
.rH ^
CO P
CD O
p
pi 60
o3 P
©
P3
CD ©
rH
rP
P
i — i
P
o
[H
p
r^
S3
nH
£
©
rH
P
&H
CD
Ph
n
rH
rH
03
rP
03
©
rP
2
rP
P
©
rP
p
o
P
P
P
P
©
rO
P
• rH
m
• rH
Ph
o
CD
Pi
P
P
CO
p
03
P
©
O
©
CD
o3
r
H
HJ
rd
nd
H^
CD
n
00.
r— (
i 1
P
t/J
'*3
P
o
H
■s
CO
rO
13
o3
H>
P
o
P
©
^
=+H
p
o
P
60
o
r3
©
rH
P
rH
CD
np
CD
U
P
CO
rG
H3
a
CO
03
3
P
P
P
c3
p
cd
CD
B
'2
^
rrl
QJ
5h
3
bJD
rH
©
o
Ph
o
CD
CD
3
CD
O
OD
©
"HH
rl
oo
03
o
u
CO
03
P
"^
CD
O
H3
P
03
CD
a
©
p
CO
•+J
03
Hi
rP
Ph
^T
CD
00
a
O
1
rP
'o
g
hS
CD
H3
CD
O
P
CD
CO
p
O
CD
rP
Li
oo
O
o
p
>1
CI*
CD
CD
CO
GD
CO
rQ
^P
IS
a
'm
o
rH
bO
03
+3
a
~P
co
p
G
03
CD
H^>
'&
H
Ph
CT 1
HS
rJ
Ph
* r " '
QJ
cr;
P
ao
rD
03
■+3
rH
O
CD
CD
H
CD
O
P
cc
CD
O
©
r^l
rri
p
H
rO
O
P
CD
o3
P
©
rP
p
P
P
p
U
03
P
60
rl
H
P
o
P
P
CD
a
n3
CD
!<
rH
fH
©
CD
P
on
r^
CD
CD
P
1— 1
DO 00
O
Analytical Study of the Alternating Current Arc. 359
Fig. 25.
(From 'Electrician,' Dec. 20, 1895, p. 247.)
5U!I
True Power in Watts Expended in the Arc.
amount of mean spherical illumination can be obtained, if that
power is supplied in a continuous current form than if it is
supplied in an alternating form. It has been suggested that,
with proper carbons and under proper conditions, the alter
nating and continuouscurrent arc should give the same mean
spherical candlepower for the same expenditure of power, as
has been shown to hold true in the case of the incandescent
lamp by the investigations of Profs. Ayrton and Perry. We
think, however, that no a priori reasoning can apply in the
case of the arc. It is perfectly clear that, owing to the
interval of cooling that elapses, the mean temperature of the
two carbons in the case of the alternatingcurrent arc must
be less than the temperature of the positive crater of the
continuouscurrent arc, and that therefore the result we have
obtained is only w 7 hat might be expected. If the question is
asked, how do we account for the difference in efficiency ? the
answer must be that the energy absorbed in the case of the
alternatingcurrent arc is radiated at a lower temperature,
and the two arcs were therefore exactly in the same condition
as regards comparison that two incandescent lamps would
be, both taking the same total power but worked at different
temperatures, and therefore different watts per candle, and
therefore giving different mean spherical candlepowers with
the same total power expended in them.
It may be observed in all cases the alternatingcurrent arcs
we employed in our experiments took 16 to 17 amperes, and
360 Prof. W. Ramsay and N. Eumorfopoulos on the
the power was varied by varying the potential difference of
the carbons, and the alternatingcurrent arc lamp used was
one which effected this variation automatically, even although
the power expended in the arc was varied from 200 to 600
watts. In order to complete the comparison of the continuous
and alternatingcurrent arcs, it will be necessary to compare
the behaviour as regards illuminatingpower of alternating
current arcs, taking the same mean power but formed with
larger currents and less carbon potential differences ; that is
to say, comparing alternating arcs of equal powerabsorption,
but taking very different currents and therefore having dif
ferent lengths. We hope to extend this investigation to
cover these additional questions at some future time.
The above observations have necessitated an enormous
number of photometric and electrical measurements, and we
have in the above work been very efficiently aided by Messrs.
L. Birks, W. H. Grimsdale, A. M. Hanbury, E. K Griffiths,
and others, to whom our thanks are due.
XXXY. On the Determination of High Temperatures with
the Meldometer. By William Ramsay, Ph.D., F.R.S.,
Professor of Chemistry, University College, London, and
N. Eumorfopoulos, B.Sc, Demonstrator of Physics, Uni
versity College, London*.
THE meldometer, an instrument devised by Dr. Joly, has
been sufficiently described by him (Froc. Roy. Irish
Acad. 3rd series, ii. p. 38, or Chem, News, vol. lxv.), and we
need therefore only give a very brief account of it here. The
essential part of the instrument is a length (about 10 centim.)
of thin uniform platinum ribbon, about 1 millim. wide. This
is heated by a current of adjustable strength, and the increase
in length of the ribbon is measured by a delicate micrometer
screw, the ribbon being kept gently stretched by a small
spring. The temperature of the ribbon is, of course, lower
where the two forceps hold it ; but if it is suitably cut at
each end nearly to a point, a length of, say, 6 centim. in the
middle may be made of a very uniform temperature, as can be
proved experimentally by taking the reading of the melting
point of the same substance at different points along the
ribbon.
An infinitesimal quantity (scarcely visible with the naked
eye) of the substance to be melted is placed on the ribbon and
viewed with a lowpower microscope. The small quantity of
* CommuDicated by the Physical Society : read February 14, 1896.
Determination of Temperatures with the Meldometer. 361
the substance required enables it to be purified very com
pletely. The current is then put on and increased rapidly
until the substance melts, and thus an approximate reading is
obtained. This is repeated more cautiously, to obtain an exact
reading. Several readings can be taken by remelting the
same, and also by using fresh substance, the latter method
being usually adopted.
To translate the readings into temperatures, it is necessary
to standardize the instrument by taking the readings with
substances of known meltingpoints. Of these, unfortunately,
there are none known with certainty beyond about 350° C.
One reading is obtained by taking the temperature of the air,
another is the meltingpoint of potassium nitrate (339°) , and
for a third one the meltingpoint of potassium sulphate (1052°)
was adopted, for reasons that will appear. The meltingpoint
of silver is irregular, apparently because of absorption of
oxygen, and consequent spitting. Gold can be used, and
also palladium. As, however, the expansion of the ribbon is
almost a linear function of its temperature, and as the obser
vations hitherto taken did not extend beyond about 1050°, it
was considered unnecessary to take readings with palladium,
the general character of the expansion of the ribbon being
already known from Dr. Joly's observations. The question
next arises, what is the meltingpoint of gold ? There have
been two or three determinations of value, which unfortu
nately differ from one another.
M. Yiolle (C. R. 1879) determined it by a calorimetric
method, and obtains as a result 1045° (on the airthermo
meter).
Messrs. Holborn and Wien (Wied. Ann. xlvii. and lvi.,
1892 and 1895), who give 1072°, compared a thermoelement
with an airthermometer, and then used the former for deter
mining the meltingpoints of silver, gold, and copper. This
was done by inserting in a porcelain crucible the thermo
element, and also two platinum wires connected by a wire of
the substance whose meltingpoint was to be taken. The
platinum wires formed part of a circuit containing a battery
and a galvanometer. When the wire melted, the circuit was
broken, and the temperature read at the same moment with
the thermoelement. The melting of the substance must in
general lag a little behind the thermoelement ; and as no
mention is made of the rate at which the temperature was
raised, it is difficult to know how far the results can be trusted.
They obtain, however, very concordant readings.
Besides these, there are two determinations with the plati
num pyrometer : one by Professor Callendar (Phil. Mag. Feb.
362 Prof. W. Kamsay and N. Eumorfopoulos on the
1892), and the other by Messrs. Heycock and Neville (J. C. S.
Trans. .1895, p. 160). By a somewhat violent extrapolation*
from a formula which, as far as we know, has been compared
with the airthermometer only to about 625°, they obtain : —
Heycock and
Callendar. Neville.
Freezing or meltingpoint of silver ... 982° 961°
,, „ gold 1091° 1062° f
Callendar then, taking the meltingpoint of silver as 945°,
makes the meltingpoint of gold 1037°. Now Violle's deter
mination for silver is 954°, which, using Callendar's formula,
would give for gold 1049°. Holborn and Wien's value for
silver is 968°. In view of the variance between the numbers,
it was determined to take Violle's value, though the correct
value may be a few degrees higher (compare Le Chatelier,
C. JR. cxxi. p. 323, 1895). In any case, allowance can easily
be made, when further researches have determined the true
meltingpoint.
The gold used by us was a very pure specimen obtained
from Messrs. Johnson and Matthey. The salts used, with the
exception of some of the iodides, were pure specimens, pre
pared specially by ourselves. In a few cases the salts so
prepared were recrystallized, and meltingpoints were taken
both of the recrystallized salt and also that obtained from the
mother liquor. No difference could be detected, and hence
no further mention is made of these determinations.
Gold on melting alloys with the platinum, and hence must
destroy to a certain extent the uniformity of the ribbon. A
* In Holborn and Wien's last paper {he. cit.) the resistance of pure
platinum is determined at different temperatures with their thermo
element, and their results cannot be expressed quite satisfactorily ("nur
ungeniigend") by means of Callendar's formula. The resistance of two
of their pure platinum wires began to differ beyond about 900°, while
agreeing below this temperature. They therefore consider this property
unsuited to extrapolation.
t In a recent paper (J. C. S. Trans. 1895, p. 1025) Heycock and
Neville state that " Callendar did not rigidly follow the method of cali
bration, which was afterwards developed by Griffiths and himself, and
which we have always adhered to. That method requires that the
resistance of the pyrometer should be determined at three standard tem
peratures," viz., at 0°, 100°, and the boilingpoint of sulphur, 444 c, 5.
And lower down they say, u If Callendar had standardized his thermo
meter on the boilingpoint of sulphur . . . . ; " hence they infer that Callen
dar did not use the boilingpoint of sulphur for this determination. But
the determination of this boilingpoint is given in Phil. Trans. 1891, i. e.,
it was published before the paper referred to, and the latter is also later
than another paper (Phil. Mag. July 1891, p. 109), in which the boiling
point of sulphur is directly referred to. We do not quite see how to
reconcile the various statements.
Determination of Temperatures with the Meldometer. 363
remedy for this is to dust the ribbon lightly with finely
powdered talc. This, however, is not very satisfactory, and
interferes somewhat with the observations ; but with care,
fairly good results can be obtained, as the following numbers
show: —
With talc. Without talc.
NaCl 799° 792°
K 2 C0 3 .... 883 880
KC1 752 762
Ba(N0 3 ) 2 ... 583 575
It was then found that the meltingpoint of potassium
sulphate is very little different from that of gold, viz., 7
degrees higher or 1052°; and this salt was afterwards used
instead of the gold ; thus there is no need to use talc.
Some meltingpoints are sharply marked, others are not.
In these cases the lowest point was taken at which spreading
over the ribbon could be detected.
For purposes of comparison determinations by other ob
servers are given; some determined the meltingpoints {e.g.,
Carnelley, and Meyer, Riddle and Lamb), others the freezing
points (Carnelley, Le Chatelier, Heycock and Neville, and
McCrae). The data are taken from the following references: —
Carnelley (calorimetric method): J. C. S. Trans. 1876,
p. 489 ; 1877, p. 365 ; 1878, p. 273.
Le Chatelier (thermoelectric method, assuming melting
point of gold as 1045°): Bull. Soc. Chim. t. xlvii. p. 301 ;
C. R. t. cxviii. pp. 350, 711, and 802.
V. Meyer, Eiddle and Lamb : Ber. xxvii. (1894) p. 3129.
— In this method the salt has been previously fused in a
platinum tube with a wire down its centre, and to this
wire is attached a weight passing over a pulley. "When
the salt melts, it is pulled out by the weight, and the
temperature is determined at the same moment by an
airthermometer.
J. McCrae (Wied. Ann. lv. p. 95), relying on Holborn and
Wien's results, standardized his thermoelement with
boiling diphenylamine (304°) and boiling sulphur
(444°*5). For his thermoelement he used platinum
against an alloy of platinum rhodium, and also against
an alloy of platinum and iridium. The numbers in
brackets given below refer to the latter, and the others
to the former. It will be noticed that they do not agree
perfectly. This may, of course, be due to his metals not
being of the same purity as those of Holborn and Wien.
The latter also from their observations find that the iridium
alloy is not as well suited as the rhodium alloy.
364 Prof. W. Ramsay and N. Eumorfopoulos on the
Salts of Lithium.
The lithium carbonate bought as pure gave a conspicuous
sodium coloration to the Bunsen flame. It was purified
by successive precipitation and washing, until it gave a
brilliant carmine coloration to a Bunsen flame, free from
yellow fringe, showing an absence of sodium. The melting
points are not very well marked as a rule.
Earn say &
Eumorfopoulos. Carnelley. Le Chatelier.
Li 2 S0 4 853 818 830
Li 2 C0 3 618 695 710
LiCl 491 598
LiBr 442 547
Lil below 330 446
It will be noticed that there is here no agreement between
the results of different observers. Our results are as a rule
about 100 degrees lower than Oarnelley's, except that of the
sulphate. Our resistances did not allow us to take the
meltingpoint of lithium iodide, as we had not arranged for
temperatures below 330°, but its meltingpoint is below that
of potassium nitrate (339°). Lithium iodide is very hygro
scopic, so that on placing it on the ribbon it quickly liquefies;
then on putting on the current, it becomes solid and then
liquid again at the temperature given above, i, e., it has
melted, and remains so indefinitely.
Salts of Sodium.
The salts of sodium and potassium were prepared from the
bicarbonate, precipitated by carbonic acid from a solution of
the pure carbonate. The meltingpoints are well marked,
though all are not equally so, e. g., the iodides.
Meyer,
Bamsay & Le Clia Kiddle Heycock &
Eumorfopoulos. Carnelley. telier. & Lamb. Neville. McCrae.
Na Q S0 4 ... 884 861 860 863 883 883
Na 2 C0 3 ... 851 814 820 849 852 861 (854)
NaCl 792 772 778 815 ... 813
NaBr 733 708 ... 758 ... 761
Nal 603 628 ... 661  695(668)
The agreement between Messrs. Heycock and Neville's
results and our own is only apparent, as they find 1062° as
the meltingpoint of gold, while we assume 1045°. The fact
that they are determining freezing and not meltingpoints
may introduce some difference.
Determination of Temperatures with the Meldometer. 365
Salts of Potassium,
Meyer,
Ramsay & Le Cba Riddle Heycock &
Eumorfopoulo9. Carnelley. telier. & Lamb. Neville. McCrae.
K 2 SO*... 1052 ... 1045 1078 10665 1059(1166)
K 2 C0 3 ... 880 834 860 879 ... 893(885)
KC1 762 734 740 800 ... 800
KBr ... 733 699 ... 722 ... 746(709)
KI 614 634 640 685 ... 723(677)
M. Le Chatelier gives an earlier value 1015° for potassium
sulphate, and also 885° for potassium carbonate. Our value
for potassium sulphate is 7 degrees above that of gold.
Messrs. Heycock and Neville's is 4*5 above their value for
gold, but they mention that there may be an error of 2 degrees,
due to the alkalinity of their potassium sulphate.
Salts of Calcium, Strontium, and Barium.
The salts of calcium, strontium, and barium were prepared
from their carbonates precipitated from the purified nitrates.
Their meltingpoints are not well marked, especially those of
calcium ; and here again the iodides are less well marked
than the chlorides.
Meyer,
Ramsay & Le Oha Riddle McCrae,
Eumorfopoulos. Carnelley. telier. & Lamb.
Ca(N0 3 ) 2 499 561
CaCl 2 710 719 756 806 802
CaBr 2 485 676
Oal 2 575 (?) 631
Sr(N0 3 ) 2 570 645
SrCP 796 825 840 832 854
SrBr 2 498 630
Sri 2 .. 402 507
Ba(N0 3 ) 2 ... 575 593 592
BaCl 2 844 860 847 922 916(941)
BaBr 2 728 812
Bal 2 539
Calcium chloride is very difficult to observe, as it slowly
softens. We found it practically impossible to take the
meltingpoint of calcium iodide, as it is exceedingly hygro
scopic, and, on heating, it is almost immediately oxidized.
We do not think it can be above the value given, though it
may be below it.
Salts of Silver and Lead.
These were prepared by precipitation. Their melting
points are well marked.
Phil. Mag S. 5. Vol. 41. No. 251. April 1896. 2 C
366 Determination of Temperatures with the Meldometer.
Ramsay & n n
Eumorfopoulo.. Carnelley.
Ag 2 SO* 676 654
AgCl 460 451
AgBr 426 427
Agl 556 527
PbSO 4 937
PbCP 447 498
PbBr 2 363 499
PbP 373 383
A curve is appended with the meltingpoints of the salts of
potassium marked on it. In drawing this curve there must
l 2 3
Number of revolutions of the micrometerscrew.
The Magnetic Field of any Cylindrical Coil. 367
of necessity be a small uncertainty, as there is no datumpoint
between 339° and 1052° ; but this cannot amount to more
than a very few degrees.
In conclusion, we may point out that plotting our meldo^
meterreadings against other observers' meltingpoints does
not give a smooth curve.
University College, London.
XXXYI. The Magnetic Field of any Cylindrical Coil.
By W. H. Everett, B.A.*
APPLYING Ampere's formula for the magnetic force
at any point due to an element of current, the force
perpendicular to any plane circuit, carrying a current i, is
found to be, at any point P,
od^O
h 2 y<
h being the distance of P from the plane of the circuit, and
r, 6 the polar coordinates of any point of the circuit referred
to the projection of P as origin.
The longitudinal force at any point due to a current in a
cylindrical coil, or solenoid, is given by a second integration.
It is the sum or difference of two terms, each of the form
= inh I
dO
\/r* + h 2 '
where h denotes the distance of the point from an end plane
of the solenoid, and n the number of turns per unit length
of h. The depth of the coil, normal to the cylindrical surface,
is assumed to be inconsiderable. The limits of integration
are and 2ir for any point whose projection, taken parallel
to the axis, falls within the solenoid.
Similarly, the transverse force at any point, due to a sole
noid, is found to be
~R — in£( . T r ) ds,
VvV + V VV + A 2 V
the summation being vectorial.
The latter two formulae can be readily applied, for approxi
mate calculation, to a cylindrical coil of any crosssection,
* Communicated by the Physical Society, being abstract of paper read
November 8, 1895.
2 C 2
368 The Magnetic Field of any Cylindrical Coil.
including coils of circular and rectangular sections. But in
the case of rectangular coils the formulae become integrable.
Let p be the perpendicular distance of any point P from
one of the faces of the rectangular coil, and a one of the two
parts into which the corresponding side of a crosssection is
divided by the perpendicular from P. Then the longitudinal
force at P is given by the algebraic sum of sixteen terms,
each of the form
v .  C a dx
\Jo (afi
= msin
(# 2 +j9 2 ) V^+p^ + h*'
ah
And the transverse force at P is the resultant of eight terms,
each of the form
J W# 2 +i> 2 +V yV+j^ + V/ '
2 ^{pz + hf ' a + V^+p + V/
The first formula in the paper can be used to find an ex
pression for the force due to a circular current, at any point
P in the plane bounded by the circle. Draw any chord
through P, and call its segments r, /. Write c for the
distance of P from the centre, b for half the minimum chord
through P, and a for the radius. Then for the force at P
the formula reduces to
= gsl V« 2 c 2 sin 2 <9 .dd.
This can be written
$ being the perimeter of the ellipse with a and b as semi
axes, and having some value between 2wa and 4a, according
to the position of the point considered.
[ 369 ]
XXXVII. A Method of Determining the Angle of Lag, By
Arthur L. Clark, S.B., Prof of Mathematics and Physics,
Bridgton Academy ; North Bridgton, Maine, U.S.A.*
THE power or rate of expenditure of energy at any given
instant of time, on an electrical circuit, may always be
found from the equation
W = EI;
where W is the power in watts, E the E.M.F. in volts, and
I the current in amperes. But if the average power is
desired this formula is not general. It suffices only where E
and I are constant or nearly so.
It is a problem at the present time to measure the power
expended on circuits through which flow alternating currents
whose E and I vary harmonically with the time. In this
case the formula becomes
W =~y cos ;
where <£ is the difference in phase or is the angular magni
tude of the delay of the rise of I behind E. E and I are the
maximum values of the E.M.F. and current respectively.
It is obvious that cos is a very much desired value, and
different methods for determining it have been conceived.
There have been several phaseindicators brought before the
scientific world during the past year or two, but of these
there are very few, if any, which furnish a convenient means
of accurately measuring a difference of phase.
The instrument herein described is the outcome of ex
tended investigation carried on by the author during the past
year at the Worcester Polytechnic Institute. Many of the
different forms of apparatus which depend upon the inter
ference of sound waves, vibrating wires, &c, were constructed
and experimented upon, with unsatisfactory results.
It was found that indicators which are influenced to a
marked degree by small variations of the vibrationperiod are
of little value. As this variation interferes seriously with
results, and as no dynamo furnishes a current absolutely
periodic, such an indicator is worthless commercially.
The wellknown Lissajous's figures have been used at
different times as a means of determining the angle of lag,
and are the basis of the herein described apparatus. The
current from the dynamo passes through a single loop of wire
* Communicated by the Author.
370 Prof. A. L. Clark on a Method of
clamped at one end, and carrying a small mirror on the other
end, which is free to vibrate. This loop is suspended between
the poles of a magnet (electro or permanent), so that with
every change of direction of the current through this loop, it
will tend to rotate one way or the other according to Max
well's rule, i. e. y " Every portion of the circuit is acted upon
by a force urging it in such a direction as to make it enclose
within its embrace the greatest possible number of lines of
force."
Now if a beam of light falls on the mirror, the reflexion
will be drawn out into a line by the vibration of the mirror.
This beam of light coming from this mirror falls on a second
mirror, arranged as the first but actuated by another current
and with its plane of vibration perpendicular to that of the
first. In the resultant reflexion we find our means of mea
suring the amount by which one mirror leads the other, or, in
other words, by how much the phase of the current in the first
leads that in the second.
We will call the direction of vibration of the beam of
light as given by each mirror alone the axes of X and Y
respectively. That is, the axis of X is the figure from the
first mirror while the second is stationary, and the axis of Y
that from the second while the first is stationary.
The equation of a simple harmonic motion of amplitude a
along the axis of X may be expressed
x — a sin 0,
where 6 is a linear function of the time.
Also the equation of another harmonic motion of amplitude
b, along the axis of Y, whose time differs from 6 by an
amount , is
?/ = 6sin (0 —
.
Combining by eliminating 6 since
sin 0= — , cos 6=. ~ \/a 2 —w*
a a
the resulting equation is
y — ~ {x cos (j) — s/d^—x 2 sin ) ,
an equation in x and y, independent of the periodic time.
This equation is the equation of an ellipse. The resultant
reflexion, then, is an ellipse whose shape depends upon a, b,
and ({>.
Determining the Angle of Lag. 371
The equation of the long diameter of the ellipse is
y= x.
u a
Then since the short diameter is perpendicular to this, its
equation is
a
y= b *.
Treating either of these equations as simultaneous with the
equation of the ellipse, the coordinates of the intersection of
these diameters with the curve may be found, from which
may be deduced the lengths of the diameters in terms of — \/l—x 2 sin (ft,
the equation of a family of ellipses whoso parameter is (ft.
The equation of the diameters is
y=±x.
Combining this with the equation of the curve, there re
sults as the squares of the coordinates of intersection
2 _ sin 2 (ft 2 _ sin 2 (ft
y ~2(l + cos(ft) ; X ~2(l + cos)*
The upper sign in the denominator belonging with the
positively sloped and the lower with the negatively sloped
diameter. The squared length of the semidiameters is the
sum of these squares, or
sin 2 (ft
d
1 + cos (ft'
The whole diameter is double this semidiameter, so calling
D : the positively and D 2 the negatively sloped diameter,
2 _ 4 sin 2 = l, from which = 0, or the currents are in phase. If
the ellipse becomes a circle, D^=zD 2 2 and the numerator be
comes 0, consequently cos = 0, and
} we have a practical means
of determining the difference of phase.
Worcester Polytechnic Institute,
Worcester, Mass.
XXXVIII. A Note on Mr. Burch's Method of Drawing
Hyperbolas. By F. L. 0. Wadsworth, E.M., M.E.,
Assistant Professor of Physics, University of Chicago* .
IN the January number of the Philosophical Magazine
Mr. Burch describes a very simple and convenient
method of drawing an hyperbola by the use of two similar
triangles. This method is very similar to one which I have
been using for some time and which I have described in my
lectures for the past two years, although I have never pub
lished it. Mr. Burch's invention of the method antedates
* Communicated bv the Author.
Mr. Buich's Method of Drawing Hyperbolas. 373
mine, however, by several years, as lie states that he first
used it in 1885, while it first occurred to me in 1893. In
the present note I only wish to call attention to the fact that
the particular construction described is only one example of
a general class of solutions of this character, and to describe
two or three others which are, I think, equally simple and
convenient.
In general, if in any two similar triangles two dissimilar
sides are kept constant and the other sides varied, it is evident
that these two varying sides, which are proportional to the
two fixed sides, will be asymptotic coordinates of an hyperbola
of which the modulus is the product of the two constant sides.
The simplicity of the corresponding graphical or mechanical
tracing of the curve depends simply upon our choice of tri
angles and choice of sides. In the method of construction
which I most frequently employ, the two similar triangles
have a common vertex at the origin o, fig. 1, and the two
sides ob, bo and od, de parallel to the asymptotes of the re
quired hyperbola. Then if we put
ob = x,
de=y f
bc=l,
we have at once
xy = od.
Hence if we draw a series of triangles in each of which be
374 Prof. F. L. 0. Wadsworth on
is unity and od is constant and equal to —  — , the sides ob,
de, of any pair represent coincident values of x and y in the
corresponding hyperbola. These coincident values laid off
along their respective axes give a series of points on the
curve from which it is easy to trace the curve itself. In
practice, the whole operation may be rapidly and easily per
formed by means of a Tsquare and a single triangle of which
one angle is equal to the angle yox between the axes *. First
draw the line mn (fig. 2) parallel to and at unit distance
from the axis of x, and the line dg parallel to the axis of y
a? + b 2
and at a distance from it equal to —  — . Then to the points
c d c" on the first line draw the lines oc oc 1 oc" &c. by the aid
of the edge of the Tsquare or an auxiliary ruler, and the
lines cb, c'b' ', c n b" by means of the triangle or bevelgauge.
Project the points of intersection e, e' ' , e" of the first set of
lines with the line dg upon the second set of lines, giving us
the points s, s', s" on the required hyperbola. This method
is particularly rapid and convenient in plotting rectangular
hyperbolas on crosssection paper, the only instrument then
necessary being a rule to draw the radial lines oc, oc', &c.
If desired, an instrument can easily be constructed on these
lines to trace the curve mechanically, but generally the
graphical process is more rapid. A whole series of contours
to the thermodynamic surface (pv = const.) can be drawn by
this process in a very few minutes, the same set of lines oc,
oc', oc", and cb, c'b', c"b", answering for all the curves.
2nd method. — Make the vertices of the two similar triangles
coincide at c instead of as before, and make ab = oc = x, and
cd=y (fig. 3). Take a point a on the axis of y at unit dis
tance from the origin and draw from it the lines ac, ac' , ac"
to points on the axis of x, and the lines be, b'c', b"c" parallel
to the y axis as before. Mark off a distance equal to
fd = — j — on the edge of a triangle (or bevelgauge), Q, of
which the angle at d is equal to the angle yox, and slide this
triangle along each of the lines be, b'c', b"c", &c, until the
point / intersects the corresponding line ac, ac', or ac". The
points d, d', d" will then evidently be points on the hyperbola.
In practice it is not necessary to draw the lines be, b'c', &c.
at all, it suffices to place a Tsquare whose blade is parallel
to the axis of y, so that its edge passes through any of the
* It is convenient to use for this purpose an ordinary steel bevel
gauge the two blades of which may be adjusted to the required angle.
Mr. Burch's Method of Drawing Hyperbolas. 375
points on the axis of x ; slide the triangle along this edge
until the point / falls on the corresponding line from a, and
f/c.3.
mark the position of the vertex d of the triangle (see fig. 4) .
This method is^ perhaps even more rapid and convenient than
the first, as it involves the drawing of only one set of lines
ac, ac f , &c. Like the first, it involves the use of only a triangle
and a Tsquare.
3rd, method. — Let the vertices of the two similar triangles
coincide at b (fig. 5). Then if be = unity we have as before
To determine the points on the curve graphically in this
case, we need a triangle or bevelgauge of an angle equal to
yox and a parallel ruler. The side be of the triangle or bevel
gauge should be of unit length and a distance fb= a
laid off on the side bq. The triangle is placed with its side
bq coinciding with the axis of a, and one edge of the parallel
ruler is brought against the point c of the triangle and a pin
placed at the origin. The other blade of the parallel ruler is
then moved out until it passes through the point / on the
376
Prof. F. L. 0. Wadsworth on
horizontal edge of the triangle, as in fig. 6. The point e at
which the blade of the ruler intersects the side be of the
FfG. 6.
triangle will be one of the points on the required curve, the
others of which may be found by sliding the triangle along
the axis of x, always keeping the two edges of the parallel
ruler in contact with the three points o, c, and f. This method
is somewhat simpler mechanically, but not quite so rapid and
convenient as either of the preceding.
It is evident that there are six other possible solutions to
be obtained by combining the sides ac, ae (fig. 1) with each
pair of adjacent sides. But these solutions are unsatisfactory
graphically, because simultaneous values of x and y will be
represented by lines inclined to each other at an angle dif
ferent from the angle between the axes. If the angle at a is
made equal to the angle between the axes, we have one of the
three solutions already considered.
The use of two similar triangles in the graphical, and more
particularly the mechanical tracing of curves, is of wide ap
plication. By their aid we may always express the product
or quotient of two variable quantities geometrically as the
length (tensor) of one of the sides. Other applications of
this principle will be found in a recent paper of the author's
on the mountings of concave grating spectroscopes*.
* "Fixed Arm Concave Grating Spectroscopes," F. L. 0. Wadsworth.
AstroPhysical Journal, vol. ii. p. 370 (Dec. 1895).
Mr. Burch's Method of Drawing Hyperbolas. 377
The Use of the Quadruplane as a Hyperbolagraph. — The
two asymptotic coordinates of any hyperbola evidently form
two sides of a parallelogram of constant area. Hence any
hyperbola can be readily traced by the use of the Sylvester
Kempe quadruplane linkage, the four vertices of which lie at
the four angular points of a parallelogram of constant area
and constant obliquity*.
The product of the adjacent sides of this parallelogram, or
as Sylvester calls it the " modulus " of the cell, is equal to
B2 _ A2 sin aiS in«. 2 =
sin^tf
where B and A are the distances between the pivotal points
of the long and short links of the cell, a x and a 2 the angles
adjacent to the line joining the pivotal points, and the angle
subtended by this line at the intervening vertex. If we make
each link of the cell symmetrical we have
ai = « 2 = 9019/2,
and M _ B 2 A 2
" 4sin 2 0/2
The obliquity of the parallelogram is equal to 0.
Hence to describe an hyperbola whose axes are a and b we
must make the modulus of the cell equal to the modulus of
the hyperbola, or
4
and the angle equal to the angle between the asymptotes, or
2ab
= sin'
a 2 + 6 2
Then if one vertex of the cell is fixed and an adjoining
vertex is moved along a straight line (the edge of a Tsquare
or straight edge for example), passing through the fixed
vertex, the vertex diagonally opposite the latter will describe
the required hyperbola, having the fixed point as origin and
the straight line as one of the asymptotes. To describe
different hyperbolas it is necessary to be able to vary both
and M. The first may be easily done by making each link
in two halves, pivoted together at the vertex of the link with
a divided sector and clamp, by means of which the desired
* See " The History of the Plagiograph," Sylvester, ' Nature/ vol. xii.
p. 214 ; also Kempe, ' Lecture on Linkages/ p. 25 et seq.
378 Mr. R. W. Wood on a
angle 6 may be laid off. The modulus of the cell is best
changed by varying either A or B. This may be conveniently
done by making each of either the long pair or the short pair
of links like the legs of a pair of proportional dividers.
A small model of an instrument on the above lines has been
constructed and found to work very easily and accurately.
Like all hyperbolagraphs, however, the range of motion is
limited (although larger in this than in most forms), and a
considerable amount of time is necessary for the preliminary
adjustment. For these reasons I have generally found one
of the preceding graphical solutions more rapid and con
venient, especially when a number of curves are to be drawn
on the same sheet.
This application of the quadruplane, which occurred to me
recently while making an application of the Peaucellier
linkage to a concave grating mounting *, seems so simple and
obvious that I feel sure it must have occurred to others as
well as myself; but as I have not been able to find any sug
gestion to this effect in any of the papers on the subject that
I have examined, I have ventured to present the foregoing
description as another illustration of the practical application
of the beautiful geometrical discovery of Prof. Sylvester and
Mr. Kempe.
Ryerson Physical Laboratory,
University of Chicago, U.S.A.,
January 1896.
XXXIX. A Duplex Mercurial AirPump.
By R. W. Wood f.
IN working with highly exhausted tubes, such as are used
for the production of the Rontgen rays, one of the
difficulties met with is the speedy deterioration of the vacuum
due to the liberation of gas from the electrodes and the glass.
If the tube be thoroughly heated, while on the airpump, this
trouble is partially remedied, but even with this precaution
the tubes are not very durable and have to be pumped out
frequently. In order to overcome this difficulty I have con
structed a new form of mercurial airpump, which can be
made on a very small scale and attached permanently to the
Rontgen tube. By this arrangement, any traces of gas that
* " On the Use and Mounting of the Concave Grating as an Analyzing
or Direct Comparison Spectroscope," The AstroPhysical Journal/ vol. iii.
p. 47 (Jan. 1896).
t Communicated bv the Author.
Duplex Mercurial AirPump. 379
make their appearance can be easily pumped out. The
apparatus is so compact that it can be held in the hands while
in operation, not requiring mounting on a board. The pump
is very simple, and a glance at the accompanying diagrams
will make its construction clear. It will be seen to consist of
two bulbpumps joined at the base by a Utube of glass (fig. 1) .
The pumping is done by rocking the apparatus, the mercury
filling the exhaustbulbs A A alternately. This duplex action
makes the pumping very rapid, for one bulb exhausts while
the other fills, there being no lost time. The traces of gas
liberated in the dischargetube are pumped over into the ex
hausted bulbs B B, where they are stored, being prevented
from returning by the mercury which remains in the W traps
between A and B. The upper bulbs are joined by a tube E,
which has a small lateral tube P blown into it ; this arrange
ment being necessary for the preliminary exhaustion.
Mercury is first introduced through P until the bulbs A A
are half full. A gentle rocking of the apparatus is necessary,
as the fluid is held up in the bulbs B by the compression of
the air in A.
When enough mercury has been introduced, the apparatus
is placed in the position shown in fig. 3, when the fluid
should stand at the level indicated in the diagram. The side
tube P is now drawn out into a thick walled capillary in a
blastlamp, in order to facilitate the subsequent closing of the
apparatus. This tube being connected with a good mercury
pump by means of a wellgreased, thickwalled rubber hose,
380 Mr. R. W. Wood on a Duplex Mercurial AirPump.
the apparatus is exhausted as completely as possible. During
the exhaustion it must be supported in the position shown in
fig. 3 ; otherwise the air escaping from the dischargetube
will throw the mercury violently against the top of the bulbs.
It is a good plan to carefully heat the bulbs and the discharge
tube by means of a Bunsen burner while the pump is in
action, in order to drive off moisture. The current of a fair
sized inductioncoil should also be passed through the dis
chargetube for a few minutes to rid the electrodes of air as
completely as possible. It will be found that the vacuum
cannot be made perfect enough to give a Crookes dark space
of more than an inch, owing to the leakage through the
rubber hose. The capillary part of P is now heated to fusion
in a small flame which hermetically seals the entire apparatus.
The comparatively poor vacuum in the dischargetube can
now be made as perfect as is possible with any mercurypump
by slowly rocking the apparatus, holding it by the bulbs A A.
If the pump is properly made, the traps hold and require no
attention : if not, a little dexterity is required, to prevent the
mercury from running out into the bulb, and they have to be
constantly watched. Care must of course be taken that the
airbubble, compressed into the trap by each stroke, is driven
entirely around the bend and into the reservoir B. If for
any reason the pump requires to be subsequently opened, it
must be placed in the inclined position and a file scratch
made on the tube P. A bit of redhot glass pressed against
the scratch will cause a crack through which the air will
slowly enter. If the tube be broken open suddenly, the
mercury will be forced over into the dischargetube. I con
Intelligence and Miscellaneous Articles. 381
structed my apparatus with the Rontgen tube projecting to
one side, as shown in the side view, fig. 2. This makes a
support for the pump so that it will stand alone. A pump of
this description in connexion with an ordinary " Darkspace  "
tube makes a very convenient piece of lectureroom apparatus
for showing the character of the discharge at different pres
sures. By tipping the pump far enough the upper trap can
be emptied, and the air stored in B returned to the discharge
tube again, showing the phenomena at higher pressure.
Owing to the absence of rubber connexions and stopcocks,
the mercury remains always clean and there is no leakage.
I am now constructing a pump on this principle on a large
scale for general laboratory use in which the rocking motion
is to be effected by waterpressure, which, if found serviceable,
will be described in a subsequent paper. The chief objection,
of course, is that the entire pump is in motion, which makes
its connexion with a stationary receiver somewhat difficult.
This can perhaps be done by bringing the exhausttube into
coincidence with the axis of rotation, and using a rubber tube
surrounded with mercury as a joint.
The small pump can be ordered with or without the
Rontgen tube from Herr Glasblaser R. Burger, Chaussee
str. 2 E, Berlin, Germany.
Berlin : Physikalische Institut.
XL. Intelligence and Miscellaneous Articles.
NOTES OF OBSERVATIONS ON THE RONTGEN RAYS.
BY HENRY A. ROWLAND, N. R. CARMICHAEL, AND L. J. BRIGGS.
rPHE discovery of Hertz some years since that the cathode rays
* penetrated some opaque bodies like alaminium, has opened up a
wonderful field of research, which has now culminated in the
discovery by Rontgen of still other rays having even more
remarkable properties. We have confirmed, in many respects,
the researches of the latter on these rays and have repeated his
experiments in photographing through wood, aluminium, card
board, hard rubber, and even the larger part of a millimetre of
sheet copper.
Some of these photographs have been indistinct, indicating a
source ol ! these rays of considerable extent, while others have been
so sharp and clear cut that the shadow of a coin at the distance
of 2 cm from the photographic plate has no penumbra whatever, but
appears perfectly sharp even with a lowpower microscope.
So far as yet observed the rays proceed in straight lines, and all
efforts to deflect them by a strong magnet either within or without
the tube have failed. Likewise prisms of wood and vulcanite have
Phil. Mag. S. 5. Vol. 41. No. 251. April 1896. 2 D
382 Intelligence and Miscellaneous Articles.
no action whatever so far as seen and, contrary to Eontgen, no
trace of reflection from a steel mirror at a large angle of incidence
could be observed. In this latter experiment the mirror was on
the side of the photographic plate next to the source of the rays,
and not behind it as in Eontgen's method.
We have, in the short time we have been at work, principally
devoted ourselves to finding the source of the rays. For this
purpose one of our tubes, made for showing that electricity will
not pass through a vacuum, was found to give remarkable results.
This tube had the aluminium poles within l mm of each other, and
had such a perfect vacuum that sparks generally preferred 10 CU1 in
air to passage through the tube. By using potential enough,
however, the discharge from an ordinary Euhmkorff! coil could be
forced through. The resistance being so high, the discharge was
not oscillatory as in ordinary tubes but only went in one direction.
In this tube we demonstrated conclusively that the main source
of the rays was a minute point on the anode nearest to the
cathode. At times a minute point of light appeared at this point
but not always.
Added to this source the whole of the anode gave out a few
rays. From the cathode no rays whatever came, neither were
there any from the glass of the tube where the cathode rays
struck it as Eontgen thought. This tube as a source of rays far
exceeded all our other collection of Orookes' tubes, and gave the
plate a full exposure at 5 or 10 cm in about 5 or 10 minutes with a
slowacting coil giving only about 4 sparks per second.
The next most satisfactory tube had aluminium poles with ends
about 3 cm apart. It was not straight but had three bulbs, the
poles being in the end bulbs and the passage between them being
rather wide. In this case, the discharge was slightly oscillatory
but more electricity went one way than the other. Here the
source of rays was two points in the tube, a little on the cathode
side of the narrow parts.
In the other tubes there seemed to be diffuse sources, probably
due in part to the oscillatory discharge, but in no case did the
cathode rays seem to have anything to do w T ith the Eontgen rays.
Judging from the first two most definite tubes, the source of the
rays seems to be more connected with the anode than the cathode,
and in both of the tubes the rays came from where the discharge
from the anode expanded itself towards the cathode, if we may
roughly use such language.
As to what these rays are it is too early to even guess. That
they and the cathode rays are destined to give us a far deeper
insight into nature nobody can doubt. — American Journal of
Science, March, 1896.
NOTE ON "FOCUS TUBES " FOR PRODUCING A'RAYS.
BY R. W. WOOD.
The tubes for producing the a?rays which are furnished with a
concave kathode for focussing the kathode rays on the glass, in order
Intelligence and Miscellaneous Articles.
383
to diminish the size of the source and increase its intensity, have
the fault that, owing to the great heat developed, the glass is very
apt to crack. I have had some success with a tube which 1 made
in which the kathode hangs as a pendulum from the centre of a
spherical bulb, by the slow rotation of which one brings a fresh
and cold surface into the focus continually, thereby avoiding over
heating. The concave kathode hangs by an aluminium wire from a
short cylinder of aluminium fastened into a glass tube, through
w T hich a platinum wire passes which lies in the axis of rotation.
The anode is also in the axis of rotation, so that the connexions
with the coil can be easily made. My tube has the fault that
many kathode rays emanate from the upper surface of the concave
plate and are lost. This might be overcome by covering the kathode
with a cap of glass. As a suggestion for further experimenting,
this note may be of interest to persons working with the new
rays.
Berlin, March 8, 1896.
NOTE ON ELEMENTARY TEACHING CONCERNING
FOCAL LENGTHS.
To the Editors of the Philosophical Magazine.
GrENTLEMEN,
"With respect to Prof. Lodge's question on page 152 as to the
simplest convention of signs in dealing with focal lengths for junior
students, it appears to me to be a matter which can be best settled
by one who has had experience in teaching two or more methods.
Hence, as this is a qualification to which I can lay no claim, I
forbear to dogmatize.
It would seem to me, however, undesirable, apart from weighty
reason, to teach an elementary student to let the algebraical sign
of a given line depend, not upon its direction, but upon the physical
384 Intelligence and Miscellaneous Articles.
nature of the optical image at which the line terminates. Still
there is a manifest advantage in the convention proposed, since it
allows the relation of focal length and conjugate focal distances to
be represented by a single equation for both mirrors and lenses.
I give below the adaptation of the graphical method to the
convention of signs proposed by Prof. Lodge (and used in Granot's
'Physics'), viz. : — Distances to real images to be considered positive
and distances to virtual images negative.
The fixed points through which the rotating lines pass are now
in the righthand upper quadrant both for convex lenses and
concave mirrors, and in the lefthand lower quadrant for concave
lenses and convex mirrors.
That is, referring to Pig. 1, page 61, Jj x is now shifted so as to
coincide with M a which remains unmoved, and L ffl similarly is
made to coincide with M, r .
Tours faithfully,
Univ. Coll., Nottingham, Edwin H. Baeton.
SOLUTION AND DIFFUSION OF CERTAIN METALS IN MERCURY.
BY W. J. HUMPHREYS.
The investigation, of which this is a summary, was begun with
the object of determining the extent to which these phenomena
differ, if at all, in this case from the solution and diffusion of non
metallic solids and liquids.
The method of investigation was to fill a vessel of constant cross
section with pure mercury, put on its surface a freshly amalgamated
piece of the metal to be examined, and after allowing it to stand
a definite length of time in a place free from external disturbances
and of fairly constant temperature, to remove from known depths
below the surface samples of the amalgam and analyse them.
The metals examined were lead, tin, zinc, bismuth, copper, and
silver, and the results indicated that there is no essential difference
between the solution and diffusion of these metals in mercury and
the same phenomena in any other case.
Probably the most interesting results were those given by copper
and silver, both of which dissolved to a much less extent than any
of the other metals examined, but diffused more rapidly. At 28°
0. the silver dissolved to the extent of only about one part in two
thousand, and the copper to a still less extent — about three parts
in a hundred thousand ; while the rate of diffusion of the silver
was about twenty millimetres per minute, approximately sixty
times that of copper and fully six hundred that of zinc.
This investigation, of which the details will soon be published,
was suggested to me by Dr. J. W. Mallet, P.E..S., of the University
of Virginia, and carried out there under his supervision during the
months of August and September, 1895. — John Hopkins University
Circulars. Pebruarv 1896.
THE
LONDON, EDINBURGH, and DUBLIN
PHILOSOPHICAL MAGAZINE
AND
JOURNAL OF SCIENCE.
[FIFTH SERIES.]
MAY 1896.
XLI. On the Laws of Irreversible Phenomena. By Dr.
Ladislas Nat ANSON, Professor of Natural Philosophy
in the University of Cracow *.
IT can scarcely be doubted that the theory of dissipation of
energy is still in its infancy. Reversible phenomena
are well understood, but they do not involve dissipation at
all ; and what is known about irreversible phenomena is
merely the qualitative aspect of their general laws. In fact,
of the general quantitative laws of irreversible phenomena
we are as yet utterly ignorant. Now I venture to think
there is a general principle underlying irreversible phenomena
which is easily seen to be consistent with fact in various cases
well investigated : it is an extension of Hamilton's Principle,
and (with much diversity, of course, as to form and gene
rality) has been stated by Lord Rayleighf, by Kirchhoff J,
by v. Helmholtz §, and by M. Duhem . It seems that propo
* From the Bulletin International de V Academie des Sciences de Cra
covie, Mars 1896. Communicated by the Author.
t Proceedings of the London Mathematical Society, June 1873.
The Theory of Sound/ i. p. 78 (1877).
X Vorlesungen iiber Math.Physik. Mechanik, 1876, Vorlesung xi.
§ BorchardtCrelle's Journal f. Mathematik, Bd. c. (1886); Wissen
schaftliche Abhandlungen, Bd. iii. p. 203 ; ibid. Bd. ii. p. 958 ; Bd. iii.
p. 119.
 Journal de Mathematiques de lioumlle Jordan (4) vol. viii. p. 269
(1892): vol. ix. p. 293 (1893); vol. x. p. 207 (1891). Seo further,
Prof. J. J. Thomsons ' Applications of Dynamics to Physics and Che
mistry,' London, 1888, where the fundamental standpoint is a very
similar one, the chief object of investigation being, however, the theory of
reversible phenomena.
Phil. Mag. S. 5. Yol. 41. No. 252. May 1896. 2 E
386 Prof. L. Natanson on the
si ti oris equivalent to those indicated by these investigators
could be enunciated in the form of a simple and very general
formula ; we venture to think that the fundamental principle
which it embodies is worth attention. Besides, it seems to
afford the proper foundation for an attempt to arrive at some
deeper insight into the laws of dissipation of energy.
Part I.
§ 1. Introductory . — Conceive a system : it may be either
finite or infinitely small ; it may be an independent system,
or it may be only a part of some other system. Let the state
of the system, at time t, be determined by the values of
certain variable quantities, q () and of their first differential
coefficients with respect to the time, s i or dgjdt. We shall
suppose that the energy of the system consists of two parts,
the first of which, T, is a function of the q { and the s fj homo
geneous of the second degree with respect to the s v and the
second, say U, is a function of the g. only. Let 3 denote the
absolute temperature of the system : $ may be an independent
variable, or otherwise it must be a definite function of the
variables. Suppose that the quantities q v s i received certain
arbitrarily chosen infinitesimal increments 8q v Ss i ; the energy
T will then become T + oT, and U will becoine U + SU. Let
then %P.Bq i be the work done on the system reversibly,
during the transformation, by extraneous forces, and let SQ
or ItRjbq. be the quantity of heat simultaneously absorbed by
the system from the exterior ; P { will then be the generalized
or Lagrangian extraneous " force " in the " direction " of
the variable q { , and R^ will be the " caloric coefficient/' as
it is called by M. Duhem, or the generalized "thermal
capacity " of the system with respect to the variable q..
With respect to the quantity SQ we now make the following
assumption, which we shall find is in accordance with fact.
Let us suppose that every variation Bq. t.ikes the special value
dq { or sflt ; then the values of the variables q { will become
9i + dq { ; the energies T and U will become T + dT, U + dU ;
the work done by external forces will be XF i dq i ; and the
quantity of heat absorbed will be dQ or 2R^fy f . If now the
variables be allowed to return to their primitive values q v T
and U will resume their former values T, U, the external
work — 2P.tf^ will be done, but the quantity of heat absorbed
will generally not be dQ but a different quantity, say djQ.
Write
and
therefore
We have
and
Laivs of Irreversible Phenomena
2d'Q=+dQ rfiQ, .
+ d«Q=+dQ d'Q;
4Q=d«>Q+d'Q;.
387
(1)
(2)
(3)
(4)
(5)
thus d°Q is the reversible, and d'Q the irreversible part of
the heat absorbed. Now, if we assume that these quantities
are of the form
we may consider the new quantities
SoQ = 2RJ^; 8'QrrSR^,
(6)
(7)
5°Q + 8 / Q gives again £Q. Let us generally define SQ, 5°Q,
S'Q to represent the expressions which result if in the ex
pressions of the quantities c/Q, d Q, and d ! Q (which we sup
pose to be empirically known *), variations Sq { are substituted
in place of the corresponding differentials dq v
§ 2. Statement of the Principle. — Let us consider a given
period of time, from t = t to t = t 1 . Let Sq., £*., BT, 8U,
SP^S^, as usual, represent variations which, between the
limits t = t and t~t 1} are functions of the time susceptible of
being differentiated, and which vanish at these limits them
selves ; finally, let 8Q, 8°Q, B'Q be the corresponding in
finitesimal expressions calculated as above stated. The
following principle seems then to hold in physical phenomena:
between t = t and t = t ± events which occur in the system
must be such that the equation
JU
dt{BTbJJ + tFfiq i + BQ t }=0
(10
is satisfied. For brevity, this, when necessary, will be re
ferred to as the Thermokinetic Principle.
* To write down the expressions of dQ and d'Q, a much greater
number of variables would evidently be required in most cases than to
write d°Q j thus in most cases many of the coefficients R° will be equal
to zero. A similar remark applies to the coefficients P { , dT/dfy dT/d.^,
and $Ufdq t .
2 E 2
388 Prof. L. Natanson on the
§ 3. Lagrangian Equations. — From (I.), remembering the
definitions laid down, we obtain by a wellknown calculation
These equations, a thermokinetic extension of Lagrange's
wellknown dynamical equations, have been given implicitly
by Helmholtz and explicitly by M. Duhem ; the form they
take in an important particular case had been previously ex
plained by Lord Hayleigh.
§ 4. Conservation of Energy. — Considering a real trans
formation dq v ds { , multiply each of these equations by sflt
respectively, and add ; we find
dT + dU2P^dQ = (1)
The principle of conservation of energy in its general form is
thus seen to follow from the thermokinetic principle. That
inversely the thermokinetic principle cannot be deduced from
conservation of energy is an obvious proposition which scarcely
requires special mention.
§ 5. Free Energy. — We shall suppose in the following
(except when the contrary is expressly stated) that one inde
pendent variable is the temperature ; and accordingly we
shall use g t to indicate all the other variables. That work is
not required for merely changing the temperature of a system
is an experimental fact ; hence, when the variables •&, q., and
s. receive increments &$, 8q { , Bs v the work done on the system
will be still %T i 8q i (in our present modified notation) and no
term including && will appear. Variables with such properties
attributed to them have been employed by Lord Kelvin as
long ago as 1855 ; they have been often adopted in general
thermodynamical investigations. M. Duhem calls them
" normal " variables.
Let us suppose that 3, g. represent a system of u normal "
variables. Write
2 H % " 2B  %i=2 ii &?i  • • • (1)
The function V, if it exists, will be called the free energy of
the system, because, as we shall find hereafter, V defined by
equation (1) will agree in the case of Reversible Thermo
Laics of Irreversible Phenomena, 389
dynamics with what, from Helmholtz, received that designa
tion. Equation (I.) accordingly becomes
£*{«ts'8 ?4 +spa«(?4)»+8'Q J =° ( 2 )
Now let us further assume that the following equations are
true : —
H=°< g" ds (3)
they are found to hold good in all cases of which we have
precise knowledge ; lastly, let us suppose that there is no
term containing 8§ in the expression for 8'Q. (With respect
to this point compare § 12.) Equation (2) may now be
divided into
^r;=o, w
and
d*
j"*«ft { STS gs^ + SP^+S'Q J =0. . (III.)
This equation expresses the principle in a form similar to
that of equation (I.). It is a useful equation, owing to the
readiness with which it admits of application in various cases,
but its abstract generality is of course much more restricted
than that of the fundamental equation.
§ 6. Reversible Dynamics. — In Dynamics properly socalled,
i. e. in Reversible Dynamics, ideal phenomena of motion are
dealt with, and the notion of temperature is not taken into
account. Therefore, in Reversible Dynamics a function Y can
be considered, depending on the remaining variables q. only,
which does not differ, except by a constant, from the " poten
tial energy " U ; this is a remark already made by M. Duhem.
Of course it must be restricted to the Dynamics of points and
of rigid bodies, since, for instance, in Hydrodynamics and
Aerodynamics the difference between the quantities V and U
is variable and depends on the compressibility of the fluid.
From (III.) we obtain, leaving out the irreversible term
5'Q, the fundamental principle of Reversible Dynamics.
§ 7. Electromagnetic irreversible phenomena. — Energy stored
in the aether can be transferred to matter and converted into
heat ; this phenomenon, when it occurs, is a thoroughly ir
reversible one.
390 Prof. L. Natanson on the
Here, therefore, we may put R°=0 and ^U/d 2 /*(§i * 61 )] cos (**) ~K^ + dy) ° os (wy)
""% + ^) C0SM; ' ' (3)
p y = "Kai + dy) cos ^ f ^~ 2 Kdy "~^ cos (?w)
"<^ + s) cos M ; • (4)
p° = i\& + §W C0S M <57 + Si) cos to)
+ [^2 M ^_jfl)]ooB(«). . (5)
If now a system &p, 8?/, &e of infinitesimal virtual displace
ments be imposed upon the fluid, the temperature being kept
constant, then the work IF fig. done by extraneous forces
will be
jJdSfo&B +p y hy +p z Sz) +ffida dy dz p(XBx + YBy + Z&);(6)
the variation of the energy T will be
8T = \\\ dx dy dz p(u8u + v8v + wSw) ; . . . ( 7 )
the variation of the energy V, which in Hydrodynamics it is
usual * to call " intrinsic ;; energy, will be
* See, for example, that otherwise excellent treatise ' Hydrodynamics '
by Prof. Lamb, ed. 1895, pp. 1112, 469, 507. It is not with the true
intrinsic energy U, but with the free energy V that we are here con
cerned ; the customary use of the word " intrinsic " seems, therefore, to
involve a serious error.
Laws of Irreversible Phenomena.
393
and, lastly, the quantity of heat which must be " absorbed "
in order to compensate the effects of viscosity will be
+ a vay a*Aay W
+2 vb^ aJva* + a*/
+ 2 W + ayAa* "3y/'
= — jjj" d# cfy dz 2^<
M^)
In order, therefore, to comply with the rule respecting
S'Q we have to write
S'Q=$dxdydz2ini
j. i/S? a^VaS*? a& \
+ 2 V^ + a#Aa* a*/
+ 2 la* + aWVa* + aW'
Kio)
i
Let us now verify whether in the present case our general
principle applies. From (1) we find
'W^JtpJ+^g + " V2M+ ^ ]S * !=o; (it)
{ +[ ]%/ + [ ]* )
now
(12)
394
Prof. L. Xatanson on the
so that from (8), (10), (12), and again from (3), (4), (5) we
obtain
*p
>] + &[® + £>]
+§§dxdydzp {
+
a*
= 8V + yQ+jydS(p,^+p f ^+^&) (13)
Further, we see that
fc*S$*.4,* l @.l. + %* + $«.)£*«. (Ml
because d#
T 2 = i JJJ dx 2 dy 2 dz 2 p 2 {u 2 2 + v 2 2 + w 2 2 ) ,
SP ^ = IST^'i ^1 ^ pi ( x i ^1 + Y i 5 ^i + z i ^1)
+ j]J dx 2 dy 2 dz 2 p 2 (X 2 8x 2 + Y 2 &/ 2 + Z 2 &zr 2 )
+ f(TdS 1 p 1 {cos (n^) S^i + cos [n^y) hy x + cos (n^) o^}
+ JJT ^^2^2 { cos ( n 2 x ) ^2 + cos (n^y) 8y 2 + cos (w^) 6> 2 } . (8)
The quantity of heat generated in time dt by diffusion may
be written
JJJ <&e
(2)
By
J
MH,:
**,=
(3)
Mr. Heaviside and H. Hertz, it is well known, have con
structed the whole of Maxwell's Theory upon two systems of
equations, one of which is the system (2) above, whilst the
second follows at once from (1) and (3). We shall take
A , A , A z to be the independent variables ; that is the choice
* Sitz. Berl. Akad. 12 Mai 1892; Wiss. Abh. Bd. iii. p. 476.
also Boltzinann, Vorlesungen iiber Maxwell's Theorie, vol. ii. p. 7.
See
eJ>JB
= 0.
398 Prof. L. Natanson on the
which Lord Kelvin, Prof. Boltzmann,and other writers adopted
when endeavouring to find dynamical analogies for electro
magnetic phenomena. The part of the energy, called T,
which depends on the quantities dAJdt, dAJdt, dkjdt, will
be then the electric energy
T= a ^tf^T, (7)
and
 fWff & ^ G ( F » 8 A, + F y SA J+ F,8A,) = PW# r . (8)
J 'o J h
Then, from the wellknown MaxwellHelmholtz principle,
on the continuity of properties oil surfaces of separation
between different media, and from the equations (3), we
obtain
iJ>m***{(g.$)M£#>'.
f (BA y aA,\BSA,_/aA I _ BA.N38A,
\ fix B.y / Bi/ V ~dz ^x ) ~dz
^ V "dz ~dx / ~$x V dy d? / ^
=  *dt8\J (9)
>
i
I
J
Laws of Irreversible Phenomena. 399
If, therefore, the general principle is applicable here, the
terms in (6) containing 47701^, &c, should reduce to
P 1 dt h f
J to
Q (10)
Now the quantity of energy which becomes absorbed from the
aether and converted into heat is, for the time dt and the
volume \\\dxdy dz,
Jjftf.r^(feC(E l rfA l +E/A y + E / /AJ; . (11)
hence
8'Q = Jjj*^ dy dz C (E^A, + E y 8 k y + E z 8 A J :. (12)
and thus the principle contained in (I.), or in (III.), is again
seen to hold good.
Part II.
§ 14. Introductory. — The foregoing naturally raises the
question, Does a general law exist concerning the infinitesimal
expressions d'Q and 8'Q, which have been found to charac
terize dissipation of energy in the various particular cases
discussed ? 1 venture to answer this in the affirmative ;
but the hypothesis I advance does not profess to be more
than a conjecture and an approximation.
Let us consider in every particular case the quantity
? =  2Fs ^ (iv.)
In the case of irreversible Dynamics, § 9, the function F is
well known, and has been called by Lord Rayleigh the
"Dissipation Function ;" I should suggest that this term
be extended to all cases covered by equation (IV.).
Let us imagine a material (or at any rate partly material)
system. Suppose that it is not in equilibrium, and observe,
in a quantitative maimer, the disturbances which its state
involves. Let it be isolated so as not to be disturbed by ex
traneous action. We know from experience that under such
circumstances the disturbances in the system must finally
subside and tend to disappear. This general behaviour may
be called the coercion of disturbances, because of the contrast
it offers with inertia. (See Phil. Mag. for June 1895, p. 509.)
For definiteness let us consider a continuous body. Let
dxdydzp be the mass of an element dxdydz, and let
dxdydzpf represent its dissipation function, so that F, the
dissipation function for the portion ^^ dxdydz of the bodv,
400 Prof. L. Natanson on the
be = \udxdy dzpf. Then, generally speaking, F is suscep
tible of three kinds of variation, and dF/dt is the sum of three
terms : — 1. A surface integral relating to the action between
the body and the exterior world through the boundary of the
body; 2. A volumeintegral expressing " action at a distance "
between the body and the exterior world ; and 3. A volume
integral representing " coercion," i. (^+ /? ?)j and (w + f), / the mean value of /within an
element, and D/Dt the rate of " coercion." Then
hence
= (Tf d$p{l;f cos (nx) + 97/ cos {ny) f 5/ cos (nz) }
+ tfJ<*^ P (x.g + YJ + 7>f w y^d«dy dzp % (2)
The three terms on the righthand side refer to the three
kinds of variation as above stated.
The assumption we propose to examine is that the third, or
coercive term DF/D£ is always proportional to the value
of F. Thus, writing r for a constant period of time,
DF 2F
This equation, we shall find, is general ; in the neighbour
hood of states of equilibrium at least it is exactly fulfilled.
The period of time r was first considered by ClerkMaxwell ;
Laws of Irreversible Plienomena. 401
in an important case it received the name of the modulus of
the time of relaxation* and may, without inconvenience, be
called so in other similar cases.
Equation (V.) may be verified in various cases, which we
shall take in order.
§ 15. Irreversible Hydrodynamics. — From § 10 we have
Writing p xx , p xy for the usual components, we have
*^— Kg**)'*,— ^+g> • (2)
and four other equations of the same form. These equations,
it is well known, must be fulfilled if the dynamical equations
of Navier, Poisson, Stokes, and Maxwell are to be true ; they
may be described, therefore, as being in agreement with ex
perience, and so also may be equation (1). Hence
f= u^dydA ^y+^pr+^*n (3)
Again, if the disturbance is not a very violent one, we have
the equations f
P(p«j>) _ 2 t fd^_ ie \. 2Ec__(3!? . M m
and four other equations, to be written down from symmetry ;
it may be well to point out that they are " kinematical "
equations, therefore independent of any particular molecular
hypothesis. Now, if we put
(5)
£=T.
P '
we obtain from (2) and (4)
T)(p xx —p) pxx—p Qpyz
Bt t ' D*
pyz
(6)
* Philosophical Transactions, 1867, p. 82. See also 'Treatise on
Electricity and Magnetism,' third edition, vol. i. p. 451.
t Philosophical Transactions, 1867, p. 81.
Phil. Mag. S. 5. Vol. 41. No. 252. May 1896. 2 P
^02 Prof. L. Natanson on the
and four similar equations ; and from (3) and (6)
{P**P) — TV +\PnP) Tv
F= ^dxdydz{^ + (P**P) B{ % P) ~
44k
+ 2w 2^+2» 5£«+2p 5^
whence, by (3), we infer that DF/D*=2F/t, as stated
above.
The value of t in air, at the temperature 0° C. and normal
pressure, is approximately 2.10" 10 of a second (Maxwell, Phil.
Trans. 1867, p. 83). We may also compare the relative
values of r in two fluids. Jn doing so we may assume, in
accordance with Prof. Van der Waals' leading idea, that the
values of t would bear a constant proportion if they were
calculated for u corresponding " states of the flnids *. Hence
the coefficients of viscosity will likewise,^ in corresponding
states of two fluids, bear a constant numerical ratio f.
§ 16. Diffusion.— The signification of the symbols being
the same as in § 11, we find the dissipation function of diffu
sion to be
F = i^d X dydzAp lP2 {(y 2 u i y + (v 2 v 1 Y+(iv^ic l y}. (1)
The theory of diffusion can be deduced, in the case of two
gases, from " kinematical " equations and from the following
equations " of coercion," in which D/Dt refers to the total
coercive action of both gases :
^=Ap 2 K« 1 );i§=A /0l K« 2 ), . (2)
and four other equations of similar form. If the dynamical
equations of Maxwell and Stefan are true, equations (2) must
likewise be fulfilled ; they may be said therefore to agree
with experience. Let us now pass to the usual case of slow
and quiet diffusion (Maxwell, Phil. Trans. 1867, pp. 7374).
If we write S for the temperature, R for the gaseous con
stant, we shall find the value of the coefficient of diffusion,
or h say, to be RS/Afo + ft) ; and, if p=p l + r» the charac
* See Kamerlingh Onnes, Algemeene Theorie der Vloeistofen, Tweede
Stuk, p. 8, 1881. . .
f See Kamerlingh Onnes, ' Communications from the Laboratory oi
Physics at the University of Leiden,' no. 12, p. 11, 1894.
Laws of Irreversible Phenomena. 403
teristic period t for the coercion of the disturbance will be
1 (Pl + P2) h
MPI+P2)
(3)
In a system composed of nitrogen and oxygen, at 0° 0. and
normal pressure, the value of t (from v. Obermayer's experi
mental results) is about 4*5 x 10~ 10 of a second. Returning
to (2) we obtain
D(U 2 — Mi) U 2 — U X . .
Dt " T ' W
and two other equations which may be written down from
symmetry ; hence (1) reduces to
F = h^dxdydzApw { K Ml ) DK ^ Ml)
+ fa*i) ^ +K^i)~ A ^ — ~j> • • ( 5 )
and this gives
DF_ 2F
D* ~ t
(6)
Let us verify that, as stated above, 2F is the rate at which,
owing to diffusion, heat is being irreversibly generated. First,
from conservation of energy, we have
VtM 9 , !+A(i4+iiS+i«!+e+i«+Ci)J
Then, from (2) we obtain
^ j]J dx dy dz \p x (u\ + v* f w]) =$$dx dy dz Ap x p 2
X{w,(w 2 — Wi) + ^ 1 fv 2 Vi) + wiiw^—wx)}, . (8)
^ jJJ d# dz/ tfe ^p 2 (m + v\ + 1^) =JJj^ % dz Ap 2Pl
x {^2(^1 — w 2 ) +v 2 (v } — v 2 ) + w 2 (w 1 — w 2 )} y . (9)
whence by (7) it follows : —
= jjy 2 ™i) 2 }
= 2F . (10)
and this proves the proposition.
2F 2
404 Prof. L. Natanson on the
§17. Electromagnetic Dissipation. — The electromagnetic dis
sipation function is
Y=±$dxdydzQ{W x + W + WX ... (1)
the symbols being defined as in § 13. The disturbance settles
down obeying the wellknown equations
K ^ = 4 w CB,; K^ = 4,rCE, ; k5J = 4,tCE ; (2)
they are therefore the electromagnetic u coercion " equations.
If we take t = K/47tC, as has been done by Maxwell and
many others, we see that
F=iJXf < /^y^ T c(E^ + E,^ + E^) . (3)
DF 2F
D« = V W
Prof. J. J. Thomson has shown * that for water with
8'3 per cent, of H 2 S0 4 , t cannot differ much from 2. ID 11 of
a second ; and for glass at 200° C. from about 10~ 5 of a
second.
§ 18. Irreversible Dynamics. — In the case of § 17 the energy
we have called T is proportional to the dissipation function F;
the same holds in § 16 if we have p 1 u 1 + p 2 u 2 = Q (see Maxwell,
Phil. Trans. 1867, pp. 7374). Hence, in such cases equa
tion (V.) becomes DT/Dt= — 2T/t. Again, in the Irreversible
Dynamics of § 9, if the additional dissipative forces — R/ be
proportional to the corresponding components of momentum,
the same proportionality holds. For example, let
*'££ w
represent the additional dissipative force acting in the ^.direc
tion ; then T = tF ; and since from (5), § 9, it is easily shown
that
DT
W = ~ 2F ' ( 2 )
* ' Notes on Recent Researches in Electricity and Magnetism,' 1893,
§ 32.
Laws of Irreversible Phenomena. 405
D f Dt being the rate of variation of the kinetic energy arising
from the dissipative forces, we see that, in this case,
DF/D*=2F/t and DT/D*=2T/t. . (3)
Cf. Lord Rayleigh, • The Theory of Sound/ vol. i. p. 78.
§ 19. Dissipation Function of Conduction. — In the Philo
sophical Magazine for June 1895, p. 506, it was shown that
in conduction of heat the dissipation function is of the form
m=§$***v
(i)
the symbol being employed to denote 3(Z 2 + v 2 + i 2 )' From
(12) and (39) in the paper referred to, we have
"D*^' uf bp ^y> T5T~ 5i ^" (2)
P^ = ^ ; P^= 5 A^;^ = ^/>r;(3)
Bt k P x Vt k P * Bt k H 2 ' K }
equations (2) are the " kinematical/' and equations (3) the
" coercive " equations of the problem. They must be fulfilled
in order to make the equations hold :
pr '~ k ^x' pr >~ k dy' pr ~ V' " ' {)
and, therefore, to secure applicability for Fourier's equation.
The time of relaxation we define as
r=k/5p, ....... (5)
neglecting differences p—p^, &c. From (1) and (4) we
obtain
»iJ^**i{CpO , +&»»v) , + (prJ , }> • (6)
and from (1) and (2) we have :
whence, by (5), we find again
DF 2F
w=v ( 8 >
§ 20. Connexion between the periods t. — Let r p be the
406 Dr. J. Shields on a Mechanical Device for
characteristic period r for conduction of heat in a given gas,
and let t m denote the period relating, for the same gas, to
internal friction. The coefficient of conductivity in Fourier's
equation, as usually written, is ^c v k, k denoting the same
quantity as in § 19. Now in the Kinetic Theory of Gases it is
shown that this coefficient is equal to
Tfo 1 V» W
if 7 be written for the ratio c Jc v of specific heats, and fi for
the coefficient of viscosity [for example, see Prof. Boltzmann's
Vorlesungen iiber Gastheorie, equations (238), (54), and (57)].
Hence
T E = I(71>M 0r =! T M> ' • • ( 2 )
since, strictly speaking, our calculation requires the gas to
be monatomic. In a similar manner may all the periods of
relaxation, corresponding to the various powers of coercion
of a given body, be mutually connected ; and every such
simple equation, if it holds, is equivalent to a definite physical
law.
XLII. A Mechanical Device for Performing the Tempera
ture Corrections of Barometers. By John Shields,
D.Sc, Ph.D.*
THE height of the barometer is generally reduced to 0° C.
by means of the formula
Vi R 1+ ^
where B is the reduced height, B^ the observed height at the
temperature t, and ft and 7 the coefficients of expansion of
the mercury and scale respectively ; or, since fi and 7 are in
general both very small, we may write
B„=B,[1 (57)0
In order to facilitate the reduction, tables containing the
corrections corresponding to definite temperatures and observed
heights have been compiled, and in the laboratory it is only
necessary to consult such a table, and if necessary perforin a
simple interpolation in order to find the correction which
* Communicated by the Author.
the Temperature Corrections of Barometers. 407
must be subtracted * from the observed height to reduce it to
0°0.
Graphic methods are sometimes used to obtain the tem
perature correction, and one of the best of these, which was
first brought to my notice by Prof. Ramsay, is described in
the Appendix as it seems to be little known.
During the course of an investigation it was necessary for
me to read and correct the barometer several times daily,
and as this operation became rather tedious I was induced to
make a barometer which indicated the height and the cor
rection simultaneously. The construction of the barometer
presents no great difficulties, and as it is extremely useful in
its new form I now beg to lay a description of it before the
scientific world. It can be read with certainty to 0*1 millim.,
which is sufficiently accurate for most purposes. Whether an
improved method of reading and better workmanship than I
have been able to bestow upon it would make it suitable for
meteorological observations must be left for meteorologists to
decide. For ordinary laboratory work, however, it meets all
the requirements. It is not necessary to know the temperature
at all ; and by mentally subtracting the correction as indicated
by the correcting instrument from the observed height (also
obtained directly by setting the barometer), the observer is
enabled in one entry to write the corrected height of the
barometer in his notebook.
The most suitable form of barometer to employ with the
correcting instrument is that described by Dr. J. Norman
Collie f, but an ordinary syphon barometer may also be
adapted for the same purpose. The lower end of the baro
meter is cemented or otherwise securely fitted into a brass
cap A, fig. 1, to which is attached a rod B, which moves
vertically in a guide in order to prevent the barometer from
rotating when it is raised or lowered by means of the screw G.
The barometer itself is kept in a vertical position by the
guides D D which are attached to the framework. The back
of the framework consists of a long, narrow board, the lower
end of which is shown at E, and to which the nut carrying
the screw is fixed. A plateglass mirror F, carrying the
graduations, is firmly screwed down on the main frame by
means of the pictureframe moulding Gr Gr, which is planed
down at the back to such an extent that the mirror is held
tightly clamped in position. The plateglass mirror is care
fully graduated between 700 and 800 millims., and has also
* For temperatures below 0° C. the correction must be added.
+ Trans. Chem. Soc. (1895), p. 128.
a
408 Dr. J. Shields on a Mechanical Device for
a zero line etched on it, but the space
between zero and 700 millims. may be
left ungraduated.
This particular form of mounting the
barometer, independent of the correcting
instrument which has yet to be described,
is in itself very useful ; as by setting the
lower meniscus of the barometer at the zero
line by means of the screw at the bottom ^
of the frame, the uncorrected height can
be read off directly, and this obviates the
necessity of taking down the upper and
lower readings nnd adding or subtracting
as the case may be.
Before proceeding to show how the £
device for indicating the amount of the
temperature correction can be attached to
a barometer mounted in this way, it is well
to note that the areas of the upper and
lower reservoirs of the barometer are
supposed to be equal, and are in fact
approximately so, if these reservoirs are
cut from adjacent parts of the same piece
of tubing. Assuming now that the baro
meter is accurately set, and that the
pressure of the atmosphere then changes,
if the pressure rises or falls n millim.,
then, on again adjusting or setting the
barometer, any point on the stem will ob
viously be raised or lowered n/2 millim.
Should, however, the cross section of the
capillary tube, which connects the main
stem of the barometer with the lower
cistern, be large when compared with the
cross section of the lower cistern, then the
above relation will not hold good.
This source of error may be eliminated
either by making the cross section of the
capillary small, or by selecting a lower
cistern with a proportionally larger area.
It may also be eliminated, if necessary,
in graduating the scale of the correcting
instrument, but any slight error intro
duced in this way has scarcely any ap
preciable effect on the accuracy of the
readings.
the Temperature Corrections of Barometers. 409
The precision of the complete barometer is limited not so
much by the accuracy of the temperature correction, as this
can easily be made to read correctly to 001 millim., but by
the precision with which the barometer can be set and by the
accuracy of the graduations. As has already been mentioned,
the combined error in setting the barometer and reading the
scale at the top should not exceed 0*1 millim. Of course all
danger from the above source is removed if the common
syphon form of barometer is employed; but as Collie's
modification presents other advantages which are clearly set
forth in his paper * , his form of barometer is to be preferred,
especially as it is only necessary for the purposes of the
correcting instrument that the above relation between n and
w/2 should hold good within a millimetre or two.
In designing the correcting instrument use has been made
of the fact that the variation in position of a point on or
attached to the stem of the barometer is proportional to the
variation of the height of the barometer. The point is repre
sented by a horizontal thread of mercury H (figs. 1 and 2),
contained in an ungraduated thermometer which is firmly
attached to the stem of the barometer in a horizontal position.
Behind the horizontal thread of mercury is fixed a scale or
small plate of curves K, in such a position and drawn in such
a manner that the position of the horizontal thread of mercury
(the ordinates) indicates approximately the height of the
barometer. The correcting instrument is shown on a larger
scale in fig. 2, The distance from the top to the bottom of
the plate of curves, i. e. from L to M, is actually 50 millims.,
but this, from what we have already said, represents altogether
100 millims., the position of L corresponding to a barometric
height of 800 millims., whilst M corresponds to 700 millims.
The position of the horizontal thread of mercury at the time
of setting the barometer thus corresponds approximately to
the actual height of the barometer. A series of lines 1, 2, 3, 4,
&c. millim. are drawn or engraved on the plate K, so that the
position of the meniscus of the horizontal thread of mercury
gives the temperature correction directly in millimetres. The
method of drawing these lines requires some explanation. It
is desired to reduce the height of the barometer to 0° C.
Obviously, then, the line of zero correction must lie
immediately behind the point corresponding to 0° C. on the
horizontal thermometer, and it must furthermore be vertical,
as no matter what the height of the barometer may be the
correction at 0° C. must always remain zero.
* Collie, loc. cit.
410 Dr. J. Shields on a Mechanical Device for
Before the position of the other lines can be fixed, it is
necessary to ascertain the length in millimetres corresponding
to 1° G. of the horizontal thermometer. Let this be n millim.
Fig. 2.
12 3 4
rrrn
We have already seen that the formula for reducing the
height of the barometer to 0° C. is
B =B,[l(/3 7 )*],
hence the correction which must be subtracted from the
observed height is
die Temperature Corrections of Barometers. 411
In order to fix the position of the top of the lines giving the
corrections 1, 2, 3, 4 &c. millim., all that we have to do is to
make B t =800, and calculate the temperature t, which would
make the above expression equal to 1, 2, 3, 4 &c. millim.
Thus, for the top of the line representing a correction of
4 millim., we have
(/3y).800.*=4.
For all ordinary purposes we may make ft — 7 for mercury
and a glass scale = 0000181 0000009 = 0*000172, hence
0000172x800"
But since the thermometer is not graduated, we must multiply
the value of t so found by n to get the distance in millimetres
of the point 4 from the point 0. Similarly the tops of the
other lines may be found, but in general it will be found
sufficient to calculate the greatest correction only, and then to
divide the distance between it and into the required number
of equal parts. In the same way the corrections correspond
ing to a barometric height of 700 millims. (the lower ends of
the lines) may be obtained by making B^=700. Then, since
the correction is proportional to the height of the barometer,
straight lines joining 1 and 1, 2 and 2, and so on will repre
sent the corrections between 700 and 800 millims. The spaces
between and 1, &c. may also be subdivided into halves or
tenths if necessary.
The scale is either drawn or engraved on paper, a plate
glass mirror, or other convenient material, and then mounted
on a bridge in front of the stem of the barometer and behind
the correcting instrument. The final adjustment is made by
moving the correcting thermometer into the proper position
before clamping it tightly to the stem of the barometer. A
piece of wood or other soft material interposed between the
stems of the barometer and thermometer prevents any risk of
breakage on screwing up. In adjusting the correcting
instrument or thermometer, care must be taken that, firstly,
the zero point of the thermometer is precisely in front of the
line of zero correction, and, secondly, that the thread of the
mercury is truly horizontal, and that its position between the
700 and 800 millim. fines of the correcting scale corresponds
as nearly as possible with the actual height of the barometer
at the time.
For convenience, the completed barometer should be sus
pended on a vertical wall with a good light falling on it. In
412 Dr. J. Shields on a Mechanical Device for
order to take a reading it is first gently tapped, the lower
meniscus is then set exactly at the zero line by means of the
screw at the bottom, and the temperature correction as
indicated by the correcting instrument at once read off lest
the heat of the body should cause any alteration ; the height
of the barometer is then observed at the top of the instru
ment, which, after subtraction of the temperature correction,
gives the barometric height reduced to 0° C.
By considering the correcting instrument it is obvious that,
the temperature remaining constant, the rise or fall of the
barometer is accompanied, after setting the lower mercury
meniscus to zero again, by an upward or downward displace
ment of the horizontal thread of mercury, and, consequently,
to an increase or decrease in the correction. Similarly, a
rise or fall of temperature is accompanied by an increase or
decrease in the correction.
The instrument I have just described is one out of four
possible modifications. Both scales may be fixed whilst the
barometer and thermometer are displaced simultaneously or
vice versa. Again, the barometer and correcting scale may
be fixed whilst the other parts are adjustable or vice versa.
The first two will probably be found most useful. Distinct
advantages might be gained by fixing the barometer with the
thermometer above it, and at the same time etching the
barometer and correction scales on the same piece of plate
glass mirror, which could be placed behind them and be
moved vertically by a set screw at the bottom of the
instrument.
University College, London.
Appendix.
Fig. 3 illustrates a convenient graphic method for obtaining
the temperature correction of a barometer or other column of
mercury.
The ordinates represent the height of the mercury column *
and the abscissae the correction in millimetres. The correction
for any temperature not represented by a diagonal line
passing through the origin is easily obtained by a graphic
interpolation. As fig. 3 is too much reduced in size to be of
any value, a diagram of convenient size may be obtained by
plotting the ordinates on a piece of curve paper ruled into
* In this particular case the height is supposed to be measured with a
glass scale. If a brass scale is used the correction is always about
GO per cent. less.
the Temperature Corrections of Barometers.
413
inches and tenths of inches, each inch representing 100 millim.,
whilst each millimetre of correction is represented on the
lower horizontal axis by two inches. A series of lines, each
Fig. 3.
8° 10' 12° (4° 16° 18° 20' 22° 24° 26° 28° 30°
4.0 MM.
representing a difference of 2° C. of temperature, are then
ruled through the origin to the points on the upper horizontal
axis marked 8°, 10°, 12°, &c. The position of these points is
easily obtained from the formula
(/3— 7) B^ = correction in millim.,
by substituting for t the values 8°, 10°, 12°, &c, for B, the
value 800, and for /? and 7 the coefficients of expansion
of mercury and the material of which the scale is made
respectively.
[ 414 ]
XLIIL An Addition to the Wheatstone Bridge for the Deter
mination of Low Resistances. By J. H. Reeves, M.A.,
City and Guilds of London Central Technical College *.
WHILE it can be assumed with fair certainty that
even in moderately equipped laboratories there will
be found one sensitive galvanometer and a good set of resist
ancecoils arranged in the form of a Wheatstone bridge, it is
by no means so common to find a convenient method for
determining low resistances, such as, for example, a Kelvin
bridge. The piece of apparatus which forms the subject of
this paper is a comparatively cheap addition to the ordinary
bridge, and enables the resistances of exact metrelengths of
even thick wires to be directly measured with almost as much
ease as larger resistances can be determined with the ordinary
Wheatstone bridge. The method also possesses the advantage
that all the measurements are made in terms of a standard
wire with fixed contacts, and, therefore, not subject to the
wear which accompanies the frequent use of a slider ; further,
in the case of copper wires no temperaturecorrection is
needed.
The apparatus is represented in fig. 1. On a mahogany
Fig. 1.
G H
baseboard are stretched close to one another two wires — one,
ABCD, being the standard of comparison, while the other,
EFGH, is the wire to be tested.
A and E are two massive pieces of brass which can be
joined together by a plug P. D and H are smaller pieces of
brass with bindingscrews attached.
The standard BC is permanently fixed to A and D, whilst
two clamps F and G fixed respectively to E and H form the
means of fixing in its place the wire to be tested.
KL and NM are two brass springs which pass over but do
not touch BC. They are provided with bindingscrews at
K and M, and with two knifeedges L, N exactly one metre
apart, which press on the wire FG when in position, while
* Communicated by the Physical Society : read March 13, 1896.
Determination of Low Resistances. 415
two screws S, S' will raise the springs when required for
inserting or removing this wire.
At points B, C in the standard are soldered two short
lengths of wire terminating in bindingscrews Q and T. The
gauge of the standard wire, which is of copper, and the
distance apart of B and C are so chosen that the resistance
between B and C is O'Ol ohm at the temperature of the ware
at the time of its final adjustment, which should be noted.
The method of determining these points is given later on.
The arrangement is a variation of the Kelvin bridge, with
this difference : that in the latter the measurements are made
by varying the length between the knifeedges of sliders
which press on the standards, of which several are required :
whilst in the former only one standard resistance is employed,
and the measurements are effected by the alteration of the
other resistances in the arrangement. If the whole apparatus
had to be bought, the Kelvin bridge would be the cheaper,
but the new Addition utilizes the box of coils, which as before
mentioned is sure to be available, and only requires in addition
one standard to be made and adjusted.
The theory of the Kelvin bridge is to be found in the text
books. In Gray's ' Absolute Measurements/ for example,
vol. i. p. 359, it is proved that if fig. 2 represents the usual
arrangement, and, if R, r, x, y, a, b, s are as marked, the
following relationship must hold in order to obtain balance: —
R x a
y
Fig. 2.
V
s
I
Further, it is shown that when s is small compared with
a and b the accuracy of the equation
R_ x
r ~~ y
is not affected by a small want of equality between the ratios
a , x
r and — .
b y
In addition to the galvanometer and set of coils there is
416
Mr. J. H. Reeves on an Addition
required a slidewire bridge, which may be of a rough descrip
tion but should have a resistance of an ohm or so. At the end
of the paper it is shown how this latter and the Addition
can be united in one.
The whole arrangement is joined together as shown in
fig. 3, the dotted lines indicating the temporary connexions.
By comparing this with figs. 1 and 2, no detailed description
is necessary, as R, r, x, y, a, b, s are lettered to correspond.
The battery may with advantage be a storagecell with two
resistances in series with it. One, p, may be an adjustable
carbon resistance, while p l may be constructed of wire and
should have a considerable resistance. This latter terminates in
mercury cups m and n, so that by joining these cups together
with a short connector the resistance can be easily short
circuited.
The resistances x and y are resistances unplugged from the
box of coils forming the ordinary Wheatstone bridge ; y is
the 1000ohm coil of one of the ratio arms, and x is in the
adjustable arm. If such a set of coils be not available, y may
be any 1000ohm coil, and x any box of coils containing
resistances up to 5000 ohms. The two leads joining x and y
to B and B' respectively should be as stout and short as
possible. The resistances from d, d' to I form the a b of
fig. 2.
To make a measurement, insert the wire whose resistance
is required in its proper clamps, taking care that it lies quite
straight between them. Its diameter and material being
known, the resistance of one metre of it can be approximately
to the Wheatstone Bridge. 417
calculated. Calling the resistance B l5 choose x x in the box
so that
x 1 B x
Tooo <>or
Eemove the ping P, and, including p r in the battery cir
cuit, obtain balance by shifting the slider I, the galvanometer
being also shunted if necessary. The arrangement is now an
ordinary Wheatstone bridge,
a 4 B x __ x x _ III
b\r y r '
or a K, B • . i
— = — x = — approximately.
Next insert the plug P, shortcircuit p r so that now a
strong current passes when the battery circuit is closed, and
obtain balance by altering x. Let the new value be x 2 . Then.
n T?
since approximately r = — , we have, by the formula quoted
above,
E _ x 2
r ~~ y'
'■' R=xr very approximately.
j
In any case this new value of E is very much nearer the
true value than the approximate value E x . If the value of x
has to be but little altered, it may be taken as the true value
of E, but if, on the other hand, the first estimate was con
siderably in error, and, in consequence, a large alteration
has to be made in x in order to obtain balance, the much
closer value thus obtained must be taken as the first approxi
mation, and both experiments repeated.
Now, since the wires are so close to one another, their tem
perature can be called the same, and, if the tested wire be of
copper, their temperaturecorrections are the same. There
fore, if E be a certain fraction of r at one temperature, it will
be the same fraction at any other. But in the actual bridge
on which the experiments described later on were first
made, this temperature was 17°' 7 C. Therefore, substituting
numerical values in the last equation,
^I^o^^^lopoo 0111118 ^ 17070 
Of course, if E be not made of copper, the temperature
must be noted and allowed for.
The accuracy and sensibility of this method depend on
three conditions : —
Phil. Mag. S. 5. Vol. 41. No. 252. May 1896. 2 G
418
Mr. J. H. Reeves on an Addition
(1) The accuracy of the ratio  ;
if
(2) The accuracy of the value of r ;
(3) The sensibility of the galvanometer.
As far as condition (1) is concerned, x and y being taken
from a good box have an error of certainly less than O'l
per cent.
Condition (2) is discussed later, and the effect of condition
(3) can be best seen by reference to tests made in the labora
tory of the Central Technical College with the apparatus
arranged as in fig. 3, except that, at the start, the battery
consisted of a single Daniell cell, the galvanometer being a
fourcoil reflecting one having a resistance of 687 ohms and
a sensibility of about 400 scaledivisions per microampere.
Table I.
Tests made on high conductivity copper, R=
100,000
ohms.
No. of
wire.
S.W.G.
Diameter,
mils.
X.
Observation.
R.
Specific
Gravity.
Specific
Resistance.
Conductivity.
Matthiessen's
Standard.
f
4324
^ div. deflexion left.
1
22
2807 \
[
4326
4328
2618
Small „ right.
1^ div. „ right.
1 „ „ left.
I 004325
1
8841
1616
1003
20
355 j
2620
2622
Small „ left.
^ div. ,, right.
1 002621
8918
1572
1031
18
488 [
(
1421
1422
802
i „ „ left,
i „ „ right.
3 „ „ left.
1 0014215
1 0008032
8912
1602
1011
16
642 \
803
i „ „ left.
8907
1572
1031
\
804
2 „ „ right.
J
801 J
515
2 „ „ left.

14
516
Small „ left.
10005163
8888
1566
1034
517
1 div. ,, right.
J
12
1032 [
314
315
1 , „ left.
1 „ „ right.
j 0003145
8904
1585
1022
—
1069 [
291
292
1 „ „ left.
5 „ „ right.
j 0002916
8940
1580
1026
—
1038 
308
309
H „ „ left.
1 „ „ right.
1 0003086
8870
1588
1024
—
104*2 {
309
310
2 „ „ left.
 „ „ right.
] 0003098
8891
1580
1026
Matthiessen's standard has been taken to be a specific resistai
ice (resistance
per cubic centimetre at 0° O.) of 1620 International Microhms.
Towards the end of this table it will be seen 1
;hat even a
sensitive galvanometer barely allowed an accur
acy of 0*1
red, and a
per cent. The Daniell cell was therefore remo^
small a
ccumr
dator with a resistan
ce of 2 o
bms in
series wi1
h
to the Wheatstone Bridge*
419
it substituted. The last three wires were then again tested,
and a marked improvement was at once seen, for now an
alteration of 1 ohm in x produced a difference of deflexion of
1015 scaledivisions, thus giving the interpolated figure
correct to the first place given, i. e., R correct to 1 part in 3000.
To further test the capabilities of this method, a strand
cable composed of 7 copper wires, each of No. 16 S.W.Gr.,
was taken, the strands being flattened to pass under the
springs, which had not been constructed for a wire of such a
large diameter. The results, using the same cell and resist
ance, are shown in the next table.
Table II.
X.
Observation.
E.
117
6 divs. deflexion left.
118
H » „ left.
0001183
119
3 „ „ right.
Here only 4J divisions correspond with a change in x of
1 ohm : thus an error in reading of J a division means an
error in R of 0*1 per cent. The effect of using a still larger
current was next tried, the 2ohm coil being removed, and the
two resistances /?, p' of fig. 3 substituted. An ammeter was
also included in the batterycircuit, so that the actual current
could also be measured.
Now, on passing a strong current both wires heat, and if
the wire r has the smaller diameter it will heat the faster.
Therefore its resistance will also increase the faster, and hence
the resistance of the wire R will apparently diminish, and
vice versa. In order to see how much this diminution might
amount to, the resistance p 1 was shortcircuited, and p was
adjusted until the batterycurrent was 6 amperes, and the
results are shown in Tables III. and IV.
Table III.
With the current only continued long enough to measure
the deflexion : current 6 amperes.
X.
Observation.
E.
118
119
3 divs. deflexion right.
23 „ „ left.
00011812
2G2
420
Mr. J. H. Reeves on an Addition
Table IV.
After the current had flowed continuously for 3 minutes
6 amperes as before.
X.
Observation.
R.
119
118
117
28 divs. deflexion left.
4 „ „ left.
20 „ „ right.
00011784
The difference between these two results is about £ per
cent., while the deflexions are more than sufficient for an
accuracy in It of 0*1 per cent.
A few days later these experiments were repeated with a
current of 2 amperes. The slight discrepancy in the value of
R was due to the fact that in the interval the cable, which
was not quite straight, had been removed. On replacing it a
slightly less length must have been included between the
knifeedges. As, however, these experiments were being
made solely to find the sensibility of the measuring arrange
ment, no attempt was made to straighten it. The results are
shown in Table V., experiment (a) being for the short dura
tion of current, and (b) the result after the current had flowed
continuously for 3 minutes.
Table V.
Current 2 amperes.
Experiment.
X.
Observation.
R.
f
119
16 divs. deflexion left.
]
(a)
{
118
117
3 „ „ left
10 „ „ right.
1
0001178
(ft)
{
118
117
4 „ „ left.
9 „ „ right.
i
0001177
From the above results it will be seen, firstly, that 13
divisions correspond with 1 ohm difference in x, thus making
the fourth significant correct, in other words, an error in R is
less than 0*1 per cent. ; and secondly, the passage of a current
of 2 amperes only produced in 3 minutes a difference of like
amount.
No larger cable was tried, nor was any wire smaller than
to the Wheat stone Bridge. 421
No. 22 S.W.G., but between these limits an accuracy of 0*1
per cent, was easily obtained so far as the sensibility of the
method was concerned.
The accuracy of the standard now remains to be considered.
This can be best seen from a description of the method by
which the correct length between the points B, was de
termined.
In order that this length might be of a convenient amount
to suit the dimensions of the baseboard, it should have been
made of copper wire No. 17 S.W.Gr. ; but a piece of this gauge
which was bought for the purpose proving unsatisfactory, a
piece of No. 16 was taken and stretched till of the required
diameter.
One, Q B, of the two short arms of fig. 1, was soldered on,
the other, T C, being left loose. The whole was then screwed to
a rough board, and the wire annealed by passing through it
a current of about 50 amperes till it was too hot to touch and
then allowing it to cool, the operation being repeated ten
times. The arrangement represented in fig. 4 was next made.
Fig. 4.
C \K±lw> • *
STfel M
By following out this figure, it will be seen to be the same
arrangement as fig. 3, except that now the place of the plug
is taken by the gap P. The standard was a 1ohm coil of
manganin wire, constructed by Messrs. Nalder, Bros. Its
terminals rested in mercurycups from which the various leads
were taken, the ends of all wires dipping in the mercury being
freshly amalgamated. The leads joining the 10 and 1000
ohm coils together and to the bindingscrews T and M respec
tively were short and stout.
The arm T C was then put approximately in position, and
good contact at C was maintained by pressure. To avoid
thermoelectric effects this pressure was made with a piece of
wood, and in order to see if any such effects were present a
commutator, not shown in the figure, was put in the battery
circuit. It was then found that if balance had been obtained
422 Determination of Low Resistances.
for one direction of the current, a reversal produced no appre*
ciable deflexion, thus showing that this piece of wood had en
tirely prevented such effects. As a precaution, however, the
current was reversed each time balance was obtained, and in
no instance was any deflexion caused by the reversal.
The experiments were then conducted in the same order as
before. With the gap open, balance was obtained by moving
the slider I. The gap was then closed and balance obtained
by altering the position of the arm T C. The gap was next
opened, and the slider moved till again no deflexion was noted.
This motion was very slight, and the alteration, after closing
the gap, was not found to have produced any observable want
of balance. The correct position of C was thus determined.
In order to see with what degree of accuracy this position
had been arrived at, the arm was shifted 1 millim. in both
directions, and the want of balance was indicated by a de
flexion on either side of zero of one division ; hence, as
the length between B and C was over one metre, the error
in the position of C was well under 0*1 per cent.
The arm TO was then soldered to the wire, and the tests
repeated. No deflexion was observable with the ratio ^r«,
but on adding a 1ohm coil to the 1000, making the ratio
r^r, the want of balance produced nearly 2 divisions de
flexion. The wire was then screwed in its proper place and
there tested in a similar fashion with the same result, viz.,
with the ratio tttftr no deflexion was noted, while with the
ratio TKryi a deflexion of over 1 division showed the want of
balance. A thermometer lying beside the wire indicated
17°7 C. Thus the resistance between B and C was O'Ol ohm
at 17°* 7 C, with an error of less than 0*1 per cent.
Thus on all three conditions an accuracy of 0'1 per cent,
was obtained. This simple apparatus is, therefore, capable of
measuring the resistances of metrelengths of wires between
the limits of No. 22 S.W.G. and a stranded cable of 7
No. 16's (and probably over a still greater range) with an
accuracy throughout of 0*1 per cent., an accuracy quite suffi
cient for all commercial purposes.
Although in the above simple form of The Addition a slide
wire bridge is necessary, it may happen that this latter is not
Absorption Spectrum of Iodine and Bromine. 423
available. At a slight additional cost this slidewire can be
made an integral part of the apparatus,
sented a rather more elaborate design.
Fig. 5.
In fig. 5 is repre
The slidewire is
^4
x
x±j
T H *~7
V7*
B~
easily recognized. The contactpiece of the slider S must be
made adjustable so as to be able to press on either part of
this wire. The circle K represents an ordinary Wheatstone
bridge key.
It will be seen that this design is very selfcontained. All
the connexions to be made consist of : a battery, with its
supplementary resistance, to the bindingscrews B, B', a
galvanometer to g and S, and two resistances between X, Y
and Y, Z
coils, and
sistance.
The former, marked R 1? is an adjustable box of
the latter, R 2 , is a single coil of 1000 ohms re
XLIV. On the Absorption Spectrum of Solutions of Iodine
and Bromine above the Critical Temperature. By R.
W. Wood *.
IN examining solutions of iodine above the critical tempera
ture with a spectroscope, I have found that the fine lines
which characterize the absorption spectrum of gaseous iodine
may be either present or absent, depending on the amount
of the solvent present.
These lines are not present in the spectrum of iodine
solutions, and their disappearance under the abovementioned
conditions seemed to be due either to the pressure exerted by
the vapour, or to something akin to solution.
The tube containing the liquid was heated in an iron tube
provided with two vertical slits, cut opposite each other, for
the passage of a light ray, which was subsequently analysed
with a large spectroscope of high dispersion.
* From the Zeitschrift fur Phys. Chemie. Communicated by the
Author,
424 Mr. K W. Wood on\the Absorption. Spectrum of
For the preliminary investigation four tubes of similar
size were prepared (1, 2, 3, & 4, fig. 1) containing equal
amounts of iodine, but successively increasing amounts of
bisulphide of carbon.
Fig. 1. Fig. 2. Fig. 3.
AA
i
XI
^
12 3 4
These tubes were successively heated until their contents
were homogeneous, and their absorption spectra observed.
No. 1 showed the lines almost as distinctly as iodine alone :
in No. 2 they were fainter but still visible : it was with
difficulty that they could be seen in No. 3, while in No. 4 they
were entirely absent. No. 4 was then opened, and a little more
iodine added. On reheating, the lines appeared. Variations
in the temperature had no apparent effect. A mixture which
at 300° snowed the lines faintly, showed no change when
heated to 350°.
A large number of experiments were tried with varying
amounts of iodine, and with various solvents such as chloro
form, liquefied sulphur dioxide, and water, and all were found
to act in the same way. It was difficult to get reliable results
with water owing to its action on the glass with formation of
iodides.
To determine the effect of a greater pressure with less
density, an apparatus (fig. 3) was constructed of glass, in which
ressure could be developed by the electrolysis of water.
he long arm, which contained the iodine, was heated in the
5
iodine and Bromine above Critical Temperature. 425
iron tube, and a current of electricity sent through the water
in the short arm. The absorption spectrum was watched for
an hour and a half, at the end of which time the apparatus
exploded, but up to the very end the fine fines lost nothing
in distinctness. The pressure was calculated from the time,
the current strength, and the capacity above the liquid, and
was found to have been about 250 atmospheres, or more than
double the critical pressure of bisulphide of carbon. This
indicates that the disappearance of the lines is due to the
density of the vapour rather than to its pressure.
The quantitative investigation of these phenomena was
next undertaken. A powerful arc light was substituted for
the incandescent lamp, and a lens so arranged as to throw
an image of the " crater " on the heated tube. By this
arrangement, the spectra of much denser solutions could be
observed. A tube provided with a long capillary neck, of
the form shown in fig. 2, was constructed and carefully
graduated. The contents of this tube could be varied without
altering its volume by cutting off the tip of the capillary, and
by warming or cooling the tube cause the liquid to run out
or in; the tip could then be sealed once more. This operation
could be repeated about 60 times before using up the capillary,
only about 2 mm. being removed at each filling. A certain
amount of iodine and bisulphide of carbon being introduced,
the tube was sealed, heated, and examined. If the lines were
present in the spectrum, a little more of the bisulphide
was added, and this was continued until the lines just
disappeared, indicating complete solution. The density was
determined by noting the amount of fluid as measured by
the graduations, since, when the contents are homogeneous,
these values are proportional. Just before complete solution
the lines are so faint as to be invisible in the stationary
spectrum, but by moving the telescope to the right and to
the left, they could be detected, the eye being more sensitive
to a moving faint object than a stationary one. By using
this device, the density necessary to just cause the disappear
ance of the lines could be determined with considerable
accuracy.
The iodine was measured in the following manner. A
saturated solution in OS 2 was made at 12° and a capillary
pipette (p, fig. 2) was dipped into it. The fluid rose to a
certain height, which was marked. The iodine solution was
then washed out of the capillary into the small tube t by means
of a drop or two of CS 2 put into the wide top of the capillary.
This was immediately transferred to the graduated tube in
the manner described. The amount held by the capillary
426 Mr. R. W. Wood on the Absorption Spectrum of
pipette made only a small drop of the saturated solution, and
held 0*00031 gr. of iodine. This was determined by filling
and emptying the pipette ten times, and determining the
iodine volumetrically, with sodium thiosulphite.
The graduated tube was divided into 20 parts, and the
amounts of iodine that could be present with amounts of CS 2
varying from 1 to 10 divisions, without showing the iodine
gas spectrum, were determired. With the tube halffull of
CS 2 , it was possible to add iodine until the tube was quite
opaque ; consequently densities greater than *5 that of liquid
CS 2 could not be investigated.
The values found directly are not in shape for discussion,
since as we increase the density of the vapour we also
increase its amount ; in other words, we are working with
varying amounts of solvent as well as varying densties.
To reduce these varying amounts of solvent to unity is
very simple, and the results are given in the following table.
For the various densities 8 are given the amounts of iodine %
which can be mixed with 1 gram of the CS 2 vapour without
the lines appearing in the spectrum • or, in other words, the
amounts of iodine which 1 gram of CS 2 will dissolve at
different densities.
If the tube contain m grams CS 2 with m ! grams of iodine,
and if v be the volume of the tube in c.cm., then
8
m
X =
mf
m *
1 Gram CS 2 .
S(H 2 0=1)
•05
x>
iodine in grs.
•00300
•10
•00325
•15
•00350
•20
•00375
•25
•00440
•30
•00514
•35
•00600
•40
•00686
•45
•00741
•50
•00851
The values plotted on coordinate paper (Hg t 4), the 8
values as abscissae, the x as ordinates, show that the solvent
power increases rapidly with the density. Any point on the
.plane to the left of the curve represents a mixture which
shows the iodine lines in the spectrum, any point to the right
a mixture in which they are invisible. If the tube holds a
Iodine and Bromine above Critical Temperature, 427
mixture represented by some point on the curve, and a
portion be removed, B is thereby diminished, while % remains
unchanged. The point corresponding to this mixture lies to
Fig. 4.
•0085
•0080
•0075
•0070
•0065
•0060
•0055
•0050
•0045
•0040
•0035
•0030
WFJL,
h'
^m<
Mi
■n
YA
■B
iP^J
ss
•05 10 15 '20 25 "30
Iodine in CS 2 vapour.
•50
the left of the curve, and the lines should appear. This was
found to be the case.
Similar investigations were made with bromine, the absorp
tion spectrum of which is very similar to that of iodine. A
different method of measuring the halogen was adopted,
however. Seven drops of fluid bromine were brought into
428 Mr. R. W. Wood on the Absorption Spectrum of
the tube, and bisulphide of carbon added up to the 11th mark.
The absorption spectrum showed no trace of the lines. Half of
the contents was then removed, and the amount of bromine in
this determined. On sealing and reheating the tube the
lines were distinctly visible, as was to be expected, and CS 2
was added little by little until the lines just disappeared.
Half of this new quantity was then removed, and the same
process repeated. In this way data were obtained from
which the following table has been made. The curve is
shown in fig. 5.
1 Gram CS 2 .
0=1).
X, bromine in grs.
X (Bromine).
X (Iodine).
05
•0182
61
10
•0200
61
15
•0241
69
20
•0263
70
25
•0299
68
30
•0350
68
35
•0415
69
40
•0499
73
45
•0623
8*4
50
•0802
94
In the third column of the table is given the ratio of the
X values for bromine and iodine for the corresponding 8
values. A given amount of CS 2 vapour at any density from
•05 to '40 will dissolve from 6 to 7 times as much bromine
as iodine. Liquid bromine and CS 2 appear to be miscible in
all proportions.
These investigations show that to a certain amount of CS 3
vapour a certain definite amount of iodine vapour (depending
on the density of the OS 2 ) can be added without causing the
lines characterizing the spectrum of iodine vapour to appear.
If more iodine vapour be added, the lines are at once seen.
The conclusion that one naturally draws is that the iodine
molecules bind themselves in some way to the CS 2 molecules,
and are incapable of exercising the selective absorption
peculiar to the molecules of pure iodine gas. On this
supposition, we may look on the curves in figs. 4 and 5 as
solubility curves, and may consider, in a certain mixture of
CS 2 and iodine or bromine vapour, the halogen as existing in
two states, one part dissolved in the CS 2 vapour, and the
other free.
Hannay and Hogarth have shown (Proc. Roy. Soc. xxx.
pp. 178 & 484, 1880) that nonvolatile solids in solution are
Iodine and Bromine above Critical Temperature. 429
not precipitated when the solutions are heated ahove the
critical temperature, but remain dissolved in the vapour.
I have made a rather hasty quantitative investigation of
this phenomenon, and have obtained curves similar to those
obtained for iodine and bromine by the optical method. An
Fig. o.
•080
•075
•070
•065
■060
•055
•050
045
•040
•035
•030
•025
•020
~1
j
:■ 
1

'■'■ \: ..
■ : ; . 1
Wf^ : 
J
I !
; ■ 
■
1
1
1 1
 *>
■'
/
}
/
: '
c
/
:
i , "
j ' "_ :
£
I
7.
:'~ ■' ' ":
; " ;

m
r J


y^
r
w
.
•05 10 15 20 "25 30 35
Bromine in 0S 2 vapour.
•40
•45
•50
ethereal solution of Hgl 2 , and an alcoholic one of KI, were
used. In the first case the tube held a solution so strong,
that when the contents became homogeneous above the
critical temperature, a portion of the salt was thrown down
430 Absorption Spectrum of Iodine and Bromine'.
on the wall of the tube, which was then inverted quickly,
and cooled at the bottom by an airblast : the vapour con
densed here, and the salt remained above on the wall. The
Fig. 6.
•0070
•0065
•0060
•0055
•0050
•0045
•0040
•0035
•0030
•0025
BZfl
MfloM
•0020
•0015
•001()
•0005
>jB
Hi''
•05
•20
tube was then opened, and the amount of dissolved solid
determined. In the second case a more accurate method was
used ; *02 gr. of the salt (KI) was brought into the tube,
and alcohol added little by little until no precipitation occurred
above the critical temperature.
Substitution Groups whose Order is Four, 43l
The tube was heated in an airbath and illuminated by a
beam from an arc light, so that the faintest crystal film could
be easily observed, and the amount of alcohol necessary to
completely dissolve the salt in the gaseous state very
accurately determined.
Half of the contents was then removed, the tube resealed
and heated. A thick film appeared on the wall, which
corresponds exactly to the reappearance of the lines in the
spectrum, in the experiments with iodine and bromine.
More alcohol was added until the vapour had the density
requisite for the solution of this amount of salt.
The following values were reckoned for 1 gram of solvent
at different densities : —
1 Gram Ether.
1 Gram Alcohol.
8 (H0 2 =1).
X (Hgl 2 in
grs.).
5(H 2 0=1).
x (KI in grs.).
•020
•0010
•084
•00083
•053
•0016
•147
•00106
•080
•0024
•195
•00195
•114
•0050
•231
•00270
•133
•0068
•273
•00450
The curves plotted from these values (fig. 6) are quite
similar to the curves for iodine and bromine as determined by
the spectroscope, which is not unfavorable to the supposition
that the halogen vapour is in part dissolved in the vapour of
the bisulphide of carbon. If the dissolved substance is
volatile at the temperature used, as is the case with iodine
and bromine, the undissolved portion is in the state of a free
gas ; if nonvolatile, as in the case of Hgl 2 and KI, it is
precipitated as a crystalline film on the wall.
In conclusion I wish to thank Prof. Warburg for the means
of carrying on the investigations which he has placed at my
disposal and for the interest that he has taken in the work.
Berlin, Physikalische Institut.
XLV. The Substitution Groups whose Order is Four,
By G. A. Miller, Ph.D*
IT seems proper to say that Professor Cayley began the
enumeration of all the regular substitution groups of a
given order since he determined these groups for the first
order that presents any difficulties, viz., for the order 8f«
* Communicated by the Author.
t Phil. Mag. vii. (1854) pp. 4047 and 408409; xviii. (1859)
pp. 3437.
432 Dr. Gr. A. Miller on the Substitution
Later he gave a list of all the regular groups whose order
does not exceed 12 together with a geometrical representation
of them*. Kempe had previously given such a listf, but his
results were not quite correct.
Since all groups are isomorphic to regular groups J and
two distinct regular groups cannot be simply isomorphic §,
it is clear that the enumeration of all such groups within
certain regions is very important. Complete enumerations
for the first part of the two following series of orders have
been published : (1) when the order of the groups is the
product of a given number of prime factors  , and (2) when
it does not exceed a given number If.
Two more comprehensive enumerations with respect to
order may be mentioned, (1) the enumeration of all the tran
sitive groups of given orders, and (2) the enumeration of all
the groups of given orders. The latter of these includes the
former, and each of them includes the regular groups. It
may happen that the transitive groups of a given order are
also regular. This is, for instance, the case when the order
is a prime number, or the square of a prime number. When
the order is a prime number (p) there is one group for every
degree which is a multiple of p : L e. there are n groups of
order p whose degree does not exceed np, n1 of these are
transitive, n being any positive integer. It should be observed
that the number of the transitive groups of a finite order is
always finite, while that of the intransitive groups of any order
is infinite.
When the order of the groups is a composite number, the
problem of determining all the possible groups becomes more
complex. We shall confine our attention to the groups
whose order is four. Since none of the transitive constituents
of these groups can be of an odd degree, we see that the
degree of such a group must be even and not less than four.
We may therefore represent the degree by 2n.
To find all the cyclical groups of degree 2n we have
only to construct a 1,1 correspondence between a cyclical
tran sitive groups/ a. ^^ J and a 2,1 correspondence between
* American Journal of Mathematics, xi. (1889) pp. 139157.
t Phil. Trans, clxxvii. (1886) pp. 3743.
% Jordon, Traite des Substitutions, p. 60.
i § Netto, ' Theory of Substitution Groups ' (Cole's edition), p. 110.
i Holder, Mathematische Annalen, xliii. (1893) pp. 301413; Cole
and Glover, American Journal of Mathematics, xv. (1893) pp. 191221 ;
Young, ibid. pp. 124179.
If Miller, Comptes Rendus, cxxii. (1896) pp. 370372,
Groups whose Order is Four. 433
each one of these groups and a group of the second order
whose degree is 2ra — 4a. The number of such groups for a
given value of n is therefore equal to the largest value of a,
and the individual groups may be given by assigning to a the
successive integers beginning with unity.
To find all the noncyclical groups of degree 2n we may
construct a 1,1 correspondence between a fourgroups*, and
(1) a 2,1 correspondence between each one of these groups and
a group of the second order whose degree is 2n — 4a, (2) a
1,1 correspondence between each one of these groups and a
group of the fourth order which is of degree 2/i — 4a and
contains n — 2a systems of intra nsitivity. The number of the
groups of the first one of these two types is the same as that
of the cyclical groups, and the individual groups may be
given in the same way. The number of groups of this and
the cyclical type is therefore twice the largest value of a.
The groups of the second type of noncyclical groups
present somewhat greater difficulties. Here a may assume
the value zero in addition to its values in the two preceding
cases. We shall first determine the groups when a is zero ; •
i. e., we shall first seek all the
Groups which contain 2n Elements and n Systems of
Intransitivity .
The average number of elements in the substitutions of
such a group is nf, and the number of elements in all of its
substitutions is 4w. The number of systems of two elements
is therefore 2n. These 2n systems must occur in three sub
stitutions. If the smallest number of systems in any one of
these three substitutions is represented by S, we have
For each value of S which satisfies this relation there must
be at least one group, since we have only to use the remaining
systems for the second generating substitution in order to
construct such a group.
In general we have the following : —
* Bolza, American Journal of Mathematics, xi. (1889), p. 297.
t Frobenius, Crelle's Journal, ci. (1887) p. 287.
Phil Mag. S. 5. Vol. 41. No. 252. May 1896. 2 H
434
t)r. Q. A. Miller ore the Substitution
Value
ofS.
Number of
Groups.
Number of systems of two elements
in the substitutions.
1
1
1 n1
n
2
2
J2 7i2
\2 n1
n
n— 1
3
2
/3 n3
\3 n2
n
n1
f 4 n— 4
n
4
3
? 4 n3
n1
(4 n2
n2
(5 n5
n
5
3
^5 n4
n — 1
(5 n3
n2
•
•
• •
•
r
Tm n— ra
n
1 \ m 1 1
(m even) ~ + 1
m n — m + 1
1 : •
n1
m <
m
\m re 2
Tm n — m
 m n— m + 1
m
n
(m odd) — ~ —
n1
k.
m + 1
Im n 2
m — 1
re r
The groups for the same value of S are all distinct ; but it
may happen that two groups which correspond to different
values of S are identical. This can, however, not occur so
long as the value of S satisfies the relation
Q =n
b< 2*
Identical groups can therefore only occur when the value of
S is such that
n Q == 2n
2 (BGi )([t]2 +1 )
few 01 *)(G ] +2 )  1 11 ^(KHI]) 2 
By means of this formula we can readily determine the
number of groups which contain 2n elements and n systems
of intransitivity for any particular small value of n. The
individual groups may be found by assigning to S the
successive
integers from 1 to o and rejecting the identical
groups according to series (A) when the value of S satisfies
the relation
The groups of the second type of noncyclical groups,
which correspond to the other values of a, are found in
exactly the same way. Their number may therefore be
found by means of the given formula provided we use instead
of n the following series in order,
n— 2, Ti—4, Ti—6, . . ., 2 or 3. ,
By adding the double of the largest value of a to the sum of
the numbers of these groups corresponding to the different
possible values of a, we obtain the number of groups whose
order is four and whose degree is 2ti. Two of the groups
are transitive when n is 2. For the other values of n all the
groups are intransitive.
Example,
It is required to find all the groups whose degree and
order are 14 and 4 respectively.
* The brackets indicate that the largest integer which does not exceed
the inclosed fraction is to be used.
2H2
436 The Substitution Groups whose Order is Pour.
To find the number of these groups we observe that the
largest value of a is 3. Hence there are 6 groups of the first
two types. To find the number of groups of the third type,
i. e. of the second type of noncyclical groups, we assign the
following three values to n : —
7, 5, 3.
Hence we have for
n=7, 2.4(43) 2 =7,
n=5, 2.3l(32) 2 = 4,
n=S, 1.3(2l) 2 = 2.
The total number of groups is 6 + 7 + 4 + 2 = 19.
individual groups are given in the following list* : —
The
Number
1.
Groups.
{ (abcd.efgh.ijkl)cyc.(mn) )dim.
2.
{ (abcd.efgh) eye. (ij.kl.mn) }dim.
3.
{ (abed) eye. (ef.gh.ij.kl.mn) }dim.
4.
{ (abcd.efgh.ijkl)^(mn) }dim.
5.
{(abcd.efgh) ^(ij .kl.mn) }dim.
6.
{ (abed) ^(ef. gh.ij.kl.mn) }dim.
7.
(ah) (cd. ef.gh.ij.kl.mn) .
8.
(ab.cd) (ef.gh.ij.kl.mn).
9.
(ab.ed) (cd. ef.gh.ij.kl.mn).
10.
(ab.cd.ef) (gh.ij.kl.mn) .
11.
(ab.cd.ef) (ef.gh.ij.kl.mn).
12.
(ab.cd.ef. gh) (gh.ij.kl.mn).
13.
(ab.cd. ef.gh) (ef.gh.ij.kl.mn) .
14.
{ (abed) 4 [ (ef) (gh.ij.kl.mn)] } u .
15.
{ (abcd)±\_(ef.gh) (ij.kl.mn)] } hl .
16.
{ (abcd) 4 [(efgh) (gh.ij.kl.mn) ] } u .
17.
{ (abcd) 4 [ef.gh.ij) (ij.kl.mn)] \ u .
18.
{ (abcd.efgh)±[(ij) (kl.mn) ] } u .
19.
{(abcd.efgh\[(ij.kl)(kl.mn)] }j x .
* The notation is that which Professor Cayley used and explained in
his articles in the Quarterly Journal of Mathematics, vol. xxv.
Alleged Scattering of Positive Electricity by Light. 437
When only the number of the possible groups for a given
degree is required, and when n is a large number, it is very
desirable to avoid assigning so many different values to n as
are necessary if we employ the given formula. By observing
that all the fractions in this formula are increased by integers
when n is increased by 6, we may readily find the following
formula. By means of it we can find the number of
groups (N) directly for any value of n. m represents any
positive integer, and a x represents the largest value of a, i. e.
the largest integral value of x which satisfies the relation
n
r
When n=§nij
„ n=6m + l,
„ n=6m + 2,
N=m(3m 2 + 6m + l)
^r m(6m 2 + 15m + 5)
2
N = 3m(m + l)(m + 2) + l
„ ti= 6m + 4,
„ n= 6m + 5,
Hence there are
N=(m + l)(3m 2 + 9m + 4)
H _ 3(m + l)(2m 2 + 7m + 4 )
2
4(96 + 60 + 5) , _. „_ . , R .
— a + 24 = 64:6 groups of degree 50,
249.84.85 + 1 + 500 = 1,778,361 „ „ 1000, &c.
Ziiricli, Switzerland, March 1896.
XLYI. On the alleged Scattering of Positive Electricity by
Light. By J. Elster and H. Geitel *.
THE question whether light which facilitates the passage
of negative electricity from a conductor into the sur
rounding gas can, in like manner, accelerate the discharge of
positive electricity is not without significance for the proper
apprehension of the photoelectric process.
* Translated from the Ann. der Physih und Chemie, Bd. lyii. (1896) ;
from a separate impression communicated by the Authors.
438 Profs. J. Elster and H. Geitel on the Alleged
If, in fact, this action of light can be shown to take place,
then it is no longer possible to believe that we have here to do
with a specific phenomenon of the kathode, and, moreover,
the view that the photoelectric process depends upon the
discharge of the one (the gaseous) coating of an electric
double layer which is continually renewed at the surface of
contact between the conductor and the gas must be put to
experimental proof, since the nature of the electricity escaping
in light must always be the same as that which the gas in
contact with the conductor itself takes. From this point of
view, therefore, it would not be intelligible that one and the
same conductor in the same atmosphere should give off both
electricities more easily in light than in darkness. Now
experiment shows that the illumination of a negatively
charged surface, with proper choice of light and of substance
illuminated, causes an active discharge of electricity into the
surrounding gas, whilst the corresponding phenomenon for
positive electricity — if it takes place at all — must be much
more insignificant. Thus Hrn. Stoletow and Righi have not
Jbeen able certainly to recognize the action of ultraviolet
light upon positively charged surfaces, and we ourselves have
so far not been able to observe any loss of positive electricity
in light which was not sufficiently well accounted for by the
usual loss of electricity or by the sources of error to be more
definitely spoken of in what follows.
A paper by Herr E. Branly has recently appeared*, in
which the acceleration of the electric discharge by ultraviolet
light is maintained to hold good also for positive electricity.
On account of the importance of the subject we have repeated
the experiments described in this paper, and with the arrange
ments which seemed to us best suited to exclude sources of
error, and following the method of Herr Branly as closely as
possible in essential points. After we had failed in obtaining
the same result as Herr Branly, we tried whether the alkali
metals, which are so sensitive to ordinary light with negative
electrification, would show a photoelectric discharge also with
positive electricity. In what follows we venture to report
upon the results obtained in these experiments.
The most obvious method of observing the scattering of
electricity in light, which method was also employed by Herr
Branly, is to connect the electrified surface to be examined
with an electroscope, and to judge of the loss of electricity
produced by the light in a given time from the decrease in the
divergence of the leaves. This method has the disadvantage
that, on account of the high tension employed, the whole of the
* Compt. Bend. cxx. p. 829 (1895).
Scattering of Positive Electricity by Light. 439
electrified system of conductors must be extremely well
insulated so that a feeble action of the light may not be
hidden by the loss of electricity not connected with the action
of the light. But apart from this, there is a disturbing cause
arising from the fact that each time the observer changes the
sign of the electric charge, there is a return current from the
insulating supports of that electricity which had passed to
them during the previous electric condition.
Much less exposed to these sources of error is the arrange
ment which Herr A. Eighi and we ourselves have often em
ployed in photoelectric experiments, especially if it is necessary
to recognize feeble action. In this method the electrical
measuring apparatus, together with the conductor to be
illuminated, are at the commencement of the experiment at
zero potential, and the strength of the action of the light is
measured by the velocity with which the potential becomes
equal to that of a conductor kept at constant potential, which
stands opposite to the illuminated surface at a small distance
from it.
The arrangement of the experiment was as follows : — The
ultraviolet light was furnished by the spark of a condenser,
which was connected with the poles of an inductioncoil
actuated by 4 to 8 large Bunsen elements, the spark having
a maximum length of 18 centim. The current was broken
by means of a Wagner's hammer with platinum contacts,
and the spark of the condenser was taken between two alu
minium wires at a distance of 2 millim. The galvanic battery,
the inductioncoil, sparkspace, and all the necessary con
nexions were placed in the open air in front of the closed
window of the observing room. One of the panes of the
window was replaced by a plate of thin iron connected to
earth, which was provided with a circular opening, in which
was inserted a quartz lens of 50 millim. diameter. Since the
focus of this lens coincided with the spark, a parallel beam of
ultraviolet light was formed by the lens within the room,
whilst at the same time the electrostatic action of the induction
coil and of the electrified air from the spark was shut off from
the room. Within the room, at a distance of about 25 centim.
from the window and at right angles to the beam of light, was
placed a piece of ironwire gauze with a mesh of about
1 millim., and parallel to this, at a distance of 2 to 4 millim.,
the insulated plate of the substance to be examined. From
this a wire went to the quadrant electrometer (sensitiveness,
1 volt = 23 divisions), whilst the wire gauze was charged
to a potential of about 525 volts by a battery of several
hundred Leclanch6 cells. According as the wire gauze was
440 Profs. J. Elster and H. Geitel on the Alleged
charged with positive or negative electricity, the plate parallel
to it must become charged with negative or positive electricity.
If now the earth connexion of the electrometer was removed,
then as soon as a passage of electricity took place between
the gauze and the plate, the change of potential in the latter
could be read off on the electrometer. It is to be observed
that with a positive charge of the gauze, the plate to be tested
has negative electricity on its surface, and that therefore from
this clean metallic surface in ultraviolet light a free discharge
towards the gauze was to be expected and a consequent
positive charge of the plate. We used for the experiment a
disk of amalgamated zinc, also similar pieces of ziuc covered
with a thin layer of paraffin or tallow, and also a plate of
wood covered with tallow. According to Herr Branly, such
surfaces covered with paraffin or tallow suffer a greater loss
of positive than of negative electricity when exposed to light.
"We observed the deflexion of the electrometerneedle
which took place in one minute, both with positive and with
negative charge of the gauze, and both in the dark and when
illuminated with light from the spark. The Wagner's hammer
was so arranged that it came into action of itself upon closing
the current, it was therefore only necessary to keep the
current closed for one minute. The results of a series of
measurements are brought together in the following table.
The numbers give the change of potential of the plate,
measured in volts, which took place in one minute : each
number is the mean of two readings : —
Amalgamated
Zinc Plate.
Paraffined
Zinc Plate.
Zinc Plate covered
with Tallow.
Wood Plate
covered with
Tallow.
Dark.
Illumi
nated.
Dark.
Illumi
nated.
Dark.
Illumi
nated.
Dark.
Illumi
nated.
Gauze T
positive J
Gauze \
negative J
+040
016
+138
(+123)
040
+054
022
+069
016
+079
009
+038
006
+058
052
+052
052
As was to be expected from what has been said, we have in
this series of observations evidence of the great photoelectric
dispersion from a plate of amalgamated zinc charged with
negative electricity.
We were not able to expose the plate for a full minute to
Scattering of Positive Electricity by Light. 441
the light, as the deflexion of the electrometer on the scale
could not then be read off. Therefore the plate was exposed
only five seconds, and the deflexion — reduced to volts — was
multiplied by 12*. But beside this action of the light —
undoubted and already known — the numbers show no other.
There are, it is true, deflexions of the electrometerneedle in
the dark as well as in the light, which, however, in no case
reach the amount of one volt, and which on account of their
inconstancy are to be referred to an irregular passage of
electricity from the gauze to the plate, probably caused by
the dust of the air.
Only in two cases is this feeble transference of electricity
slightly greater in light than in the dark, viz. with a nega
tively charged gauze opposed to a plate of amalgamated zinc,
and with the gauze positive and the plate of paraffined zinc.
If one wishes to find in this a proof of an action of light,
then only the first case can be taken to show a photoelectric
dispersion of positive electricity. But here also a sufficient
explanation is to be found in the fact that the ultraviolet
light reflected from the polished surface of the amalgamated
zinc strikes the side of the gauze turned towards it and pro
duces a passage of negative electricity from it to the plate,
so that in this case the photoelectric discharge is not from the
positively charged plate but from the negatively charged
gauze.
We have now to describe an experimental arrangement in
which this action of the reflected light appears perfectly
clearly. All the observations show that the paraffined or
greased surfaces are not photoelectrically sensitive ; in no
case is the scattering of electricity from these found to be
greater with a positive charge than with a negative charge
or in the dark.
The small deflexions of the electrometer observed in the
* A more accurate calculation of the change of potential during an ex
posure of one minute would be obtained by use of the formula
hit.
V Vl =V.e
in which V denotes the potential of the charging battery, V, that of the
illuminated plate, k a constant, J the intensity of the light, and t x the
time of exposure. From this we should have for two times of exposure
t x and t 2 and the corresponding potentials v x and v 2 :
M^HM^}
from which v 2 can be easily calculated. In the foregoing case we obtain
for v 2 the value shown in brackets, J 123 volts.
442 Profs. J. Elster and H. Geitel on the Alleged
dark, which always indicated a loss in the charge of the
gauze, show that the instrument is too delicate for experi
ments such as these, in which a thin plate of air is exposed to
a fall of potential of more than 100 volts per millimetre. We
have therefore repeated these experiments with the much less
sensitive aluminiumleaf electroscope, and were able to make
the charge of the gauze and the time of exposure twice as
great as before. But then also the photoelectric discharge
took place only when the gauze had a negative charge.
Thus the charge of an amalgamated zinc plate rose in fivo
seconds to 400 volts, of an oxidized zinc plate in two minutes
to 190 volts, and with greased or paraffined zinc or wood
plates the potential remained at zero, irrespective of the sign
of the charge of the gauze.
From the result of this experiment we concluded that in
the experiments of Herr Branly some unsuspected source of
error must have existed. In order to discover what this may
have been, we have repeated the experiment of Herr Branly,
essentially according to his arrangement as far as was possible
from the data which he gives. The sparks of the induction
coil were taken within a box of sheetiron connected to
earth, in the side of which there was a quartz window.
Opposite to this was the insulated and electrified plate con
nected with an aluminiumleaf electroscope. Since there
was no gauze placed in the way of the light rays, any elec
tricity escaping from the plate must be lost in the air or partly
pass to the side of the iron box, and from there pass to the
earth.
So long as the plate was some distance (about 50 centim.)
from the quartz window, we also observed with this arrange
ment an increase of electric dispersion in light with a negative
charge. But if the plate is brought to within a few centi
metres of the window, and, consequently, near to the box, it
may happen, if the surface of the plate is covered with tallow
or with paraffin, that a positive charge decreases in light more
rapidly than a negative charge. But here, as in the above
discussed analogous case, it is to be remembered that the
positive charge of the plate collects negative electricity upon
the side of the box turned towards it by induction ; if there
fore this surface is struck by the ultraviolet light reflected
from the surface of the layer of fat, a passage of negative
electricity from it to the plate must result, and give the
same effect as if a photoelectric dispersion of positive elec
tricity from it had taken place.
This suspicion was converted into certainty by the obser
vation that the phenomenon is dependent upon the nature of
Scattering of Positive Electricity by Light. 443
the surface of the box. If this is covered with bright metal — as
tinfoil — the transference of electricity largely increases, and
becomes strikingly great if a piece of amalgamated zinc is
placed upon it.
Since Herr Branly gives no information as to the distance
of the illuminated plate from the quartz window of the metal
box, we may consider it not improbable that this was chosen
too small, and that the dispersion of positive electricity
observed by him in ultraviolet light was caused by the
deceptive action of the light reflected from the surface of the
electrified disk.
We believe that we are justified, by the results of the
experiments described, in asserting that an increase in the
dispersion of positive electricity by illumination of the elec
trified surface by ultraviolet light has not been proved.
The striking inability of surfaces of alkalimetal to retain a
charge of negative electricity in ordinary light might suggest
that a possible action of light with a positive charge might be
expected to take place most readily with such surfaces. As
we have mentioned in a previous paper *, exhausted glass
globes of which the one electrode is formed by an alkali
metal, the other by platinum, also allow a photoelectric
current to be observed in more or less distinct manner when
they are reversed, i. e. when the alkalimetal forms the
positive pole. But we had also arrived at the conclusion that
in this case the photoelectric action had its seat not at the
surface of the alkalimetal, but at the platinum electrode.
There is, in fact, formed upon the platinum a superficial layer
by condensation of the vapours of the alkaline metal from
which in fight negative electricity passes to the anode. By
heating the platinum wire with a galvanic current this layer
is volatilized, and the photoelectric cell becomes — in its
reversed arrangement — for a short time insensitive to light.
When we recently repeated this experiment with better
arrangements and greater care, we found that after the wire
had been heated there remained n small amount of sensitive
ness to light, which, perhaps, had its origin in a scattering of
positive electricity from the surface of the alkalimetal. It
seemed important to determine the seat of this action without
doubt, whether anode or kathode.
We started from the observation that the current as usually
produced by illumination of the kathode is dependent upon
the position of the plane of polarization of the incident light
with respect to the surface of the kathode f. It was to be
* Elster and Geitel, Wied. Ann. xliii. p. 236 (1891).
f ? Cf. Elster and Geitel, Wied. Ann. Hi. p. 440 (1894).
444 Alleged Scattering of Positive Electricity by Light.
expected that any discharge possibly produced by light at
the anode would also be in some way dependent upon the
direction of the light vibrations toward the surface of the
anode. The experiments made in this direction gave, how
ever, a negative result : if we allowed a ray of light to
fall through a Nicofs prism upon the fluid surface of the
sodiumpotassium alloy which formed the anode, and altered
the position of the plane of polarization by turning the Nicol,
we found the photoelectric dispersion to be independent of
the azimuth of the light.
It must be remarked that in the " reversed " arrangement
of the cell here employed the current strength, even in strong
light, is far too small to give a measurable deflexion, even on
the very sensitive galvanometer which we employed to measure
the photoelectric currents. We therefore employed the same
method which we had used in the experiments upon ultra
violet light, i.e. we connected the alkalimetal surface with
the positive pole of the abovedescribed battery, and the
opposed platinum electrode with the quadrant electrometer.
The passage of electricity through the cell betrayed itself
then by the constant increase in the deflexion of the electro
meterneedle. A constant condition of the instrument may
be attained by making an earth connexion through a very
large resistance (a pencil mark on an insulating surface) to
the wire leading to the electrometer.
Not only does the fact that the transference of electricity
is not affected by change in the direction of the light vibrations
with respect to the plane of the anode prove that the seat
of photoelectric action is not at the anode, but we have the
further evidence that this action is perceptibly increased if
the platinum wire which serves as kathode is exposed to the
direct action of the light. It even continues of the same
intensity when by inclining the bulb the alkalimetal is made to
flow over into the side bulb and is thus removed from the cell*.
Since a clean platinumwire in a vacuum shows no photo
electric action in ordinary light, its sensitiveness can only
have been communicated to it by contact with the alkali
metal or its vapour. As we see, the result of this experiment
also leads us to the conclusion that the light has acted not on
the anode of alkalimetal but on the platinum kathode made
sensitive by its superficial coating, and we might expect that
the " reversed " cell would lose its sensitiveness upon ignition
of the platinum wire. But, as we have said, this expectation
was, curiously enough, not verified by experiment. There
* The form of the cell is shown in fig., Wied, Ann, xlii. p. 564 (1891) j
see also Phil. Mag. 1896, xli. p. 220.
Tinfoil Grating Detector for Electric Waves. 445
remained then only the supposition that the inner glass wall of
the bulb had become covered with a layer, by contact with
the alkalimetal, from which negative electricity escaped when
the light fell upon it. In order to remove this source of error
also the whole wall of the cell must be maintained at the
same potential as the anode of alkalimetal, so that there could
be no fall of potential from it to the wall. We attained this
result by covering the outside of the bulb with silver by pre
cipitation; with the exception of a small space where the
kathodewire was melted into the bulb and a " window " for
the entrance of the light.
If now the alkalimetal surface and, with it, the glass wall
of the bulb was charged with positive electricity, and the
kathodewire was connected to earth, then immediately after
ignition of the wire no loss of electricity occurred in light,
not even when a beam of sunlight entered through the window
in the silver coating. Eot until after some time, when the
wire on cooling had again covered itself with a coating of
alkalimetal, could the photoelectric discharge be again ob
served with increasing distinctness.
Thus the experiments with ordinary light on surfaces of
alkalimetal in a vacuum lead to the same result as those with
ultraviolet light, namely, that the photoelectric action is
limited to the kathode.
We have pleasure in gratefully acknowledging the assistance
we have received in this work from the Elizabeth Thompson
Science Fund in Boston.
XLVII. The Tinfoil Grating Detector for Electric Waves. By
T. Mizuno, Rigakushi, Professor of Physics, First Higher
Schools, Tokio*.
§ 1. TNa paper f, which not long since I communicated to
JL this Journal, I suggested that the change of the
resistance of the grating might be due to a mechanical effect
exerted upon it by impinging trains of electric waves. In
other words, electric waves might give impulses to some of
the strips of the grating in such a way as to let leaflets on
their margins come in contact with one another, thereby
causing a diminution of resistance. In order to confirm this
view, further inquiries were carried out soon after the com
munication of the abovementioned paper.
* From a separate impression from the Journal of the College of
Science, Imperial University, Tokio, Japan, vol. ix. part 2. Communicated
by the Author.
t "Note on Tinfoil Grating as a Detector for Electric Waves,"
Phil. Mag. vol. xl. p. 497 (1895).
446
T. Mizuno on the Tinfoil Grating
§ 2. Having constructed about forty gratings and tested
their action, I found to my surprise that while some were
extremely sensitive, others were not, being even utterly
indifferent to the impulses of electric waves, although they
had all been prepared with the same care and apparently with
the same success.
This led me to undertake a closer examination of such
gratings, which gave results that throw much light upon
their nature. But before these results can be stated, it is
necessary to describe in detail my way of preparing the
gratings, because upon that their sensibility wholly depends.
§ 3. The face of a flat wooden block of convenient size, say
10 centim. on a side, was pasted over with very fine tinfoil, as
described in my former paper.
Then came cutting lines into the tinfoil, to which particular
attention was given. Along the edge of a bamboo ruler a
sharp knife, held always inclined away from the ruler, was
drawn lightly across the surface of the tinfoil. In this way
many fine parallel slits were cut in the tinfoil,
so as to make one continuous, regular, zigzag
line, as shown in fig. 1.
A few of the gratings, thus carefully pre
pared, were found to be sensitive. But
experience has taught me that success in
preparing good detectors depends, to a large
extent, upon the nature of the wood block on
which the tinfoil is pasted in the first place,
and next upon the degree of adhesion of the
foil to the wood. A soft wood is preferable
to a hard one, and the paste used should not
be thick enough to make the foil adhere too firmly.
§ 4. The majority of the slits of the sensitive gratings, when
examined under a microscope, presented such an appearance
as that shown in fig. 2. A B and C D represent
two strips of foil with the very narrow slit or gap
a b between them that has been formed by the
knife. The shaded portion indicates the slope of
the tinfoil found at one edge of each strip.
For the sake of clearness, there is shown in
fig. 3 an end view, that is, a section of the two
strips perpendicular to their lengths. The shaded
portions indicate the tinfoil strips, A B and D in
fig. 2, of which the edge of one strip, C D, extends
some distrance into the gap, a b, and forms the
slope mentioned above. Along this slope the tin
foil presents many folds or wrinkles, which seems
Fig. 2.
ArT
Detector for Electric Waves. 447
to show that the tinfoil strip was somewhat stretched along
its edge by the act of cutting it. Nonsensitive
gratings showed none of these characteristic Fig. 3.
appearances, but had the gap between the strips I /a ,
much wider, with no decided slope and no Lrn^ L^
appreciable folds along the edges of the strips.
Hence for a grating to be sensitive, it appears to be necessary
that the gaps should be narrow and their margins sloped and
in folds.
§ 5. Although I have been unable to see clearly the interior
of a gap, yet it is quite reasonable to assume that in sensitive
gratings there will be numbers of leaflets along the margins
of adjacent tinfoil strips ; and the existence of such leaflets
once admitted, the explanation of the action of the gratings
becomes clear. For in a properly constructed grating some
of the leaflets may easily come in contact with one another
under the action of the electric waves, because of the extremely
small distance between any two opposite leaflets in the narrow
gap. Then, too, it seems to me that these leaflets must be of
various dimensions and, accordingly, some of them will be
extremely sensitive, others less so but still highly sensitive,
others again only moderately so. This being the case, the
amount of change in the resistance of the grating must depend
upon the intensity of energy of the impinging electric
oscillations, for, when it is not great enough, only the most
sensitive leaflets will come into play, but when it is sufficiently
great all the effective leaflets will be brought into action.
All the experiments I have yet made are in agreement with
this representation of the matter.
§ 6. A grating, well prepared so as to fulfil the conditions
mentioned above, proves to be an extremely sensitive detector
for electric waves, as will be seen from the experiments which
I now describe.
Experiment 1. A Hertzian parabolic vibrator, ABC, was
placed horizontally with aperture turned upwards, as shown
in fig. 4. The aperture was
covered with a sufficiently large Fig. 4.
wooden plate, ADC, entirely
coated with tinfoil. A grating,
whose initial resistance was about
71 ohms, was placed at about
5 centim. from the plate and in
a vertical line with the primary
conductor, 0, radiating electric
waves of 60 centim. wavelength.
Then, exciting the primary oscillations, I always found that
448
T. Mizuno on the Tinfoil Grating
R* 5.
the resistance of the grating was diminished by from 1 to
nearly 2 ohms.
The experiment was repeated after raising the plate, ADC,
parallel to itself and keeping it at some height from the
aperture, AC. Similar changes of resistance were also ob
served in this case. This phenomenon may of course be
understood by considering the fact that some electric waves,
which pass out of the uncovered portions of the parabolic
vibrator, will, after going through the room and being reflected
from the surrounding walls, ceilings, &c, come back ultimately
to the grating in a much enfeebled state.
Experiment 2. The above experiment was modified by
placing on the plate, A D C, a
zinc box, abed, 17 centim. by
27 centim., without top or bottom
and putting the grating inside it.
In this case also, a change of
resistance was observed, though
smaller. It is then certain that
although the side effects were
got rid of, the top effect still
remained, through which traces
of waves might affect the re
sistance of the grating. The
fact that we can annul the change of resistance by completely
closing the top of the box with a metallic plate, seems especially
to favour the above explanation.
Experiment 3. The grating was connected with the Wheat
stone bridge by means of two leading wires, and at the same
time placed inside the zinc box, just as in Exp. 2. After
balance had been well established and the top of the box closed,
the primary oscillations were excited. This time, the balance
was at once destroyed and the resistance of the grating showed
an appreciable diminution, in spite of the fact of the grating
being wholly enclosed in a metallic box. Taking away one
of the leading wires the phenomenon yet remained the same,
though the change of resistance seemed somewhat smaller
than in the former case. The leading wires thus appeared to
catch up electric oscillations and guide them to the grating.
Hence in experiments with electric waves it is necessary to
keep the grating free from any exposed wires, which might
easily take up electric disturbances. Such effects due to
leading wires were observed also by Herr Aschkinass during
his researches with these gratings.
§ 7. To what extent the sensibility of the grating reaches
Detector for Electric Waves. 449
will now be quite clear from the results of the above experi
ments. It is next of great importance to describe some
experiments as to the variation of the sensibility. In its
primitive state, the grating properly constructed is so sensi
tive that it can detect even the smallest electric oscillations.
But after having been used a few times, its sensibility under
goes a sudden and decided diminution, and then remains
nearly constant. At first, when the grating is exposed to
electric waves and its resistance consequently diminished, a
single tap given to it is almost enough to restore the resistance
to its initial or primitive value. But when we have used the
grating repeatedly, we find it necessary to give it a greater
number of taps to effect this restoration. Later on, when the
sensibility has diminished to a certain value, it seems to retain
that value without any decided further change for a long time.
This variation in the sensibility may be accounted for in the
following way : — As mentioned in § 5, the effective leaflets
along the margins of the several tinfoil strips may be of
different sizes, and some of them possibly very small. The
smaller the leaflets the more sensitive to electric disturbances,
and consequently the more liable to fatigue will they be.
Hence in the primitive state such leaflets are easily affected
by even very weak electric impulses, but soon lose this
sensibility as a result both of the repeated electric distur
bances and of the mechanical taps given to them each time.
§ 8. Though the sensibility of the grating thus always
diminishes to a certain extent by a little use, still it is even
in such a state far superior to that of an ordinary Hertzian
resonator. Even where the latter fails, the grating always
shows the presence of electric waves if there be any. Experi
ments on the nature of electric waves, namely, on rectilinear
propagation, reflexion, refraction, diffraction, polarization, &c,
can all be easily carried on by means of a properly constructed
grating. Moreover, such a grating gives not only qualitative,
but also quantitative results, to a certain extent, because the
amount of diminution of the resistance depends upon the
quantity of energy of the impinging waves. Hence, I believe,
it may prove to be of great advantage to make use of such
gratings in all lecture experiments as well as in laboratory
researches on electric waves.
In conclusion I wish to express my thanks to Mr. U.
Takashima for the kind and earnest assistance he has given
me in the preparation of many of these gratings and in carry
ing out researches upon them.
Phil Mag. S. 5. Vol. 4L No. 252. May 1896, 2 I
[ 450 ]
XLYIII. Carbon and Oxygen in the Sun,
By John Trowbridge*.
IN 1887 Professor Hutchins, of Bowdoin College, and
myself brought forward evidence to show that the peculiar
bands of the voltaicarc spectrum of carbon can be detected in
the sun's spectrum. They are, however, almost obliterated by
the overlying absorptionlines of other metals, especially by the
lines due to iron. In order to form an idea of the amount of
iron in the atmosphere of the sun which would be necessary
to obliterate the banded spectra of carbon, I have compared
the spectrum of carbon with that of carbon dust, and a defi
nite proportion of iron distributed uniformly through it. The
carbon dust and iron reduced by hydrogen was formed into
pencils suitable for forming the arcf. Chemical analysis
showed that the iron was uniformly mixed with the carbon
dust ; specimens taken from different sections of the terminals
showed in the carbons which I burned in the electric arc
28 per cent, of iron and 72 per cent, of carbon.
The method of experimenting was as follows : — That por
tion of the spectrum of the sun which contains traces of the
peculiar carbon band lying at wavelength 3883*7, and which
had been almost obliterated by the lines of absorption of other
metals, among them those of iron, was photographed. The
pure carbon banded spectrum was photographed on the same
plate immediately below the solar spectrum, and the spectrum
of the mixture of iron and carbon immediately below this.
The sun's spectrum can be regarded as a composite photo
graph, and the iron and carbon can also be regarded as a
composite photograph. It was speedily seen that from 28 to
30 per cent, of iron in combination with 72 to 70 per cent, of
carbon almost completely obliterated the peculiar banded
spectrum of carbon. This proportion, therefore, of iron in
the atmosphere of the sun, were there no vapours of other
metals present, would be sufficient to prevent our seeing the
full spectrum of carbon.
The iron in the carbon terminals which I employed greatly
increased the conductivity, as will be seen from Table I.,
which was obtained in the following manner.
The carbons were separated by means of a micrometer
screw, and the current and difference of potential were mea
* Communicated by the Author.
t I am indebted to Mr. John Lee, of the American Bell Telephone Co.,
for his skill in making the carbons and for analysis of the composite
carbons.
Carbon and Oxygen in the Sun. 451
sured with different lengths of arc. Table I. gives the
results for pure carbon ; Table II. for 28 per cent, of iron
and 72 per cent, of carbon.
Length of Arc
in millims.
1
Table I.
Amperes.
27
Volts,
25
2
23
24
3
225
20
4
20
18
5
165
15
Length of Arc
in millims.
1
Table II.
Amperes.
305
Volts.
30
2
30
3
275
28
4
24
25
5
22
20
6
20
20
7
18
19
8
16
18
The length of the arc could be nearly doubled with the
same current and the same voltage by the admixture of 28 per
cent, of iron. The light was apparently greatly increased,
but the difference in colour between the pure carbon light
and the ironcarbon light made measurements unreliable.
Moissan* has shown that the carbon in an electric oven
through which powerful electric currents have flowed is free
from foreign admixtures. Deslandres has confirmed this,
and finds only a trace of calcium present. The self purification
comes from a species of distillation of the volatile imparities.
The purest carbon is found at the negative pole. The
light of the electric furnace is due to the combustion of
carbon. Can we conclude that the sun is a vast electric
furnace ?
If the voltaic arc is formed in rarefied air or under water, its
* Comptes Bendus, cxx. pp. 12591260 (1895).
212
452 Prof. J. Trowbridge on Carbon
ty
brilliancy diminishes greatly. On the other hand, an atmo
sphere of oxygen greatly augments its vividness. The question
therefore whether oxygen exists in the sun is closely related
to questions in regard to the presence of carbon, when we
consider the temperature and light of the sun.
If suppositions also are made in regard to the magnetic
condition of the atmosphere of the sun, it is of great interest
to determine whether oxygen exists there, for oxygen has
been shown by Faraday, and later by Professor Dewar, to be
strongly magnetic.
Professor Henry Draper brought forward evidence to prove
the existence of bright oxygen lines in the solar spectrum.
Professor Hutchins, of Bowdoin College, and myself examined
this evidence, and after a long study of the oxygen spectrum
in comparison with the solar spectrum, came to the conclusion
that the bright lines of oxygen could not be distinguished in
the solar spectrum. We published our paper in 1885. I have
lately studied the subject from another standpoint ; having
carefully examined the regions in the solar spectrum where
the bright lines of oxygen should occur, if they manifest
themselves, in order to see if any of the fine absorptionlines of
iron in the spectrum of iron were absent, for it is reasonable to
suppose that the bright nebulous lines of oxygen would
obliterate the faintest lines of iron.
The method adopted by Draper for obtaining the spectrum
of oxygen consisted in the employment of a powerful spark
in ordinary air. To obtain this spark, the current from a
dynamo running through the primary of a Ruhmkorf coil was
suitably interrupted. By the use of an alternating machine
and a stepup transformer, powerful sparks can be more
readily obtained. Since the time of exposure with a grating
of large dispersion is long, considerable heat is developed in
the transformer from tbe strong currents which are necessary
to produce a spark of sufficient brilliancy. I have therefore
modified the method in the following manner. The spark
gap is enclosed in a suitable chamber, which can be exhausted.
When the exhaustion is pushed to a certain point, the length
of the spark can be increased ten or twenty times over its
length in air, and a suitable spark for photographic purposes
can therefore be obtained by the employment of far less energy
in the transformer. A pressure of eight to ten inches of
mercury in the exhausted vessel is sufficient. A quartz lens
inserted in the wall of the exhausted chamber serves to focus
the light of the spark on the slit of the spectroscope.
The following table gives the certain oxygen lines and iron
lines in the same region of the spectrum : — ■
and Oxygen in the Sun. 453
O. Fe in Sim. Intensity.
4631 462944 1
473022 4
463091 1
463161 1
4656 46547
465771
4683 468304 1
468393 2
46015 460009 In
46018 1
460211 4
4607 460484 In
460552 1
460632
460779 6
4613 461138 8
461335 4
4614*29 1
46935 469152 6
469497 1
The faintest iron lines are therefore not obliterated in the
spaces where the oxygen lines should occur.
If we examine the great absorption region about the In
line, we find that between wavelengths 393029 and 3938*55
Rowland gives eight lines which coincide with iron lines.
From the table of wavelengths of iron lines in the arc
spectrum given in the Report of the British Association for
189.1, 1 find the following lines given between these limits : —
393037 *
393122 *
393271*
393301 *
393375
393447
393481 *
393540 *
393592 *
393742 *
393816
393859
The starred lines are probably the iron lines given by
Rowland in his list of standard solar lines. The iron lines
that are not starred apparently are obliterated in the great
absorptionband near the calcium line K.
454 H. Nagaoka and E. T. Jones on the Effects
Lord Salisbury, in his address before the British Association
at Oxford, 1894, remarks : — " Oxygen constitutes the largest
portion of the solid and liquid substance of our planet, so far
as we know it ; and nitrogen is very far the predominant
constituent of our atmosphere. If the earth is a detached bit
whirled off the mass of the sun, as cosmogonists love to tell
us, how comes it that in leaving the sun we cleaned him out
so completely of his nitrogen and oxygen that not a trace of
these gases remains behind to be discovered even by the
sensitive vision of the spectroscope?"
Although we have not succeeded in detecting oxygen in
the sun, it seems to me that the character of its light, the
fact of the combustion of carbon in its mass, the conditions
for the incandescence of the oxides of the rare earths which
exist, would prevent the detection of oxygen in its uncombined
state. Notwithstanding the negative evidence which I have
brought forward, I cannot help feeling strongly that oxygen
is present in the sun and that the sun's light is due to carbon
vapour in an atmosphere of oxygen.
Jefferson Physical Laboratory,
Harvard University, Cambridge, Mass., U.S.
XLIX. On the Effects of Magnetic Stress in Magnetostriction,
By H. Nagaoka and E. Taylok Jones*.
MATHEMATICAL expressions for electric and magnetic
stresses were given first by Maxwell in his ' Electricity
and Magnetism.' In Art. 105 of this treatise the values are
found of stresses in a dielectric medium, which may be re
garded as producing the observed electric action between two
systems ; in Art. 644 corresponding expressions are found
for a magnetic medium. The expressions in the two cases
are, however, not similar ; i. e. the electric stress in a dielectric
cannot be deduced from the magnetic stress in a magnetic
substance simply by substituting specific inductive capacity
for magnetic permeability. It does not seem quite clear
whether Maxwell intended his expressions to apply to the case
of induced as well as rigid magnetization.
In 1881, v. Helmholtz f published expressions for the stresses
which are more general than those of Maxwell, since they
contain terms depending on possible changes of density in
the medium. These expressions were assumed to have the
same form for a dielectric as for a temporarily magnetized
substance, but not necessarily for a permanent magnet ; and if
* Communicated by the Authors.
t Wied. Ann. xiii. p. 400 (1881) ; or Abh. i. p. 798.
of Magnetic Stress in Magnetostriction. 455
the terms depending on changes of density are neglected, the
expressions reduce to those given by Maxwell in Art. 105.
In 1884, Kirchhoff* gave a still more general theory, in
cluding in his equations terms depending on elongations in
the direction of the electric or magnetic force, as well as
terms depending on the change of density. The fundamental
relations between intensity of magnetization I (A, B, C) and
resultant magnetizing force H («, ft, 7) are thus : —
C. \d^ oy oz J dxj
(. Xdx d# ^z / dyJ
(. \ox oy dz ) qz J "
where u, v, w are the component displacements of a particle
of the medium at (x, y, z) in the directions of the axes of co
ordinates, and K, K 7 , K" are coefficients depending on the
nature of the medium. It is to be noticed that the quantities
in the brackets are nearly equal to the susceptibility, because
the elongations and change of volume are considered to be
very small. From these and the principle of Conservation of
Energy, Kirchhoff deduces his general expressions (in Max
wellian notation) for the stresses in a substance magnetized
by induction : —
P " = (^ +K+ T) a2 + *(^ + K  K ') (a2+/32+72) '
p " = ~(s +K+ T ) ^ + *(s + K  K ')(« 2 +/3 2 +r 2 ),
/ 1 W\
P ye = P^=(i+K+^j/S 7 ,
/ 1 K"\
/ 1 K" \
P^=P sx = (i+K+^)^.
These expressions reduce to those of v. Helmholtz if it be
assumed that K" = 0, and to those of Maxwell for electric
stress (Art. 106) if we put K'=K"=0 ; and 144ttK = specific
inductive capacity of the medium.
It will be seen from these expressions that KirchhofF's
* Wied. Ann. xxiv. p. 52 (1885) j Ges. Abh., NacMrag, p. 91 (1891).
456 H. Nagaoka and E. T. Jones on the Effects
tbeor}^ makes the stress depend principally on the coefficients
K7 and K". The experiments of Villari, Lord Kelvin, and
Ewing on the effects of longitudinal stress on magnetization
show that
.&*+*>)**• Y °tr 8 = (t + *")
in iron is a quantity which may amount to 10 5 for moderate
magnetizations, so that in this case K is quite negligible
in comparison with one or both of the two other coefficients.
The preponderating influence exercised by these latter factors
will, perhaps, explain the existence of a maximum elongation
in iron and the continual contraction in nickel. Maxwell's
expressions for magnetic stress (Art. 644), in the case when
B and H have the same direction, are, however, apparently
by a coincidence deduced from the above expressions of
Kirchhoff by putting K /x = 0, K' = K = susceptibility.
About the same time Lorberg*and J. J. Thomsonf discussed
the present problem in a manner similar to that of Kirchhoff.
More recently Hertz  arrived at expressions of precisely the
same form as those given by v. Helmholtz. In comparing
his expressions with those given by Maxwell for the general
case of a magnet in which the induction and magnetic force
have different directions, Hertz says (p. 281) : —
" A difference of far greater importance (i. e. than the effect
of a change of density by electromagnetic strain) is that in
Maxwell's theory the tangential stresses V xy and V yx have
different values, while in our theory they are identical.
Under our system of stresses every material element, when
left to itself, will only change its shape ; under that of
Maxwell it will also experience a rotation as a whole. The
Maxwellian stresses cannot therefore arise from processes in
the interior of the element, and can have no place in the
present theorv. They are, however, admissible on the assump
tion that in the interior of the body in motion, the aether
remains at rest and furnishes the necessary fulcrum for the
rotation which takes place."
In attempting to calculate the change of dimensions of a
body due to magnetization we are at once placed face to
face with the question as to whether these stresses actually
* Wied. Ann. xxi. p. 300 (1884).
t Application of Dynamics, §§ 3537 (1886).
\ Ausbreitung der electrischen Kraft, p. 275 (Leipzig, 1892).
Hertz, speaking of equation (6 c), p. 284, which is obtained as a
simplified form of stress agreeing in the general case with that found by
v. Helmholtz, says that the stress is the same as that given by Maxwell
in Art. 642 of the treatise. His expression, however, is not that given in
Art. 642 but in Art. 105.
of Magnetic Stress in Magnetostriction. 457
exist in the body, or whether they, existing in a medium in
which the particles of the body are imbedded, produce modi
fied stresses in the body.
Kirchhoff supposed the surface tractions acting on a piece
of magnetized soft iron to be the same as if the stresses P xx
actually existed in the iron and in the surrounding air, where
K, K/, K ;/ are put =0, and proceeded to calculate the changes
in the dimensions of a soft iron sphere placed in a uniform
magnetic field, due to a system of stresses which satisfy the
ordinary equations of an elastic solid and at the surface of
the sphere have the above given values.
Having obtained the general solution for the strain of a
sphere, Kirchhoff gives a numerical example, neglecting the
terms affected with (K — K/) and K", supposing these quan
tities to be very small in comparison with K 2 . Kirchhoff's final
value for the elongation of a soft iron sphere is therefore pre
cisely the same as that which would be given by Maxwell's
system of stresses (Art. 644) .
Proceeding exactly on the lines of Kirchhoff, Cantone*
has calculated the variations SI and &v of length and volume
of a soft iron ellipsoid of revolution, and finds that for an
ellipsoid of great eccentricity and length /,
S/f_ P f KK' K" ■)
I ~E\ 7r+ 4K 2 2K 2 >'
Bv F f , 3(KKQ K" \ . : ■
where E = Young's Modulus for the iron (on the supposition
that the Poisson ratio = J) . He also measured experimentally
the changes of length and volume of a soft iron ellipsoid due
to uniform magnetization, and, assuming that these were due
entirely to Kirchhoff's system of stresses, deduced the values
of K' and K". He found the change of volume to be
negligibly small, and for K 7 and K" the values 44,000 and
92,000 for the mean fieldintensity (H = 33 C.G.S., H
(in iron) = 3*5, 1 = 250) which he employed.
* Mem. R. Ace. Line. ser. 4, vol. vi. p. 487 (1890) ; Wied. Electr. iii.
p. 740.
t More exactly,
?i_M!/l±^\ IH K'H 2 K"H 2
/ ~ 3E \l+20/ + 2E(l+^)  2(l + 2tf;U ~ ~2E~ '
■where 6 is a constant defined by the equation
458 H. Nagaoka and E. T. Jones on the Effects
The change of length of a softiron ellipsoid has, however,
heen investigated over a much greater range of fieldintensities
by one of us*, and the results represented by curves as
functions of magnetizing force, of the square of mag
netization, and of the former, respectively. If the change
of length due to Kirchhoff's system of stresses be calculated
from Cantone's formula, and the result subtracted from the
observed change of length, these curves (K' and K" being pro
visionally disregarded) assume the forms shown in figs. 1, 2, 3.
Since the coefficients K' and K" are not known with any
accuracy, for the iron used, the correction due to them cannot
at present be calculated. The ordinates of the fullline curves
represent therefore the changes of length produced by uniform
magnetization, corrected on Kirchhofr 's theory for the partial
stresses represented by terms containing H and K, which
were measured at the same time. The residual change of
length, represented by the ordinates in figs. 1, 2, 3, must be
due to the other unknown terms in the expressions for the
stresses, or to some change of molecular arrangement accom
panying the magnetization.
A similar process was followed by Moref, who measured
the change of length of part of an iron wire magnetized by a
coil. It was assumed, however, that there existed in the wire
B 2
a contracting stress tt , the reason given for this assumption
being that if the wire were cut in two, the two ends would
B 2
be held together by a force — per unit area. Now, accord
ing both to Maxwell's theory and to that of KirchhofF, the
. . . B 2
tension m a narrow airgap in the wire is 7, and this
has been experimentally verified by one of us over a large
range of magnetizing field % ; but both theories give different
values for the stress inside the iron. Even if the wire were
cut in two, the force holding the parts together would not be
B*
— per unit area unless the coil were also cut into two parts,
into which the pieces of wire were rigidly fixed, so that part
of the total tractive force is due to the mutual attraction of the
two coils.
* H. Nagaoka, Wied. Ann. liii. p. 487 (1894).
t Phil. Mag. Oct. 1895.
X See du Bois, Mognetische Kreise, § 104 ; E. T. Jones, Wied. Ann. lvii
p. 271 (1896) ; Phil. Mag. March 1896; Wiedemann's Electr. iii. p. 640.'
of Magnetic Stress in Magnetostriction, 459
Further, as has been already pointed out by Chree"*, the
longitudinal stress in iron is, according to both theories, a
@liSH^P^j^^^^^^^^^SB
IH^HHS
Mmmmm
—■
818 Hi
■^hB
HH
IIEIIH
3
3Q*
tension, not a pressure ; its effect is therefore to lengthen, not
to shorten a piece of soft iron along the lines of force.
* ' Nature/ Jan, 23, 1896.
460
H, Nagaoka and E. T. Jones on the Efects
•9G RHQBI nH^aB Jc*¥raB
H 2
Fi». 3.
?■" ■ '■' i
!M MSI fi[!KI
I 2
of Magnetic Stress in Magnetostriction. 461
B 2
This use of the expression ^ , incorrect according to
existing theories, appears to have been first made by S.
Bidwell*.
As to the stresses inside the iron, Maxwell's theory
H 2
(Art. 642) gives "a hydrostatic pressure — combined with
BIT v7T
a longitudinal tension ^— along the lines of force ;" while
Kirchhoff's theory gives a hydrostatic pressure
2 f 47T J b7T 2
combined with a tension
along the lines of force.
Other systems of stresses in an isotropic elastic medium,
which are equivalent to gravitational and electrostatic forces
in certain cases, are discussed by Chreef, and shown to bo
essentially different from that given by Maxwell (Art. 105).
So far there does not appear to be sufficient experimental
evidence to enable one to decide between these theories.
Calculating the strain of an anchorring according to Kirch
hoffs theory, we easily find that
I 4,*(1 + 30) ^ }
1 Sv
~3 v'
n denoting the rigidity.
As the principal factor K/ can be found from the measure
ment of effect of pressure on magnetization, the easiest method
for making the crucial test would be to try experiments on an
anchorring. Moreover, it is worth while to notice that the
volumechange would in this case amount to the order 10 ~ 5 . v
in iron and to — 10~ 4 . v in nickel, calculating the value of K'
from the experiments of Bidwell on rings of these two metals.
In conclusion, we would express our thanks to Dr. H.
du Bois, at whose suggestion the preparation of this com
munication was undertaken.
g^'  March 25, 1896.
* Phil. Trans, clxxix. p. 217 (1888).
t Proc. Edin. Math. Sjc. xi. p. 107 (189293).
[ 462 ]
L. Notices respecting New Books.
Index of Spectra. Appendix G. By W. Marshall Watts, D.Sc. t
F.I.O. Manchester: Abel Hey wood and Son, 1896.
rPHE present Appendix to Dr. Watts's ' Index of Spectra ' brings
* the record of spectroscopic work down to the present time, and
contains results of observations published within the past three
years. It opens with Eowland's table of standard wavelengths,
which is followed by an account of the researches of Eder and
Valenta. These include the sparkspectra of sodium, potassium,
and cadmium, and the line and band spectra of mercury. The
oxy hydrogenflame spectra of several metals and oxides, observed
by Hartley, are next tabulated, and the record is brought to a close
with an account of the recent work of Eunge and Paschen on helium
and parhelium, the two constituents of cleveite gas. — J. L. H.
LI. Intelligence and Miscellaneous Articles.
ON AN ELECTROCHEMICAL ACTION OF THE RONTGEN RAYS ON
SILVER BROMIDE. BY PROF. DR. FRANZ STREINITZ.
HPO Eontgen we are indebted for his great discovery of the
* property of the 47rays of exciting fluorescence and producing
chemical reductions on a photographic plate. According to his
previous experiments, these properties are the only ones which the
rays have in common with those of light. Now light alters not
only the electromotive deportment, but also the conductivity of
the silver haloids. The proofs of this were furnished by
Becquercl and Svante Arrhenius. It can therefore scarcely be
doubted that electrochemical changes will be produced by the
Eontgen rays ; of course it is a different question whether they
will be accessible to observation.
Experiments were made in both directions. In order to
establish the fact of a change in the conductivity, a method used
by Arrhenius {Wiener Berichte, vol. xcvi. p. 831, 1887) was
adopted. On a glass tube a silver wire was wound bifilar,
and then coated with ammoniacal solution of silver chloride.
After the ammonia and water had been evaporated, the glass tube
was then placed in a lighttight box, out of which the ends of
the wire projected. These ends were connected with a source
of electricity and a very sensitive galvanometer (Thorn son
Carpentier). The deflexion in the galvanometer showed variations
when discharges were passed through a Hittorf's tube near to the
box. These, however, were manifestly due to inductive actions
on the galvanometer circuit. Eor when the induced circuit w T as
open, an increase in the deflexion could not with certainty be
established — as ought to be the case, since there was an increase
of the conducting power — in comparison with the deflexion
which was obtained before the discharge was set up.
Experiments on the influence on electromotive behaviour were
attended with better success. A square platinumfoil, 2 cm. in the
side, was coated electrolytically with an exceedingly thin layer of
Intelligence and Miscellaneous Articles, 463
silver bromide. Combined in dilute potassiumbromide solutions
with a standard electrode, the electrode in question showed
sensitiveness to light. This is easily proved with the help o£
a quadrant electrometer (Luggin, Ostwald's Zeitschrift fur phys.
Chemie, xiv. p. 387, 1894). A candle placed at a distance of 25
cm. from the electrode produced in half an hour a diminution of
0022 V. in the electromotive force of the combination
Zn/Zn S0 4 aq+K 2 S0 4 aq + BrK aq+Br Ag/Pt.
At the same time it is to be remembered that the light only struck
one side of the platinumfoil, while the other, which was also
sensitized, remained dark. If now a carefully enclosed discharge
tube through which ind actionsparks passed was substituted for
the candle, a diminution in the electromotive force could also be
observed. With a small inductioncoil the change amounted to
0*017 volt in the coarse of 45 minutes, and in another experi
ment with a larger inductioncoil to 0*019 in 40 minutes.
By a corresponding increase in the delicacy of the method, the
electrochemical deportment of Bontgen rays may possibly furnish
a more convenient method of investigating them than that with
the help of photography. — Wiener Berichte Feb. 6, 1896.
TRIANGULATION BY MEANS OF THE CATHODE PHOTOGRAPHY.
BY JOHN TROWBRIDGE.
Photography by means of the Bontgen rays seems already to be
of great importance in examining certain portions of the human
body to determine the presence of metallic bodies, calcareous
formations, and fragments of glass. The shadow pictures as they
are taken at present, however, do not give the approximate
position of the shots, for instance, embedded in the flesh. They
indicate only the line in which they are situated. It occurred to
me that the principles of triangulation could be applied with
success to determine more exactly the position of the metallic
particles. I was led to this conclusion by considering Bumford's
photometer. This instrument, it is well known, consists merely
of a vertical rod placed opposite a suitable screen of white paper.
The two lights, the intensities of which are to be compared, are
placed in a fixed position, and throw two shadows of the rod on
the screen. From a measurement of the positions of the lights
when shadows of equal intensity are thrown on the screen, an
extinction of the brightness of the lights can be obtained. More
over, by measuring the distance between the shadows, and by
drawing lines from them to the lights, the position of the rod
throwing the shadows can be determined. This position is
evidently at the intersection of these lines.
I have used two Crookes' tubes with two terminals making an
angle with each other, and have employed a toandfro excitation
by means of a Tesla coil. A suitable screen of glass shielded the
sensitive plate first from one cathode and then from the other.
From the distance between the shadow pictures of a shot, for
instance, on the back of the hand and from the position of the
464 Intelligence and Miscellaneous Articles,
terminals, the height of the shot above the sensitive plate could be
estimated. It seems to me that this method promises to be of
importance in the surgery of the extremities of the body ; for the
question whether to make an incision from the palm of a child's
hand or frcu the back of the hand is an important one. Stereo
scopic pictures can also be obtained.
The use of a Tesla coil in obtaining shadow pictures is advan
tageous in certain respects, for by changing the size of the spark
gap in the primary circuit of the Tesla coil one has a great range
of electrical energy at command. This range can be still further
increased by putting the sparkgap in a magnetic field. I have
taken such pictures iu less than a minute, showing the bones in
the fingers. The tubes were, at first, destroyed by disruptive
sparks over the surface of the tube which apparently penetrated
the glass between the platinum terminals and the glass. I have
lately discovered, however, that if the terminals of the tube are
placed in a vessel filled with paraffin oil, and if the oil is kept cool by
an outside vessel filled with snow or ice, the entire energy developed
by the Tesla coil can be employed, and the tubes are not destroyed.
I have tried wooden lenses, both double convex and double
concave, in order to see whether the rays travel slower or faster
in wood than in air, but my results are negative. A copper ring
placed on a double convex lens of wood of approximately six inches
focus, and one also on a concave lens of the same radius as the
surfaces of the double convex lens, gave shadow pictures of the
ring which were of the same size and character as those of an
equal copper ring placed in air at the same distance from the
sensitive plate.
"We naturally turn to Maxwell's great treatise on Electricity and
Magnetism, to see if a hint of this new phenomenon cannot be
found there : for I believe there is no manifestation of electro
magnetism since the death of Maxwell which has not been
predicted or treated by him in one form or another in his
remarkable book. In section 792, vol. ii. of the treatise on
Electricity and Magnetism, he says: — "Hence the combined effect
of the electrostatic and the electrokinetic stresses is a pressure
equal to 2p in the direction of the propagation of the wave. Now
2p also expresses the whole energy in unit of volume. Hence in
a medium in which waves are propagated there is a pressure in
the direction normal to the waves, and numerically equal to the
energy in unit of volume. Thus, if in strong sunlight the energy
of the light which falls on one square foot is 83*4 footpounds per
second, the mean energy in one cubic foot of sunlight is about
00000000882 of a footpound, and the mean pressure on a square
foot is 00000000882 of a pound weight. A flat body exposed to
sunlight would experience this pressure on its illuminated side
only, and would therefore be repelled from the side on which light
falls. It is probable that a much greater energy of radiation
might be obtained by means of the concentrated rays of the
electric lamp. Such rays falling on a thin metallic disk, delicately
suspended in a vacuum, might perhaps produce an observable
mechanical effect." — American Journal of Science, March, 1896.
LONDON, EDINBURGH, and DUBLIN
PHILOSOPHICAL MAGAZINE
AND
JOURNAL OF SCIENCE.
[FIFTH SERIES.]
JUNE 1896.
LII. ThermoElectric Interpolation Formula.
By StLAs W. Holm an.*
IN this paper are collected the several wellknown types of
formulae for expressing the thermal electromotive force
of a couple as a function of the temperature of its junctions.
Two new formulae are also proposed. All then are tested
against the most reliable experimental data upon the subject,
and their relative merits discussed.
The Existing Formula,
Consider a simple closed electric circuit composed of two
different metals, each homogeneous in matter and temper,
the metals being in contact at two points. For simplicity,
assume the metals to be in the form of wires joined at their
ends. Let one junction be at a temperature of h°, the other
of c°, on the ordinary Centigrade scale. Let 2* £ be employed
as a suggestive symbol to denote the resultant electromotive
force in the circuit, induction being excluded from considera
tion. Then % c e is a function of h and c which involves
constants dependent upon the nature of the metals, and which
may be represented by
t)e=f{h,c).
The discovery of the natural expression for / (A, c) is not only
* Fro ai an advance proof of the 'Proceeding of tho American
Academy/ vol. xxxr. (n. s. xxiii.) p. 193. Communicated by the Author.
Phil. Mag. S. 5. Vol. 41. No. 253. Jane 1896. 2 K
466 Prof. Silas W. Holman on
of scientific importance, but is urgently needed in the develop
ment of the art of pyrometry. At present even a satisfactory
empirical formula for interpolation is lacking, the best still
being probably that of Avenarius and Tait.
The existing formulae are the five following : —
Ordinary or parabolic :
tU=at + bt 2 + cf + (1)
This is, of course, merely a series in ascending powers of t,
where one junction is at any temperature t° C, and the other
at 0° C, a, b, and c being constants. A more general form
for the case where the cold junction is at any constant tem
perature, ti°, is
2^ = a(i^ 1 )H^ 2 f 1 2 ) + c(i 3 ^ 1 3 )+ . . .
These expressions may, of course, be inverted, giving t as a
function of 2 e.
Avenarius : ^ h
X h c e={hc){a + b(h + c)\ .... (2)
in accordance with the foregoing notation.
Thomson: , r h + T c \
where t is the absolute temperature, r being that of the
" neutral point/'
Tait : f
$U=(k'k)(T h T e )\ Tn ?*+I?.} ... (4)
Both of the last two, by the substitution of £ + 273 for t,
obviously reduce to the Avenarius form.
e4 + , c =10 p +^ + 10 p ' +Q, °, .... (5)
where e h represents the thermal E.M.F. of the hot junction
and e c that of the cold junction. In view, however, of the
existence of the Thomson effect, these symbols can strictly be
interpreted only as having the meaning that e h — e c =%*e.
Bote. — With regard to the Avenarius, Thomson, and Tait
expressions, it may be remarked that they are not only
mutually equivalent, but that if t c or r, becomes 0° C. they
reduce at once to the ordinary parabolic form of two terms :
tU^at + bt' 2 .
They are all, therefore, forms which must apply if the latter
Thermoelectric Interpolation Formula. 467
purely empirical expression for the same temperature ranges
applies, and with the same closeness, so that it is unnecessary
to test more than one of the first four expressions against any
one set of data. Also the fact that the Avenarius and Tait
equations approximately conform to the observed data does
not necessarily in any material degree strengthen the hypo
theses which are adduced to show that these equations are a
natural expression of the law.
Without attempting here a further analysis of the com
ponents making up the resultant E.M.F. ^e, which is the
measured E.M.F. of the thermocouple, the proposed interpo
lation formulae will be merely developed and applied. It
may, however, be suggested in passing, that there seems to
the writer to be little hope of arriving at a close approxima
tion to the natural law except through an expression which
shall contain separate terms representing the temperature
function of the component arising at the contact of the
dissimilar metals, and that arising from the inequality of
temperature of the ends of each (homogeneous) element
(Thomson E.M.F.). The parabolic and Avenarius formulae
would comply in part with this requirement on the supposi
tion that the E.M.F. at contact varied as the first power, and
the Thomson E.M.F. in both wires as the square of the
temperature. And looked at from that point of view, the
neutral point would seem to have an explanation materially
different from that usually accorded to it.
The Proposed Formulas.
Exponential Equation. — The significance of this proposed
expression may be thus stated. Suppose the cold junction of
the couple be maintained at the absolute zero of temperature,
t=0°, and its E.M.F. to be consequently zero. Let the other
(hot) junction be at any temperature r h ° absolute. The pro
prosed equation is based on the assumption that the total
E.M.F of the couple would then be representable by
e f = m,T^.
where m and n are numerical constants. If then the cold
junction were raised to any temperature t c °, there would be
introduced an opposing E.M.F. e", which would be expressible
ty e" = niT\
The resultant E.M.F % h c e would then be e'e", and there
fore expressible by
X h c e = mTlmT n c (6)
2K2
468 Prof. Silas W. Holman on
If in any instance, as is frequently the case in measurements,
the temperature of the cold junction is maintained constant
while that of the hot junction varies, then mrr n c becomes a
constant, and it will be convenient to denote this constant by
/3 when t=273° abs. =0° 0. So that for this special case
where the cold junction is at 0°C, and the hot junction at
i° C, we have
2j*=mT w £. . . . . . • (7)
This expression is not advanced as a possible natural form
of the function f(h, c). It is essentially empirical, and is not
designed to account separately for the several distinct com
ponents entering into 2#. The fact that it closely fits the
experimental data arises chiefly from the well known adapta
bility of the exponential equation to represent limited portions
of curved lines. The equation also leads to certain inferences
which appear inconsistent with the known thermoelectric
laws, and fails to explain some known phenomena.
The evaluation of the constants m, n, and ft is unfortunately
attended by considerable labour. No application of the method
of least squares readily presents itself, but by a method of
successive approximations the values can be obtained with any
desired degree of exactitude. Only two measured pairs of
values of 2q£ and t are necessary for this approximation
method, the third required pair being furnished by Xo (8)
(1)1
\T A /
Thermoelectric Interpolation Formula. 469
„_ log(So"e + /3)1og(Sge + /3 ).
log t" — logr'
(9)
m
W~ or ~w~ (10)
By means of these the numerical values of the constants
may be calculated from those of r f , t", 2o e, &., as follows : —
1. Assume as a first approximation some value of n, say n = 1,
unless some better approximation is in some way suggested.
Substituting this value in (8), compute the corresponding
value of /3.
2. Using this as a first approximation, substitute it in (9)
and compute the corresponding value of n.
3. Using this value as a second approximation to n, insert
it in (8), and compute a second approximation to ft.
4. With this compute a third approximation to n, and so
continue until consistent values of /3 and n are found to the
desired number of figures. Then compute m by (10).
The rate of convergence is not rapid, but after one or two
approximations have been made an inspection of the rate will
enable the computer to estimate values of n which will be
nearer than the preceding approximation, and thus hasten the
computation.
Where an equation is to be computed to best represent a
progressive series of observed values of t and % e, this method
is of course open to some objections, since it incorporates in
the constants the accidental errors of the selected observa
tions from which the constants are deduced. This difficulty
can be sufficiently overcome by computing residuals between
the equation and the data, and amending the equation if
necessary to give them a better distribution.
Logarithmic Formula. — A very simple expression for inter
polation is of the general form
when m and n are constants. This serves fairly well for a
short range, t" —t', when ^ — 0° is not less than one third of
The convenience of the expression arises from two facts :
first, that its two constants are very easily evaluated either by
computation or graphically from the logarithmic expression
(whence the name)
Iog2o£=wlog£ + logm.
second, that its logarithmic plot is a straight line, since this
470 Prof. Silas W. Holman on
expression is the equation to a straight line if we regard
log 2o e and log t as the variables. If, therefore, a series of
values of 2 e and t are known for a given couple, points
obtained by plotting log t as abscissas and log 2 e as ordinates
should lie along a straight line. Thus a couple may be com
pletely " calibrated " for all temperatures by measuring 2 e
and t for any two values of t (suitably disposed) . The con
stants m and n may be computed, or a plot of log 2 e and log t
may be made, and a straight line be drawn through them.
Graphical interpolation on this line will then of course yield
the values of log t and hence of t corresponding to observed
values of 2 e, and vice versa, and, if desired, the constants
m and n. The expression for t as a function of 2 e is of
course t x
t=m.
Alloy.
m.
n.
/3
mv.
D
9
13671
1250
1517
C
10
095596
1310
1485
E
11
081734
1336
1469
A
10
057689
1377
1305
F
20
022865
1522
1167
G
30
0065990
1708
956
H
40
0063034
1720
977
478
Prof. Silas W. Holman
on
of the alloys. Table V. gives the percentage deviations of
these alloys from the exponential equation (data— equation),
and Table IV. shows the values of m, n, and /3 for those
equations.
Table V.
Holbc
rn and Wien. — Comparison oi
* Alloys
;.
t.
D.
A.
C.
E.
R
a.
H.
Average.
,°0.
154
48
40
50
44
45
273
09
26
27
22
23
35
23
26
379
+21
057
080
060
13
060
025
067
482
426
4015
003
060
4022
4046
003
584
+09
003
4019
4008
+013
4030
003
4002
680
030
4007
4013
4013
4012
016
026
774
019
006
008
0 02
012
042
046
019
862
+023
4020
4020
4020
4009
4015
011
4012
952
+020
010
011
015
4009
024
008
1038
4012
018
oio
021
027
4004
014
016
1120
015
006
009
0 02
4010
4001
4022
001
1200
40 02
4007
4010
4004
4003
1273
4067
+070
4080
+080
4043
4100
4073
1354
4038
4060
40 67
4051
4030
4100
4058
1445
080
038
050
090
040
060
a
d. 4001200
052
015
009
010
012
016
021
007
a
d. 4001445
025
Direct from
Air Th.
021
024
(024)
023
036
Chassagny and Abraham Data*. — The apparently very
careful measurements of these observers cover a range of 0°
to 100° C. with observations at 25°, 50°, and 75° only. The
range is too short and the intervals are too great to render
the work of much service in testing a general formula, but if
Table VI.
Couple.
5,100
S e.
4°
V 25 „
FeCu
10933
8951
11230
16851
8649
7089
8856
12789
6048
4961
6174
8599
3155
2591
3211
4321
FePt Eh
FeAg
FePt
* Chassagny et Abraham, Ann. de Chim. et de Phys. xxvii. p. 355
(1892).
thermoelectric Interpolation Formula.
479
its accuracy is as high as about o, 01, as it appears to be, this
iii part offsets the disadvantage. Measurements of Xo e and t
were made with four thermocouples, with the results shown
in Table VI. (international microvolts and degrees centi
grade on hydrogen scale).
The Avenarius equation was applied to these data by Chas
sagny and Abraham in the form
2,1 e = at + bt*.
They evaluated the constants from the 50° and 100° data.
With these they computed the temperatures which the equa
tion would yield by insertion of the observed values So 5 e and
2J° e. These values are given in Table VII., columns 2
and 3.
The exponential equation applied to these data for Fe — Pt
becomes
2o«=105096 * * 7360  6525*3 [Range 0° to 100° C.].
The values of t corresponding to the observed values 2o 5 e
and So 5 e are given in Table VII. It has not seemed for the
present purpose worth while to make similar computations
for the other couples, as they would not materially affect the
inferences to be drawn.
The logarithmic equation yields
2U = 19'2946 J * 970595 ;
log 2S 6=0970595 log * + 1285436.
The deviations are given in the Table.
Table VII.
Couple.
Avenarius.
Exponential.
Logarithmic.
t.
St.
t.
St.
6
mv.
100 S/e
per cent.
St.
S
mv.
FeOu
FePt Eh..
FeAg
FePt ....
FeCu
FePt Eb..
FeAg
FePt
2488
24885
2487
2487
7513
75135
75135
75135
+012
0115
013
013
013
0135
0135
0135
o
2480
7515
o
+020
015
26
+25
0037
+0032
o
+052
026
67
+43
480
Prof. Silas W. Holman on
The Noll Data. — A contribution of much permanent value
to the data on thermoelectrics has recently been made by
Noll *, who has measured So e and t for thirtytwo couples
oyer a range in most cases of 0° to 218° C. The metals em
ployed (including carbon) were usually of a high and stated
degree of purity, and consisted of eighteen different sub
stances, two of which were alloys (german silver and brass),
and the remainder samples of different degrees of purity or
hardness of the pure substances. The couples contained, as
one element, for the most part, either copper or mercury.
Temperatures were reduced to the airthermometer scale.
The Avenarius formula was applied to fourteen of the more
important of them by Noll. The deviations are given in
Table VIII.
Table VIII.
NolPs Data on Pure Metals.
/3
mv.
Av. Pet. Deviation.
Couple.
m.
n.
Avenarius.
Expon.
AuHg
46954. 10 3
2136
7504
+027
+017
AgHg
28637. 10~ 3
2206
6778
033
015
NiCu
82333. 10" l
1511
39502
030
017
(OdCu
37617. 10 11
494
407
048
340)
BrOu
24969. 10" l
1366
5311
014
013
ZnHg
82890 . 10  ^
2420
6516
015
018
PbCu
17674 . 10,
1800
4290
005
007
C Ul Hg
46726 . 10" 3
2130
7684
012
011
[FeHg
10913. 10 + *
83295.10 ~T
07220
62642]
OoHg
2166
15752
026
022
Pt,Cu
21475. 10 3
2266
7111
008
012
Pt 2 Cu
H095.10" 3
2353
5990
019
025
Si^Cu
42021 . 10"*
1667
4828
021
09
MgCu
20449. 10  *
1782
4487
015
017
AlOu
75643 . 10" 2
1590
5653
011
012
G.s.Cu
20454. 10 _1
1684
25899
006
005
Average om
itting CdOu and
FeHg .
±017
±014
The exponential equation I have applied to the same data.
It has not seemed essential to reproduce here the entire series
of data, and the deviations of both equations. They are
therefore, presented in a somewhat more digested form.
* Noll, Wied. Ann. iii. p. 874 (1894).
Thermoelectric Interpolation Formulae.
481
Table VIII. gives the constants for the exponential equation
(those for the Avenarius may be found in Soil's article), the
mean deviations ( = data— equation) for each series, and the
mean percentage deviations ( = 100 8/e). (See remark as to
use of e under " Barus Data.") Table IX. groups the per
centage deviations under their nearest values of t for exhi
biting their systematic character. The fact that the experi
mental method brought the observations all very nearly to
the respective temperatures t given in the table renders this
grouping possible. I have taken the liberty of correcting a
few obvious numerical errors, and of dropping a very few
values evidently containing a mistake.
It may, perhaps, not be out of place here to caution those
who would make use of NolFs data to their full accuracy that
his original, and not his interpolated, numbers should be
resorted to. The approximate linear interpolation which he
has employed is not as accurate as his experimental data
demand.
The logarithmic equation applied to the CuHg couple as
typical of the Noll data yields
SS«=257434 P*";
or
log 2o e=l2250 log £+0410665.
The residual^ to this expression are given in Table IX.
Table IX.
Avenarius Equation. — Data minus Equation in Per Cent.
15°.
57°.
100°.
138°.
181°.
198°.
217°.
AuHg
010
040
+005
4060
+090
AgHg
013
+ 20
+021
050
130
NiCu
026
015
022
031
117
ErCu
010
4020
4017
4035
ZaHg
003
+009
4041
4040
PbCu
003
011
4012
004
CSffg
+002
016
+001
+015
+038
CoHg
4042
056
Pfc,Cu
012
006
4009
4021
Pt 2 Ou
013
011
4067
021
Sn,Cu
020
4034
4040
4030
MgCu
+011
4022
037
4011
AlCu
+016
4018
018
+015
a. s.cu
...
001
4016
4011
Average
012
007
+009
4005
4016
012
Phil. Mag. S. 5. Vol. 41. No. 253. June 1896. 2 L
482
Prof. Silas W. Holman on
Table IX. — Continued.
Exponential Equation. — Data minus Equation in Per Cent.
AuHg
15°.
57°.
100°.
138°.
181°.
198°.
217°.
+001
012
006
006
+009
+012
080
AgHg
018
+022
+005
003
050
NiCu
026
016
+007
000
072
BrOu
006
001
+012
+020
0'26
013
ZnHg
030
+002
+014
+010
+065
004
PbOu
oio
025
+004
000
Cu x Hg
019
027
+001
003
025
CoHg
005
+001
+648
001
100
Pt^Ou
015
+003
006
+037
031
Pt 2 Cu
056
010
016
+070
Sn.Ou
004
+004
029
+012
+006
MgOu
+018
+015
052
+017
AlCu
+016
+012
024
+018
O. s.Cu
+007
001
+008
+005
Average
016
+008
+001
+003
002
+014
027
Logari
thmic I
]quatior
l. — Date
i minus
Equatic
n in Per Cent.
Ul Hg
+23
+15
040
+050
+080
Discussion of the Deviations.
Plots are given in the following diagram with temperatures
as abscissas and percentage deviations between the data and
the sundry equations as ordinates, i. e. 100 B/e where 8 = data
equation. Inspection will show that with one exception (viz.
the logarithmic equation applied to the Barus data) these plots,
whether the equation is the ordinary parabolic, the Avenarius,
the Barus, the exponential, or the logarithmic, have the same
general form, which may be imperfectly described as follows.
If the equation be made to conform to the data at 0° C.
and at two higher points, a and b, then the deviation will be
of the negative sign from to a, positive from a to b, and
negative above b. The slight departures from this general
form are clearly due either to accidental errors, or to failure
to make the equation conform to the data at all three points,
or at suitable ones. The evidence is therefore conclusive
Thermoelectric Interpolation Formulce,
483
that for all of the expressions the deviations are systematic and
not purely a accidental " in character.
One of two inferences is therefore warranted : —
1. That neither the parabolic, Avenarius, Barus, exponential,
nor logarithmic equation is the natural expression of the
function.
B**us,** J $u
Babus.
A VEN.
£xPOft.
Logart —
2. Or that the scale of temperature to which the values of t
are referred in the foregoing investigations departs from the
2L2
484 Prof. Silas W. Holman on
normal scale by an amount and system roughly indicated by
the above residual plots.
The latter inference, suggested by Chassagny and Abraham
in the interpretation of their results, does not seem to possess
much weight, notwithstanding the urgent need of renewed
elaborate experimental investigation of the relation between
the hydrogen, air, and thermodynamic scales of temperature.
As to the relative usefulness of the various expressions for
purposes of interpolation and extrapolation some further
inspection is necessary. The Barus equation 3, line C D,
shows slightly smaller deviations on the plot than do the
Avenarius and exponential, lines E E and F F. This, how
ever, is due to the fact that the data against which 3 is
tested are mean interpolated values, and hence have a sensibly
less variable error than those against which the other equations
are tested. An approximate exponential equation showed
less deviations than 3 against the same data. There seems,
therefore, to be no advantage in this equation sufficient to
offset the difficulty of evaluation of its constants.
Applied to the Barus data from 350° to 1250°, the ex
ponential equation shows deviations considerably less than one
half as great as those of the Avenarius, while those of the
logarithmic equation are so small as to lie far within the
range of the variable errors, and they moreover show no clear
evidence of systematic error between these limits of tempera
ture. For interpolation in the Barus data, therefore, the
logarithmic equation is far preferable, and must be conceded to
be representative of the data. For extrapolation it is un
doubtedly better than the Avenarius, which (as would the
exponential in less degree) would certainly give above 1000°
extrapolated values of %e too large, or of t too smalL The
advantage due to its simplicitjr is also to be noted.
Applied to the Holborn and Wien data from 400° to 1450°
the exponential equation shows (line KK) the same sort of
superiority to both logarithmic (line L L) and Avenarius
(line 1 1) that the logarithmic shows to the others with the
Barus data, but in a still more marked degree. Within the
limits 450° to 1450°, in fact, the distribution of the residuals
to the exponential is such as not to warrant of itself alone
any inference of systematic departure, especially when the
mean line M M from all the couples is considered. It will be
noted as an important confirmation of both the exactness of
the electrical measurements in the investigation and the
applicability of the exponential formula through a considerable
range of alloys (and therefore of values of m and n) that this
Thermoelectric Interpolation Formulas. 485
mean line M M is almost identical in form with the line K K
for alloy A. Relatively to the Holborn and Wien formula
(line H H), the exponential possesses a similar advantage,
with also the merit of greater simplicity of form.
It may therefore be affirmed that for interpolation between
450° and 1450° in the H. and W. data the exponential equation
is abundantly exact. For extrapolation above 1450° it would
not be entirely safe, although presumably better than the
others, since the departure between 0° and 450°, and the
similarity of the form to others, make a systematic departure
sufficiently certain.
Applied to the Chassagny and Abraham data, 0°l 00°, and
to the Noll data, 0°218° (see diagram), the Avenarius and
exponential formulae show about equal deviations, but with
the advantage slightly on the side of the former. In the case
of the Noll data, the line indicates that the systematic error
is slightly greater for the exponential than for the Avenarius
expression. The average deviations in Table IX., on the
contrary, show that for each individual equation the concor
dance is greater for the exponential than the Avenarius. This
discrepancy is due to the fact that, in order to eliminate local
accidental errors, the equations (both Avenarius and expo
nential) are not all made to coincide with the data at the
same temperatures, so that the process of averaging by which
the data for the Noll plots is obtained is not numerically rigid.
This does not, however, sensibly affect the general form of
the curve. The greater ease of computation of the numerical
constants of the Avenarius expression, and its applicability
where both t and t change, ought not to be overlooked. For
extrapolation the exponential would be safer, for the reason
that it has been shown above that for long ranges its syste
matic error is less.
The logarithmic equation fits the Noll data very badly, as
shown by the deviation in Table IX. (not plotted), and also
is much less close to the Chassagny and Abraham data than
are the others.
The general conclusion as to applicability, then, seems to
be that, while the Avenarius expression may be equally good
or better than the exponential for interpolation over short
ranges, yet for interpolation over long ranges and for extra
polation above the observation limits the exponential is decidedly
preferable. The exponential form is also preferable to the
remaining expressions with the exception noted.
The logarithmic form, although closely applicable to the
Barus data, is of more doubtful general value, yet on account
486 Prof. Silas W. Holman on
of its great convenience it may find application in industrial
pyrometry, as will be elsewhere indicated. Although failing
below 300° or 400°, it may probably be applied to the irido
or rhodoplatinum couple between 400° and 1200° C. with a
maximum error not exceeding about 5°. If extended to cover
400° to 2000° the error might rise to 15° or 20°.
More in detail it may be briefly noted by way of
summary : —
That the logarithmic equation fits the Barus data between
400° and 1250° with scarcely sensible systematic error, and
within the limits of variable errors of the data.
That the exponential equation similarly fits the Holborn
and Wien data within the limits 400° to 1445°.
That when made to coincide with the data at about 450°
and 1200° the systematic deviations of the exponential equa
tion from the Barus data, and of the logarithmic equation
from the Holborn and Wien data, are in general of opposite
sign and of roughly equal magnitude.
Barus Melting and BoilingPoint Data.
From the foregoing demonstration of its applicability, it
seems proper to apply the logarithmic formula to the Barus
thermoelectric data on meltingpoints*.
Whether the extrapolation above 1000° by the logarithmic
formula is entitled to any great weight may be questioned,
but there is no obvious reason why it is not more reliable than
by any of the others. I have employed the equation given
on page 474, which represents very closely Barus's high
temperature airthermometer comparisons, calculating thence
the temperatures t corresponding to the values of %^e given
by Barus for the various points, assuming Barus's value
2 2 °£ = 150 mv. The results are given in column 3 of
Table X. Column 4 quotes the most reliable previous
determinations of the same points by other observers. As to
which of the two columns of results best represents Barus's
work, there can be little doubt from the above evidence that
below 1000° it is the second, that is, the one computed from
the logarithmic equation. These combine both his own air
thermometer and meltingpoint work. Above 1000° the
logarithmic values are probably slightly too high.
* Amer. Journ. Sci. xlviii. p. 332 (1894).
Thermoelectric Interpolation Formulce.
Table X,
Barus Melting and Boiling Points.
487
Computed
by Eq. 3.
Computed
by Log. Eq.
Data by other Observers.
Mercury (B. Pt.)
Zinc
o
357
420
446
638
694
782
929
986
1091
1096
1435
1476
1585
1757
359
423
449
641
697
782
926
985
1090
1095
1441
1485
1597
1783
35676
41757
44453
635
930
968
954
1072
1035
1082
1054
1450
1500
1775
Callendar and Griffiths.
i> »>
>) >>
Le Chatelier.
Deville and Troost.
Holborn and Wien.
Violle.
Holborn and Wien.
Violle.
Holborn and Wien.
Violle.
Carnelly and Williams.
Violle.
Violle.
Sulphur (B. Pt.)
Aluminium
Selenium (B. Pt.)
Cadmium (B. Pt.)
Zinc(B. Pt.)
Silver
Gold
Copper
Bismuth
Nickel
Palladium
Platinum
Remark.
Review of the laborious researches which have been devoted
to the direct comparison of thermoelectric elements with the
air thermometer, mainly for the purpose of advancing the
art of pyrometry, has enforced the conviction that, at least
for the immediate future, this end would be better served by
accurate gasthermometer measurements of meltingpoints of
metals. Each such determination made upon a reducible
metal of known high purity under proper reproducible con
ditions fixes an enduring and reproducible referencepoint,
a pyrometric " bench mark." And there are enough inex
pensive metals, together with a possible system of simple
alloys, to give points of sufficient frequency. These would
then afford a convenient means of obtaining accurately known
high temperatures for purposes of study of all high temperature
phenomena, and particularly for calibrating thermoelectric,
electrical resistance, optical, or other secondary pyrometric
interpolation apparatus, — for it must be remembered that all
such apparatus is necessarily secondary, the gas thermometer
being inevitably the primary.
On the other hand, comparison with the air thermometer of
a thermocouple, or of a resistance pyrometer, or the study of
488 Mr. W. B. Morton on the
any progressive thermal phenomenon, while it possihly may
result in the eduction of a natural law, is very unlikely to
lead to anything more than the establishment of an approxi
mate equation with constants characteristic only of the
individual materials actually employed, and not transferable
to other, although similar materials. Such results are obviously
of a much more ephemeral character than the meltingpoint
measurements. Even when any pyrometer thus tested is
applied to the establishment of meltingpoints, it must at best
yield results inferior to direct application of the gas ther
mometer, except in cases where the latter is hampered by
want of sufficient quantity of the metal to be experimented
upon, — a condition which need only affect such costly sub
stances as gold and platinum.
Stated broadly, the great need of the art of pyrometry is
convenient methods of producing, or of recognizing when
produced, a series of accurately known high temperatures.
The analogous problem has been partially solved for ther
mometry at temperatures up to 300° C. by the investigation
of boilingpoints of certain chemically pure substances under
controlled pressures.
Rogers Laboratory of Physics,
Massachusetts Institute of Technology,
Boston, September, 1895.
LIII. Notes on the ElectroMagnetic Theory of Moving
Charges. By W. B. Morton, B.A*
1. rj VEIS subject has been brought into prominence re
X cently by the use which Mr. Larmor has made of
moving electrons in his dynamical theory of the aether. The
matter was investigated in 1881 by Prof. J. J. Thomson f,
who showed that a point charge moving so slowly that the
electric displacement it carries is not sensibly disturbed
generates magnetic force like a current element according to
Ampere's rule ; and by Mr. HeavisideJ, who investigated the
matter more generally in 1889, and showed that in steady
rectilinear motion at any speed less than that of light, the
lines of displacement continue to be radial but are concen
trated towards the plane perpendicular to the direction of
motion. The displacement at distance r, in a direction
* Communicated by the Physical Society: read March 27, 1896.
t Phil. Mag. April 1881, July 1889 • Recent Researches, pp. 1623.
X Electrical Papers, ii. pp. 504518; Electromagnetic Theory, i,
pp. 269274.
Electromagnetic Theory of Moving Charges. 489
making an angle 6 with the line of motion is proportional to
1
(l^sin^) 1
where u is the velocity of the moving charge and V the
velocity of light. The lines of magnetic force are circles
round the line of motion.
2. This solution of course represents the state of affairs at
a great distance from a small charged conductor of any shape.
It would also give us the distribution of charge on a moving
sphere if it were correct to assume that the lines of displace
ment meet the charged surface at right angles. This
assumption was made by Prof. Thomson and, at first, by Mr.
Heaviside, but the latter, quoting a suggestion of Mr. G. F. C.
Searle, subsequently pointed out that when there is motion
the electric force is no longer derived from a potential
function, and as a consequence does not meet the equilibrium
surface at right angles. Substituting, the correct surface
condition, he showed that the charged conductor, whose motion
would give at all points the radial distribution found for a
point charge, was not a sphere but a spheroid of certain
ellipticity.
3. It seemed of some interest to inquire what the distri
bution of charge on a moving sphere would be. The surface
density at a point of the surface is now the normal component
of the displacement at that point. By carrying the investiga
tion a step further I have found that, if the conductor be a
sphere or any ellipsoid, the ordinary static arrangement of
charge is unaltered by the motion ; i. e. the number of tubes
of displacement leaving each element of the surface is
unchanged, but the tubes no longer leave the surface at right
angles. We may imagine that the motion has the effect of
deforming the tubes, keeping their ends on the conductor
fixed. The proof of this, involving a consideration of the
general case, is here given and is followed by a note on the
energy of a moving charge in a magnetic field.
4. Suppose we have any distribution of charge moving
with uniform velocity u parallel to the axis of z, and that the
field has assumed its steady configuration. We shall denote
u 2
!— y2°y &> V Dei °g tne velocity of light. Then since we
have a steady state,
d_ d
.'■■ dt" U Tz
490 Mr. W. B. Morton on the
Also, since each element of charge produces a magnetic
field with no ecomponent, we have 7 = in the general case
also. Using these two data, the equations connecting the
displacement (f,g,h) and the magnetic force (at, ft,y) become
d/3 . df
dz dz
da . da
— =_ 4t™_^
dz dz
d/3 da, . dh
j J = — 4:7TU r=
dx dy dz
dg dh _ u da
dz'~dy~""^PTz
dh__df_ u dfl
dx dz ~ 47rV 2 dz 9
dy dx
These equations together with
df + dg + ^ =0
dx dy dz
are satisfied by
/=_#, ff =W h=t?^,
dx J dy dz
dy dx
where must vanish at infinity.
5. Mr. Heaviside points out that = constant is the con
dition holding at a surface of equilibrium. The matter may
be stated thus : — If we suppose the field to terminate at the
surface of a conductor, inside which the vectors vanish, we
must see that the " curl " relations of the field are not violated
Electromagnetic Theory of Moving Charges. 491
for circuits which lie partly inside the empty space enclosed
by the conductor. In particular, if there is a vector whose
line integral round every circuit in the field vanishes, the
lines of this vector must meet the surface at right angles.
Otherwise we should have a finite value for the integral round
a circuit drawn close to the surface outside and completed
inside. In other words, if a vector is derived from a potential
function, this function must be constant over the surface. In
the ordinary static case it is the electric force (X, Y, Z)
which is so derived ; but in the case of a steadily moving
/ Z Y
field it is the vector (X, Y, p J which meets the surface at
right angles. ^
6. Let F(xy z)—C be the equation of the charged surface.
Then (xyz) has to be constant over this surface and satisfy
dx 2 dy 2 dz 2
Put 2=£f, then is a function of x, y, ?, which is constant
when F(#, y,k£) = C, and which satisfies
dfy.dtydty
dx 2  dy 2 ^ dtp
Therefore if we regard (x y f ) as Cartesian coordinates of a
point, <£> is the potential at external points of an electrostatic
free distribution on the surface F(#, y, k%) = C The com
ponents of electric force due to this distribution, at a point
\x y f) on the surface, are
dej) dej) d(f>
dx dy d£'
This force acts in the normal to the surface, and is pro
portional to the surfacedensity at (x y J), which we shall
call d dcj>\ JdF d¥ dF\/ //rfFy , (dF\* TSFy
But d _ , d
dl~ di ;
h erefore, denoting differentiation with respect to x y z by
ub scripts 1 2 3,
(&, fe *•)= A<7'(F 1; F 2 , F 3 )/ s/W+W+1^7.
Now let o" be the surfacedensity at (x y z) on the moving
conductor F(# y z) = C, then equating a to the normal com
492 Mr. W. B. Morton on the
ponent of (J'g h)
_ /F t + g i\ + hF 3 _
VF^+F^ + Fa 2 '
(7 =
or putting in the values we have found for (f g h) in terms
of <£,
fc^ + foFa + fffaF,
VF^ + F^ + Fs 2
V F 1 2 4F 2 2 + F 3 2 '
Now the perpendicular from the origin on the tangent plane
to F(# y 2) = C at the point (so y z) is
_ gFi + ,yF 2 + zF 3
P 7F 1 8 + F 2 2 + F 8 2r
and the perpendicular from the origin on the tangent plane
to F(#, y, kty = G at (# y £) is
/(d¥\* , /tfF\* /rfFV
VF^ + F^ + PFs 2
= a?F!+yF 8 fgF 8
VF^ + F^ + FFs 2 '
CT __ A/9
If now"F(#y *) = C is an ellipsoid, then we know that
o' oc p 7 , therefore also a (x p, that is the arrangement of
charge on the moving ellipsoid is the same as if it were
at rest.
7. Applying the above to the ellipsoid (a b c), we find that
e_C" d\
^X \/(a 2 + X){b 2 + X){c 2 + k 2 X)'
Putting 6= a, c=ka, we get
Showing that, as Mr. Heaviside pointed out, the field of a
point charge is given when the conductor is an oblate
spheroid whose axes have the ratio 1 : k.
For a sphere the integral becomes
, e , + k'a
where U = \/\— k~ = xp
and 6 is given 4>y
To test the value of <£ let us make k' approach zero, i. e. the
motion becomes infinitely slow. 6 is then =r.
_ , Lt , (r4 &'a) — log(r— k'a)
Then *=8^^ lo S^ S^ "
— e ^ — e
§ira* r ~ 47rr*
494 Electromagnetic Theory of Moving Charges.
8. The mutual energy of a moving charge and external
magnetic field has been given by Mr. Heaviside for the case
of motion which is very slow compared with the 4 velocity of
radiation. It is eu A . cos (uA), S where A is the circuital
vector potential of the external field. Mr. Larmor, in the
second part of his " Dynamical Theory " (Phil. Trans. 1895,
p. 717), concludes that the same expression holds good for
motion at any speed. He seems, however, to overlook the
fact that in the general case the displacementcurrents in the
medium — being no longer derivable from a potential function
— will make their appearance in the result as well as the
convectioncurrent eu.
If (F G H) is the vector potential, the part of the energy
corresponding to the displacement currents will be
which in the case we have been considering becomes
J\ dz dz dz J
dxdz dy dz dz 1 )
But by a wellknown transformation, when we take the
integral through all space, we have
K
dx dz dy dz dz* )
' J dz \dx dy dz )
= since (F GrH) is circuital.
.*. The expression for this part of the energy reduces to
«(l^)jHgf^ = ^JHg^.
Therefore if the velocity u ceases to be negligible in com
parison with V, we have a correction of the second order in
u
the ratio ^ in addition to the expression involving the
convection current simply. It also appears from the above
that the force on the moving charge cannot, unless this term
be neglected, be expressed in terms of the magnetic intensity
at the charge, but will depend on the entire field*
[ 495 ]
LIV. On a Simple Apparatus for determining the Thermal
Conductivities of Cements and otlier Substances used in the
Arts. By Charles H. Lees, D.Sc, and J. D. Chorlton,
B.Sc, Joule Scholar of the Royal Society*.
THE following method of determining the thermal con
ductivities of bad conductors has been designed with
the object of simplifying as much as possible the apparatus
and observations required in making determinations which
are not required to be of the highest order of accuracy, but
in which errors of more than 2 per cent, are to be avoided.
When the constants of the apparatus have once been deter
mined, the only observations necessary to determine a
conductivity are the thickness of a sheet of the substance
used and the temperatures of three thermometers. It is
hoped that this simplification will lead those who require bad
conductors of heat for structural purposes to carry out their
own tests of the materials they have available.
Method. — The apparatus consists of a flat cylindrical metal
box, of 11*4 cms. diameter and 3 cms. depth, through which
steam . can be passed. The bottom of the box consists of
a circular brass plate 1*3 cm. thick in which a radial hole
reaching to the centre is bored. In this hole a thermometer
is placed with its bulb at the centre of the plate. The top
and sides of the box are covered with green baize to prevent
loss of heat as far as possible. This vessel is supported on
a circular plate of the material to be tested, which in its turn
is supported on a brass disk similar to the one forming the
base of the heating vessel, and like it provided with a radial
hole and thermometer.
The lower disk is suspended horizontally from a support
by three strings attached to three short pegs projecting from
the edge of the disk, and hangs about 30 centims. above the
table. About 10 centims. above the table a thermometer is
placed horizontally, with its bulb under the centre of the
lower plate to give the temperature of the air ascending to
the disks. The bulb of the thermometer is protected against
radiation from the lower disk by a bright metal screen.
The two surfaces of the disks which come into contact with
the material to be experimented on are amalgamated, so that
if the material is a solid, contact over its entire surface may
be obtained by using a thin mercury film between the solid
and disks. In order to determine the thickness of the
* Communicated by the Authors.
496 Dr. Lees and Mr. Chorlton on the Thermal
material used, two short pegs of brass, one vertically above
the other, project from the disks, and their distance apart is
measured by means of a wire gauge or calipers, when the
disks are in contact and when the plate to be tested is between,
the difference is the thickness of the plate of the material
tested. The under surface of the lower disk may be kept
polished, or it may be painted, in order that the heat radiated
from it may remain constant during a series of experiments.
When steam is passed through the upper cylinder the tem
perature of the upper disk is raised to nearly 100° C. ; heat
flows through the plate of material experimented on to the
lower disk, the temperature of which is therefore raised above
that of the surrounding air. It begins in consequence to lose
heat by radiation and conduction to the air, and eventually a
stage is reached when this loss of heat is equal to the heat
received from the material experimented on.
Hence if the amount of this loss is found by a separate
experiment, a determination of the temperature gradient in
the material experimented on at its surface of contact with
the lower plate, will enable the thermal conductivity of the
material to be found.
Theory.
If is the temperature at a point x above the under surface
of the lower plate, O the temperature of the air, k x the internal
conductivity, h { the external conductivity or " emissivity,"
Newton's law being supposed to hold for the limits of tem
perature used, p. the perimeter, q the area, of cross section of
the plate, the differential equation for the motion of heat in
the plate, the isothermals being assumed plane, is satisfied if
6  e^A, cosh y/tfc . x + B x sinh /y/gi . .*,
where A x and B t are constants.
The condition for continuity of flow at the under surface is
ki r —h\6 for m = 0,
ax
i. e.
vg
B^AA;
— O =A 1 \ cosh
^.^^sinh^.,}
1 V qh ,
Conductivities of Cements and other Substances. 407
Now for the lower brass disk
2?/q = 27rr/7rr 2 = 2/57 = '35l i
while ky_ for brass ="25, and it will be shown presently that
/< 1 = 0003 about. Hence a /PI* = '02, and since the thick 
 qh
ness of the plate is 1*3 cm. the maximum value of the expression
in brackets is
cosh 026 + ^ 003 sinh '026 = 100 18 ;
•25 x V2
or the temperature of the lower plate varies less than J per
cent., and for the purpose of the present experiment may be
taken to be uniform throughout and equal to the indication 6 l
of the thermometer in the centre of the disk. The tempera
ture of the lower disk is therefore given completely by the
equation
The rate of flow of heat into the lower disk, the thickness of
which is 1*3 centim., is given by k x j for #=1*3. It is
therefore dx
=h \Ar {#i 0o} ( sinh 026 + ^= . cosh '026 }
V gfc 1 I hy/Eh }
V gk x
= (^i^o) (l'00026 h + 026 *, \/^ )
= (^i^o)(/'i + 00013)
= (^i^o)(/ii + V), say, where /*/ = '00013,
If k is the internal, h the external conductivity of the medium
under test, the temperature at a point x above its under
surface is given by : —
Q O = (0i 0o) { cosh \J^r . a + B sinh y^ . x j ,
where B is a constant, the value of which is fixed by the
Phil. Mag. S. 5. Vol. 41. No. 253. June 1896. 2 M
498 Dr. Lees and Mr. Chorlton on the Thermal
condition for continuity of flow at the surface of contact of
the material and the lower plate, i. e.
v
. t .B=(tf 1 0 o )(A 1 + V).
qk
Hence in the material under test
ee o ={0e o \{cosh /K M.x+ A±V sinh m a <
L V 2* Jc /ph V qk J
V qk
If b is the thickness of the material, at iv = b we must have
6 = 6 2 , the temperature indicated by the thermometer in the
upper brass plate, i. e.
^^o = (^i~^o)(cosh A /Z\5 + A±V >sinll .MA
\ V qk k /ph V qk J
V q k
The maximum value of \ /PA . b for the materials experi
V qk
mented on is "1 ; hence the hyperbolic functions can be
expanded in ascending powers of \ / P— . b, and terms invol
V qk
ving cubes can be neglected. Thus : —
,,,., W ,.,( 1+ g, + A± v /g. s ),
v qk
or #2 — 01 & (i , 7 f . ph j\
The last term in the bracket never exceeds 4 per cent, of the
first two terms, and the final result will be affected only to a
very small extent if for h, /i A is substituted.
Hence we have
which enables Jc to be calculated from the result of the expe
riment if A, has been previously determined.
Determination of Ji 1% — To determine the value of this quan
tity the two plates were separated by a layer of air 3 mm.
thick, enclosed by a ring of badly conducting material upon
Conductivities of Cements and other Substances. 499
which the upper plate rested ; steam was passed through the
box at the top, and the temperature of the upper plate raised to
nearly 100° C. The lower plate was next heated a few degrees
above 100° 0. by means of a gasflame and then allowed to
cool, and as soon as its temperature fell to 100° C. the supply
of steam to the upper can was cut off and the hot water in
the can allowed to run out.
Both plates were then found to cool at almost the same
rate ; the difference in the temperatures of the two plates never
exceeded 1° C., the upper plate being always the warmer.
Since the conductivity of air is only about '00005 C.Gr.S.
units, the amount of heat conducted from the upper plate to
the lower is negligible, so that each plate cools independently
of the other.
The thermometer in the lower plate was watched as the
temperature fell, and the times at which the mercury passed
every alternate degree were observed ; the results are given
in the table below.
The temperature of the air was steady at 18° C.
Temperature of
lower plate.
Time.
Fall of
temp, per
second.
Loss of
heat, gram
degrees per
second.
Loss of
heat per sq.
cm., gram
degrees per
second.
h v
»i
O0 O 
°c.
°C.
h m s
96
78
3.17.42
94
76
18.37
'0370
392
•0255
•000336
92
74
19.32
354
75
43
28
90
72
20.30
340
60
34
25
88
70
21.30
325
44
23
19
86
68
22.33
308
26
12
12
84
66
23.40
301
14
04
09
82
64
24.46
290
07
•0199
11
80
62
25.58
274
290
86
03
78
60
27.12
263
79
81
02
76
58
28.30
256
71
76
03
74
56
29.48
250
65
72
07
72
54
31.10
244
56
68
11
70
52
32.32
235
49
62
12
68
50
34.0
220
33
51
02
66
48
35.34
208
20
43
•000298
64
46
27.12
197
08
35
93
62
44
38.57
187
198
29
93
60
42
40.46
180
94
24
95
58
40
42.40
169
79
16
90
56
38
44.42
156
65
07
82
54
36
46.57
146
55
•0101
81
52
34
49.15
140
48
•0096
82
50
32
51.43
2M2
500 Dr. Lees and Mr. Chorlton on the Thermal
The numbers in the 4th column are found by subtracting
alternate times in the 3rd column from each other and
dividing 4° 0. by the differences.
The numbers in the 5th column are obtained from those in
the 4th by multiplying by 1140 grams, the mass of the lower
disk, and by *093, the specific heat of brass, i. e. by 106.
The numbers in the 6th column are obtained by dividing
by the total radiating surface of the lower disk, plus half the
radiating surface of the nonconducting ring = 148*5 + 5*5
= 154 sq. centims.
The values of h are obtained from these numbers by dividing
by the excess of the temperature of the lower disk over that
of the air.
Experiments.
The following account of an experiment on a plate of glass
will show the method of treatment in each case : —
A disk of plateglass of diameter equal to that of the brass
plates — 11*4 centims. — was placed between them, and contact
made by means of thin mercury films. Good contact was
easily secured between the glass and the lower plate ; but
with much more difficulty in the case of the upper plate.
Contact was eventually obtained by covering the surface
of the upper plate with mercury, so that it adhered in pendent
drops, at the same time placing a few drops of mercury on
the surface of the glass, and then carefully lowering the
upper plate on to the glass and allowing the excess of mercury
to run out at the edges.
Steam was then passed through the box attached to the
upper brass plate, and thermometers inserted in the upper
and lower plates, the temperature of the air being indicated
by a third thermometer placed as described above.
After a time, the duration of which depended on the con
ductivity and thickness of the substance experimented upon,
the thermometers became steady and the temperatures were
observed.
2 = temperature of upper plate = 96' 1° C.
X = „ „ lower „ =924°C.
6 = „ „ air =200°C.
Thickness = *292 centim.
Conductivities of Cements and other Substances. 501
The thickness was obtained by measuring with a micro
metergauge the distance between the two pegs, first, when
the plates were in contact with each other, and, secondly,
when the glass was between them.
The following table gives the observations taken during
each experiment, and shows the method of determining the
conductivities from the observations : —
Remarks on Specimens Experimented on.
(a) The plates of Portland cement and plaster of Paris, &c, were made
by pouring the liquid cement on to a glass plate, on which three
equal beads strung together were laid, a second glass plate was
pressed down upon these beads, after a few hours the glass plates
could be removed and a plate of cement of uniform thickness was
obtained.
The plate of plaster of Paris and sand consisted of two parts
by weight of plaster of Paris to one part by weight of sand.
(b) When the substance experimented upon was a powder, the brass
plates were kept apart by three stops of wood, and the powder was
prevented from falling out at the edges by a very narrow circular
ring of fibre.
(c) The three values given for the conductivity of gardensoil refer to
three specimens of soil of the same kind but containing different
amounts of moisture ; the first value refers to dry soil, the second to
slightly moist, and the third to damp soil. The results show how
greatly a small amount of moisture affects the conductivity.
(d) The calico in the first experiment was dry, and weighed 891 grams.
In the second experiment it was exposed for some time to a damp
atmosphere, and its weight increased to 906 grams, so that the con
ductivity of calico increases 20 per cent, for 16 per cent, increase in
weight due to moisture absorbed.
(e) In the three experiments on old flannel, the flannel contained different
amounts of moisture. In the first experiment the flannel was quite
dry and weighed 8*21 grams, in the second it was exposed to a
damp atmosphere, and its weight was 8*29 grams, and in the third
experiment was damped with water and its weight increased to
9'05 grams. Thus the conductivity of flannel increased about 11 per
cent, for 1 per cent, increase in weight due to absorbed moisture,
and afterwards increased 25 per cent, more for a further increase
of 10 per cent, in weight.
The flannel in the third experiment is almost as good a conductor
as dry calico.
The new flannel, flanelette, silk, and linen were dry.
502
Dr. Lees and Mr. Chorlton on the Thermal
t~
CO
CO
o
CM
Q
rl
iO
CO
1—1
©
^
i
iO
Ki
©
^H
co
■*
CD
TtH
t^
t
©
co
IO
*k
CO
CM
CM
©
8
©
©
©
©
©
©
CM
©
o
©
©
©
©
©
©
8
8
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
©
tO
CM
00
CO
Tl^
CO
CO
CM
CM
1
CD
CD
o
*
^
n
cb
"*
CD 1
b
t
CD
iO
kQ
b
iC
CD
CD
CO
co
co
•o
u
CD
On
00
00
©
b
©
cp
t
i— i
i— l
M
©
©
t— 1
1
O CO
•*
»b
CO
TH
b
©
ob
ob
CO
Th
©
rL
ib
CD
r " H
CM
M
1—1
1—1
I— 1
■^
CM
* +
T %iO!D
b
N
t~
»o
CO
lO
CD
iO
iO
CO
CD
CO
iO
b
*
rti
<*
1 ^
"*l
"tf
■*
^