THE LIBRARY
OF
THE UNIVERSITY
OF CALIFORNIA
LOS ANGELES
UNIVERSITY of CALIFORNIA
AT
LOS ANGELES
Lk'xARY
SCIENTIFIC PAPERS
C. J. CLAY AND SONS,
CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,
AYE MARIA LANE.
: 50, WELLINGTON STREET.
F. A. BROCKHAUS.
ftrto gork: THE MACMILLAN COMPANY.
Bombast E. SEYMOUR HALE.
SCIENTIFIC PAPERS
BY
JOHN WILLIAM STEUTT,
BARON RAYLEIGH,
D.Sc., F.RS.,
HONORARY FELLOW OF TRINITY COLLEGE, CAMBRIDGE,
PROFESSOR OF XATCRAL PHILOSOPHY IX THE ROYAL INSTITUTION'.
VOL. II.
1881-1887.
CAMBRIDGE:
AT THE UNIVERSITY PRESS.
1900
[All Eightt roared.}
PRINTED BY J. AND C. P. CLAY,
AT THE UNIVERSITY PRESS.
PKEFACE.
A MOXG the Papers here reprinted several, relating to the Electrical Units,
-^- were written conjointly with Prof. Schuster and Mrs Sidgwick.
It may perhaps be well to remind the reader that at the time of these
researches the ohm was uncertain to the extent of 4 per cent., and that
the silver equivalent then generally accepted differed 2 per cent, from
the value arrived at by us.
TKRLIXG PLACE, WITHAM,
October 1900.
The works of the Lord are great,
Sought out of all them that have pleasure therein.
CONTENTS.
AKT. PAGE
79. On the Determination of the Ohm [B. A. Unit] in Absolute
Measure. By Lord Rayleigh, F.R.S., and Arthur Schuster,
Ph.D., F.R.S 1
Part I. By Lord Rayleigh 1
Part II. By Arthur Schuster 20
Adjustment of the Instruments and Determination of
Constants 20
The Observations ........ 24
Air Currents 28
Reduction of Observations ....... 30
Results 34
[Proceedings of the Royal Society, xxxn. pp. 104141, 1881.]
80. Experiments to Determine the Value of the British Association
Unit of Resistance in Absolute Measure .... 38
Measurements of Coil . . . . . . . . 51
Calculation of GK 53
Calculation of L . . . 53
Theory of the Ring Currents 54
L by Direct Experiment 55
Correction for Level ........ 63
Correction for Torsion ........ 64
Value of GK corrected for Level and Torsion ... 64
Calculation of U 64
Measurement of tan/* ........ 64
Measurement of D . 65
Reduction of Results 66
Comparison with the Standard B. A. Units ... 75
[Phil. Tran*. CLXXUI. pp. 661697, 1882.]
v jij CONTENTS.
ART. PAGE
81. On the Specific Resistance of Mercury. By Lord Rayleigh and
Mrs H. Sidgwick . . 78
[Phil. Trans. CLXXIV. pp. 173185, 1882.]
82. The Use of Telescopes on Dark Nights . . . . . 92
[Cambridge Phil. Soc. Proc. iv. pp. 197, 198, 1882.]
83. On a New Form of Gas Battery . . . . . 94
[Cambridge Phil Soc. Proc. iv. p. 198, 1882.]
84. Acoustical Observations. IV. . . . . . . . 95
On the Pitch of Organ-Pipes 95
Slow versus Quick Beats for comparison of Frequencies of
Vibration 97
Estimation of the Direction of Sounds with one Ear . 98
A Telephone-Experiment ....... 99
Very High Notes. Rapid Fatigue of the Ear ... 99
Sensitive Flames 100
[Phil. Mag. xin. pp. 340347, 1882.]
85. Further Observations upon Liquid Jets, in Continuation of those
recorded in the Royal Society's ' Proceedings ' for March and
May, 1879 103
On some of the Circumstances which influence the Scatter-
ing of a nearly Vertical Jet of Liquid . . . .103
Influence of Regular Vibrations of Low Pitch . . . 106
The Length of the Continuous Part 110
Collision of Two Resolved Streams . . . . .112
Collision of Streams before Resolution . . . . 115
[Proceedings of the Royal Society, xxxiv. pp. 130145, 1882.]
86. Address to the Mathematical and Physical Science Section of the
British Association . . . . . . . . .118
[British Association Report, pp. 437441, 1882.]
87. On the Tension of Mercury Vapour at Common Temperatures . 125
[British Association Report, p. 441, 1882.]
88. On the Absolute Measurement of Electric Currents . . 126
[British Association Report, pp. 445, 446, 1882.]
CONTEXTS. Vf.
ART. PAGE
89. On the Duration of Free Electric Currents in an Infinite Con-
ducting Cylinder . . . 128
[Brititk Aaociatio* Report, pp. 446, 447, 1882.]
90. On the Equilibrium of Liquid Conducting Masses charged with
Electricity 130
[PAH. Mag. xrv. pp. 184186, 1882.]
91. On an Instrument capable of Measuring the Intensity of Aerial
Vibrations 132
[PhO. Mag. HT. pp. 186, 187, 1882.]
92. Comparison of Methods for the Determination of Resistances in
Absolute Measure . . 134
L Kirchhoffs Method, Maxwell's Electricity and Magnetism,
759 135
II. Weber's Method by Transient Currents, Maxwell, 760 137
III. Method of Revolving Coil . .- . . . .139
IV. Method of Foster and Lippmann ... . . 143
V. Weber's Method by Damping 145
VL Lorenz's Method 145
[Phil. Mag. nv. pp. 329346, 1882.]
93. On the Dark Plane which is formed over a Heated Wire in
Dusty Air 151
[Proceeding* of the Royal Society, xxxiv. pp. 414 US, 1882.]
94. Experiments, by the Method of Lorenz, for the Further Determ-
ination of the Absolute Value of the British Association Unit
of Resistance, with an Appendix on the Determination of the
Pitch of a Standard Tuning-Fork. By Lord Rayleigh and
Mrs H. Sidgwick . 155
Details of Measurements :
Diameter of Disc 167
The Induction-Coils 168
The Distance-Pieces 169
The Induction-Coefficients 170
The Resistance-Coils 171
Appendix: Frequency of Vibration of Standard Fork . 177
Second Appendix : On the Effect of the Imperfect Insulation
of Coils . . . . 182
[PhU, Tnuu. CLXXIV. pp. 295322, 1883.]
x CONTENTS.
ART. PAGE
95. On the Mean Radius of Coils of Insulated Wire . . . 184
[Cambridge Phil, Soc. Proc. iv. pp. 321324, 1883.]
96. On the Invisibility of Small Objects in a Bad Light . . 187
[Cambridge Phil. Soc. Proc. iv. p. 4, 1883.]
97. On Maintained Vibrations . . . . . . . .188
[Phil. Mag. xv. pp. 229235, 1883.]
98. The Soaring of Birds , . . . . ... .194
[Nature, xxvu. pp. 534, 535, 1883.]
99. Distribution of Energy in the Spectrum . , . . . . 198
[Nature, xxvn. pp. 559, 560, 1883.]
100. Investigation of the Character of the Equilibrium of an In-
compressible Heavy Fluid of Variable Density . . . 200
[London Math. Soc. Proc. xiv. pp. 170177, 1883.]
101. On the Vibrations of a Cylindrical Vessel containing Liquid 208
[Phil. Mag. xv. pp. 385389, 1883.]
102. On the Crispations of Fluid resting upon a Vibrating Support 212
[Phil. Mag. xvi. pp. 5058, 1883.]
103. On Porous Bodies in Relation to Sound .... 220
[Phil. Mag. xvi. pp. 181186, 1883.]
104. Suggestions for Facilitating the Use of a Delicate Balance . 226
[British Association Report, pp. 401, 402, 1883.]
105. On the Imperfection of the Galvanometer as a Test of the
Evanescence of a Transient Current. 228
[British Association Report, pp. 444, 445, 1883.]
106. On Laplace's Theory of Capillarity 231
[Phil. Mag. xvi. pp. 309315, 1883.]
107. On the Measurement of Electric Currents . ... . 237
[Cambridge Phil. Soc. Proc. V. pp. 5052, 1883.]
108. On the Circulation of Air observed in Kundt's Tubes, and on
some Allied Acoustical Problems 239
[Phil. Trans. CLXXV, pp. 121, 1883.]
CONTENTS. xi
ART. PAGE
109. The form of Standing Waves on the Surface of Running
Water 258
[London Math. Soc. Proc. xv. pp. 6978, 1883.]
110. Acoustical Observations. V 268
Smoke-jets by Intermittent Vision ..... 268
Smoke-jets and Resonators ...... 269
Jets of Coloured Liquid 270
Fish-tail Burners 272
Influence of Viscosity. ....... 273
[Phil. Mag. xvil. pp. 188194, 1884.]
111. On the Measurement of the Electrical Resistance between
Two Neighbouring Points on a Conductor .... 276
[Cambridge Phil. Soc. Proc. v. pp. 133, 134, 1884.]
112. On the Electro-Chemical Equivalent of Silver, and on the
Absolute Electromotive Force of Clark Cells. By Lord
Rayleigh, and Mrs H. Sidgwick 278
The Fixed Coils 289
The Suspended Coil 290
Determination of Mean Radius of Suspended Coil . . 291
Calculation of Attraction ....... 295
The Silver Voltameters 297
Appendix 327
Explanation of Figures ....... 328
Notes.
Note to 25 329
Note to 26 329
Note to 27 330
Note to 30 331
Note to 32 331
Note 1 to 37 331
Note 2 to 37 332
[Phil. Tram. CLXXV. pp. 411460, 1884.]
113. Presidential Address 333
[British Association Report, pp. 123. Montreal, 1884.]
114. A Lecture Experiment on Induction 355
[British Association Report, p. 632, 1884.]
115. On Telephoning through a Cable . . . . . .356
[British Association Report, pp. 632, 633 1884.]
xii CONTENTS.
PAGE
116. On a Galvanometer with Twenty Wires. . ... 357
[British Association Report, p. 633, 1884.]
117. On Clark's Standard Cells . . . - 359
[British Association Report, pp. 651, 652, 1884.]
118. On the Constant of Magnetic Rotation of Light in Bisulphide
of Carbon ! ... 360
The Helix 367
Correction for Finite Length . 368
Appendix: Notes on Polarimetry in general . . . 378
Postscript 383
[Phil Tram. CLXXVI. pp. 343366, 1885.]
119. Optics ; .... 385
[Encyclopedia Britannica, xvn. 1884.]
120. tiber die Methode der Dampfung bei der Bestimmung des
Ohms . . .415
[Annalen dei- Physik und Chemie, Band xxiv. pp. 214, 215, 1885.]
121. On the Theory of Illumination in a Fog .... 417
[Phil. Mag. xix. pp. 443446, 1885.]
122. A Monochromatic Telescope, with application to Photometry 420
[Phil. Mag. xix. pp. 446, 447, 1885.]
123. Self-induction in Relation to Certain Experiments of Mr
Willoughby Smith and to the Determination of the Ohm . 422
[Nature, xxxn. p. 7, 1885.]
124. Professor Tait's "Properties of Matter" 424
[Nature, xxxn. pp. 314, 315, 1885.]
125. A Theorem relating to the Time-Moduli of Dissipative Systems 428
[British Association fiepwt, pp. 911, 912, 1885.]
126. On the Accuracy of Focus necessary for Sensibly Perfect
Definition , 430
[Phil. Mag. xx. pp. 354358, 1885.]
127. On an Improved Apparatus for Christiansen's Experiment . 433
[Phil. Mag. xx. pp. 358360, 1885.]
CONTENTS. Xlll
AKT. PAGE
128. Optical Comparison of Methods for Observing Small Rotations 436
[Phff. Mag. xx. pp. 360, 361, 1885.]
129. On the Thermodynamic Efficiency of the Thermopile . . 438
[PArf, Mag. xx. pp. 361363, 1885.]
130. On Waves propagated along the Plane Surface of an Elastic
Solid 441
[London Math. Soe. Proe. xvn. pp. 411, 1885.]
131. On Prof. Himstedt's Determination of the Ohm ... 448
[Pha. Mag. XXL pp. 1013, 1886.]
132. On the Clark Cell as a Standard of Electro-motive Force . 451
[Pka. Tran*. CLXXTI. pp. 781800, 1886.]
133. Testing Dynamos 474
[Electrical Review, xvm. p. 242,
134. The Reaction upon the Driving-Point of a System executing
Forced Harmonic Oscillations of Various Periods, with
Applications to Electricity . . . . . . .475
[Phil. Mag. xxi. pp. 369381, 1886.]
135. On the Self-Induction and Resistance of Straight Conductors 46
[PhU. Mag. XXL pp. 381394, 1886.]
136. On the Colours of Thin Plates ....... 498
[Edinburgh Trans, xxxm. pp. 157170, 1886.]
137. Notes, chiefly Historical, on some Fundamental Propositions
in Optics - . ..... ... 513
. Mag. XXL pp. 466476, 1886.]
138. On the Intensity of Light Reflected from Certain Surfaces at
Nearly Perpendicular Incidence ...... 522
Description of Apparatus ....... 525
Prism of Crown Glass (I) ....... 534
Prism of Crown Glass (H) ...... 537
Plate Glass Silvered Behind ...... 538
Silver-on-Glass Speculum . ...... 539
Mirror of Black Glass ....... 539
[Proceeding* of the Royal Society, XLL pp. 275-294, 1886.]
XIV CONTENTS.
ART. PAGE
139. Notes on Electricity and Magnetism. I. On the Energy
of Magnetized Iron . . . ... . . . . 543
[Phil. Mag. xxn. pp. 175183, 1886.]
140. Notes on Electricity and Magnetism. II. The Self-induction
and Resistance of Compound Conductors . . . .551
The Interrupters 553
The Induction-Compensators ...... 555
Appendix. The Induction-Compensators [p. 557] . . 577
[Phil. Mag. xxii. pp. 469500,
141. Notes on Electricity and Magnetism. III. On the Behaviour
of Iron and Steel under the Operation of Feeble Magnetic
Forces ........... 579
[Phil, Mag. xxm. pp. 225245, 1887.]
79.
ON THE DETERMINATION OF THE OHM [B. A. UNIT] IN
ABSOLUTE MEASURE. BY LORD RAYLEIGH, F.R.S., AND
ARTHUR SCHUSTER, PH.D., F.R.S.
[Proceedings of the Royal Society, xxxn. pp. 104 141, 1881.]
Part I. By Lord RAYLEIGH.
IT is generally felt that considerable uncertainty still attaches to the
real value of the ohm, or British Association unit of resistance. The ohm
was constructed to represent 10 9 c.G.s. absolute units, but according to
Kohlrausch* it is nearly 2 per cent, too great, and according to Rowland f
nearly 1 per cent, too small. On the other hand, H. Weber J has obtained by
more than one method results very nearly in harmony with those of the
British Association Committee. Influenced partly by the fact that the
original apparatus (though a good deal out of repair) and the standard coils
themselves were in the Cavendish Laboratory, I determined last June to
repeat the measurement by the method of the Committee, which has been
employed by no subsequent experimenter, and sought permission from the
Council of the British Association to make the necessary alterations in the
apparatus. In this way I hoped not merely to obtain an independent result,
but also to form an opinion upon the importance of certain criticisms which
have been passed upon the work of the Committee.
The method, it will be remembered, consists in causing a coil of insulated
wire, forming a closed circuit, to revolve about a vertical axis, and in
observing the deflection from the magnetic meridian of a magnet suspended
at its centre, the deflection being due to the currents developed in the coil
under the influence. of the earth's magnetism. The amount of the deflection
' Phil. Mag. vol. XLVII. p. 294, 1874.
t American Journal of Science and Arts, 1878.
t Phil. May. vol. v. p. 30, 1878.
2 ON THE DETERMINATION OF THE OHM [B. A. UNIT] [79
is independent of the intensity of the earth's magnetic force, and it varies
inversely as the resistance of the circuit. The theory of the experiment
is explained very fully in the reports of the Committee* and in Maxwell's
Electricity and Magnetism, section 763. For the sake of distinctness, and as
affording an opportunity for one or two minor criticisms, a short statement
in the original notation will be convenient :
H = horizontal component of earth's magnetism.
7 = strength of current in coil at time t.
G = total area inclosed by all the windings of the wire.
a) = angular velocity of rotation.
= ut = angle between plane of coil and magnetic meridian.
M = magnetic moment of suspended magnet.
</> = angle between the axis of the magnet and the magnetic meridian.
K = magnetic force at the centre of the coil due to unit current in
the wire.
L = coefficient of self-induction of coil.
R = resistance of coil in absolute measure.
MHr = force of torsion of fibre per unit of angular rotation.
The equation determining the current is
whence
~ w cos (tot -</>), (1)
7 = -jp+ftrf {GH(R cos B + L<o sin 6) + KM(R cos (0 - <) + Lw sin (0 - (/>))}.
...... (2)
If L were zero, or if the rotation were extremely slow, the current would
(apart from KM ) be greatest when the coil is passing through the meridian.
In consequence of self-induction, the phase of the current is retarded, and its
maximum value is diminished. At the higher speeds used by the Committee,
the retardation of phase amounted to 20.
To find the effect of (2) upon the suspended needle, we have to introduce
MK and the resolving factor cos (0 <f>), and then to take the average.
This, on the supposition that the needle remains on the whole balanced at <,
must be equal to the force of restitution due to the direct action of the
earth's magnetism and to torsion, i.e., MH sin $ -f MHr <j>. Thus
& + KMR \ ~ MH ( sin
* Collected iu one volume. Spon, London, 1873.
79] Df ABSOLUTE MEASURE. 3
In the actual experiment T is a very small quantity, say jfa; and the
distinction between rtj> and rsin <f> may he neglected.
If we omit the small terms depending upon T and upon MKjGH, we get
on solution and expansion of the radical
...... (4)
The term in tan 4 <f> is not given in the report of the Committee ; but, as
I learn from Mr Hockin through Dr Schuster, it was included in the actual
reductions. But the next term in tan* <, and one arising from a combination
of the correction for self-induction with that depending on M. are not
altogether insensible, so that probably the direct use of the quadratic is
more convenient than the expansion. At the high speeds used by the
Committee the correction for self-induction amounted to some 3 per cent.,
and therefore cannot be treated as very small.
If the axis of rotation be not truly vertical, a correction for level is
necessary. In the case of coincidence with the line of dip, no currents,
due to the earth's magnetism, would be developed. If the upper end of
the axis deviate from the vertical by a small angle /? towards the north,
the electromotive forces are increased in the ratio cos (/+/?) : cos/, i.e. T in
tile ratio 1 + tan I .ft, I being the angle of dip. A deviation in the east and
west plane will have an effect of the second order only. The magnetic forces
due to the currents will not act upon the needle precisely as if the plane of
the coil were always vertical, but the difference is of the second order, so
that the whole effect of a small error of level may be represented by writing
(1 + tan/.) for G in (3) or (4>
The next step is to express GK in terms of the measurements of the
coiL In order that there may be a passage for the suspending fibre and its
enveloping tube, it is necessary that the coil be double, or if we prefer so to
express it, that there be a gap in the middle. If [see figure]
a = mean radius of each coil,
n = whole number of windings,
6 = axial dimension of section of each coil,
c = radial dimension of section of each coil,
b' = distance of mean plane of each coil from the axis of motion,
a = angle subtended at centre by radius of each coil, ao that cot o = 6'/o,
4
then
K =
ON THE DETERMINATION OF THE OHM [B. A. UNIT] [79
(5)
sin 3 a 1 1 + ^- 4 , (2 - 1 5 sin 2 a cos 2 a) + ^ -- (1 5 sin 2 a cos 2 a - 3 sin 2 a) I ,
so that
GK= 27r 2 H 2 asin 3 a l + i + f
siira cos 2 a - sin 2 a
(7)
The correction due to the finiteness of b and c is in practice extremely small,
but the factor sin 3 a must be determined with full accuracy.
In order to arrive at the value of MKJGH, which occurs in (3), we
observe that the approximate value of K/G is 2 sin 3 a/a 3 ;
so that MK/GH is equal to tan p,, where /t is the angle
through which the needle of a magnetometer is deflected
when the suspended magnet (M) is placed at a distance
from it a/sin a to the east or west, with the magnetic
axis pointing east or west. In practice the difference of
readings when M is reversed is taken in order to double
the effect, and any convenient distance is used in lieu
of a/sin a, allowance being easily made by the law of
cubes.
The correction for torsion is determined by giving
the suspended magnet one (or more) complete turns,
and observing the displacement. If this be B ly reckoned
in divisions of the scale, i.e., in millimetres, and D be
the distance from the mirror to the scale reckoned in millimetres,
The correction for scale reading, necessary in order to pass from tan 2</> to
tan 0, will be explained under the head of reductions.
Corrections depending upon irregularity in the magnetic field, and in the
adjustment of the magnet to the centre of the coil, are given in the report.
They are exceedingly small. The same may be said of errors due to im-
perfect adjustment of the coil with respect to the axis of rotation.
In remounting the apparatus the first point for consideration was the
driving gear. The Committee used a Huyghens' gearing, driven by hand, in
conjunction with a governor. This, it appeared to me, might advantageously
be replaced by a water-motor ; and Bailey's " Thirlmere " engine, which acts
79] IN ABSOLUTE MEASURE. 5
by the impulse of a jet of water upon revolving cups, was chosen as suitable
for the purpose. As the pressure in the public water pipes is not sufficiently
uniform, it was at first intended to introduce a reducing valve; but on
reflection it seemed simpler to obtain a constant head of water by connecting
the engine with a small cistern at the top of the building. This cistern is
just big enough to hold the ball-tap by which it is supplied, and gives at the
engine a head of about 50 feet.
The success of this arrangement depends upon attention to principles, as
to which it may be well to say a few words. The work done by many prime
movers is within practical limits proportional to the speed. If the work
necessary to be done in order to overcome resistances, as in overcoming solid
friction, or in pulling up weights, be also proportional to the speed, there is
nothing to determine the rate of the engine, and in the absence of an
effective governor the motion will be extremely unsteady. In general the
resistance function will be of the form
in which the above-mentioned resistances are included under B. The term
in C will represent resistances of the nature of viscosity, and that in D a
resistance such as is incurred in setting fluids in motion by a fan or other-
wise. By these resistances, if present, the speed of working will be determined.
In the water impulse engine, however, the work is not proportional to the
speed. At zero speed no work is done ; neither is any work done at a speed
such that the cups retreat with the full velocity of the jet. The speed of
maximum efficiency is the half of the last, and the curve representing work
as a function of speed is a parabola with vertex directed upwards. If we
draw upon the same diagram the curve of work and the curve of resistance,
the actual speed will correspond to the point of intersection, and will be well
or ill defined according as the angle of intersection is great or small. At the
higher speeds of the coil (four to six revolutions per second) so much air is
set in motion that the resistance curve is highly convex downwards, and no
difficulty is experienced in obtaining a nearly uniform motion. But when
the speed of rotation is as slow as once a second, the principal resistance is
due to solid friction, and the requisite curvature in the diagrams must be
obtained in the curve of work. It was necessary in order to obtain a
satisfactory performance at low speeds to introduce an additional reducing
pulley, so that the engine might run fast, although the coil was running slow.
The revolving coil with its frame, and the apparatus for suspending the
magnet, were at first arranged as described by the Committee. This
description, with drawings, is to be found in the report, and it is reproduced
in Gordon's Electricity and Magnetistn, vol. I. The water engine was ready
6 ON THE DETERMINATION OF THE OHM [fi. A. UNIT] [79
about the middle of June, and towards the end of the month the apparatus
was mounted by Mr Horace Darwin. During July and August preliminary
trials were made by Mr Darwin, Mrs Sidgwick, and myself, and various
troubles were encountered.
The only point in which the arrangement adopted by the Committee was
intentionally departed from was in the connexion of the magnet and mirror.
The magnet is necessarily placed at the centre of the revolving coil, but in
their arrangement the mirror is on the top of the frame and is connected to
the magnet by a brass wire. In order to save weight, I preferred to have
the magnet and mirror close together, not anticipating any difficulty from
the periodic and very brief interruption caused by the passage of the coil
across the line of sight. A box was, therefore, prepared with a glass front,
through which the mirror could be observed, and was attached to the end of
a brass tube coming through the hollow axle of the coil. This tube itself
was supported on screws resting on the top of the frame. The upper end
of the suspension fibre was carried by a tall tripod resting independently on
the floor.
The first matter for examination was the behaviour of the magnet and
mirror when the coil was spinning with circuit open. At low speeds the
result was fairly satisfactory, but at six or more revolutions per second a
violent disturbance set in. This could not be attributed to the direct action
of wind, as the case surrounding the suspended parts was nearly air-tight,
except at the top. It was noticed by Mr Darwin that even at low speeds a
disturbance was caused at every stroke of the bell. This observation pointed
to mechanical tremor, communicated through the frame, as the cause of the
difficulty, and the next step was to support the case surrounding the
suspended parts independently. A rough trial indicated some improvement,
but at this point the experiments had to be laid aside for a time.
From the fact that the disturbance in question was produced by the
slightest touch (as by a tap of the finger nail) upon the box, while the
upper parts of the tube could be shaken with impunity, it appeared that it
must depend upon a reaction between the air included in the box and the
mirror. It is known that a fiat body tends to set itself across the direction
of any steady current of the fiuid in which it is immersed, and we may fairly
suppose than an effect of the same character will follow from an alternating
current. At the moment of the tap upon the box the air inside is made to
move past the mirror, and probably executes several vibrations. While
these vibrations last, the mirror is subject to a twisting force tending to set
it at right angles to the direction of vibration. The whole action being over
in a time very small compared with that of the free vibrations of the magnet
and mirror, the observed effect is as if an impulse had been given to the
suspended parts.
79] IN ABSOLUTE MEASURE. 7
In order to illustrate this effect I contrived the following experiment*.
A small disk of paper, about the size of a sixpence, was hung by a fine silk
fibre across the mouth of a resonator of pitch 128. When a sound of this
pitch is excited in the neighbourhood, there is a powerful rush of air in and
out of the resonator, and the disk sets itself promptly across the passage.
A fork of pitch 1 28 may be held near the resonator, but it is better to use a
second resonator at a little distance in order to avoid any possible disturbance
due to the neighbourhood of the vibrating prongs. The experiment, though
rather less striking, was also successful with forks and resonators of pitch 256.
It will be convenient here to describe the method adopted for regulating
and determining the speed of rotation, which has proved thoroughly satis-
factory. In the experiments of the Committee a governor was employed,
and the speed was determined by means of the bell already referred to.
This bell received a stroke every 100 revolutions, and the times were taken
with a chronometer. In this method rather long spinnings (ten or twenty
minutes) are necessary in order to get the speed with sufficient accuracy,
much longer than are required to take the readings at the telescope.
Desirous, if possible, of making the observations more quickly, I determined
to try the stroboscopic method. On the axis of the instrument a stout card
of 14 inches diameter was mounted, divided into concentric circles of black
and white teeth. The black and white spaces were equal, and the black
only were counted as teeth. There were five circles, containing 60, 32, 24,
20, 16 teeth respectively, the outside circle having the largest number of
teeth.
This disk was observed from a distance through a telescope, and an
arrangement for affording an intermittent view. An electric tuning-fork of
frequency about 63 was maintained in regular vibration in the usual way
by means of a Grove cell. To the ends of the prongs are attached thin
plates of metal, perforated with somewhat narrow slits parallel to the prongs.
In the position of equilibrium these slits overlap so as to allow an un-
obstructed view, but in other positions of the fork the disk cannot be seen.
When the fork vibrates, the disk is seen intermittently 127 times a second;
and if the speed be such that on any one of the circles 127 teeth a second
pass a fixed pointer, that circle is seen as if it were at rest.
By means of the various circles it is possible to observe correspondingly
varied speeds without any change in the frequency of the fork's vibration.
A further step in this direction may be taken by modifying the arrangement
for intermittent view. If the eye be placed at the top or bottom of one of
the vibrating plates, a view is obtained once only, instead of twice, during
* Proc. Camb. Phil. Soc. Nov. 8, 1880. [1899. For a lecture experiment the paper disc may
be replaced by a magnet and mirror, such as are used for galvanometers. See also Phil. Mag.
vol. xiv. p. 186, 1882.]
8 ON THE DETERMINATION OF THE OHM [B. A. UNIT] [79
each vibration of the fork. This plan was adopted for the slowest rotation,
and allowed 60 teeth to take the place of 120, which would otherwise have
been necessary.
The performance of the fork was very satisfactory. It would go for
hours without the smallest attention, except an occasional renewal of the
alcohol in the mercury cup. Pure (not methylated) alcohol was used for this
purpose, and a platinum point made and broke the contacts. Although, as it
turned out, this fork vibrated with great regularity, dependence was not
placed upon it, but repeated comparisons by means of beats were made
between it and a standard fork of Koenig's construction, of pitch (about) 128.
These beats, at pitch 128, were about 48 per minute, and scarcely varied
perceptibly during the course of the experiments. They could have been
counted for an even longer time, but this was not necessary. It was
intended at first to make the comparisons of the forks simultaneous with the
other observations, but this was given up as a needless refinement.
Some care was necessary in the optical arrangements to obviate undue
fatigue of the eyes in a long series of observations. In daylight the
illumination of the card was sufficient without special provision, but at
night, when the actual observations were made, the image of an Argand gas
flame was thrown upon the pointer and the part of the card near it. On
account of the necessity of removing the electric fork and its appliances to
a distance, the card, if looked at directly, would appear too much fore-
shortened, and a looking-glass was therefore introduced. The eyepiece of
the telescope, close in front of the slits, was adjusted to the exact height,
and the eye was placed immediately behind the slits. By cutting off stray
light as completely as possible, the observation may be made without fatigue
and with slits narrow enough to give good definition when the speed is
correct.
As governor I had originally intended to employ an electro-magnetic
contrivance, invented a few years ago by La Cour and myself*, in which a
revolving wheel is made to take its time from a vibrating fork, and it was
partly for this reason that the water engine was placed at a considerable
distance from the revolving coil. I was, however, not without hopes that a
governor would be found unnecessary, and a few trials with the stroboscopic
apparatus were very encouraging. It appeared that by having the water
power a little in excess, the observer looking through the vibrating slits
could easily control the speed by applying a slight friction to the cord
connecting the engine and coil. For this purpose the cord was allowed to
run lightly through the fingers, and after a little practice there was no
difficulty in so regulating the speed that a tooth was never allowed finally to
pass the pointer, however long the observation was continued. If from a
* Nature, May 23, 1878. [Art. 56; vol. i. p. 355.]
79] IN ABSOLUTE MEASURE. 9
momentary inadvertence or from some slight disturbance a tooth passed, it
could readily be brought back again. The power of control thus obtained
will be appreciated when it is remembered that the passage of a tooth per
second would correspond to less than one per cent, on the speed. In many of
the observations the pointer covered the same tooth all the while, so that
the introduction of a governor could only have done harm.
Another, and perhaps still more important, improvement on the original
method related to the manner of making correction for the changes of
declination which usually occur during the progress of the experiments.
The Committee relied for tins purpose upon comparisons with the photo-
graphic records made at Kew, and they recognise that considerable dis-
turbances arose from the passage of steamers, &c. All difficulty of this
kind is removed by the plan which we adopted of taking simultaneous
readings of a second magnetometer, called the auxiliary magnetometer,
placed at a sufficient distance from the revolving coil to be sensibly un-
affected by it, but near enough to be similarly influenced by changes in the
earth's magnetism, and by other disturbances having their origin at a
moderate distance. The auxiliary magnetometer was of very simple con-
struction, and was read with a telescope and a millimetre scale, the distance
between mirror and scale (about 2 metres) being adjusted to approximate
equality with that used for the principal magnet, so that disturbances were
eliminated by simple comparisons of the scale readings. During a magnetic
storm it was very interesting to watch the simultaneous movements of the
magnets.
In the mouth of September the apparatus was remounted under the
direction of Professor Stuart, to whose advice we have often been indebted.
In order to examine whether any errors were caused by the circulation of
currents in the frame, as has been suggested by more than one critic,
insulating pieces were inserted, mercury cups at the same time being
provided, so that the contacts could be restored at pleasure. But the
principal changes related to the manner of suspending the fibre and sup-
porting the box and tube. In order to eliminate tremor, as far as possible,
these parts were supported by a massive wooden stand, resting on the floor
and overhanging, but without contact, the top of the metal frame of the coil.
The upper end of the fibre was fastened to a rod sliding in a metal cap,
which formed the upper extremity of a 2-inch glass tube. Near the other
end this tube was attached to a triangular piece of brass, resting on three
screws, by which the whole could be raised or lowered bodily and levelled.
Rigidly attached to this tube, and forming a continuation of it, a second
glass tube, narrow enough to pass freely through the hollow axle of the coil,
protected the fibre as far as the box in which the mirror and magnet were
hung. This box was cylindrical and about 3 inches in diameter. The top
10 ON THE DETERMINATION OF THE OHM [fi. A. UNIT] [79
fitted stiffly to the lower end of the narrow glass tube, and the body of the
box could be unscrewed, so as to give access to the interior. The window
necessary for observation of the mirror was made of a piece of worked glass,
and was fitted air-tight.
On rny return to Cambridge in October the apparatus was tested, but
without the full success that had been hoped for. At high speeds there was
still unsteadiness enough to preclude the use of these speeds for measure-
ment. Since it is impossible to suppose that the tremor is propagated with
sufficient intensity through the floor and massive brickwork on which the
coil is supported, the cause must be looked for in the fanning action of the
revolving coil, aggravated no doubt by the somewhat pendulous character of
the box, and perhaps by the nearness of the approach between the coil and
its frame at three points of the revolution.
At this time the experiment was in danger of languishing, as other
occupations prevented Mr Darwin from taking any further part ; but on
Dr Schuster's return to Cambridge he offered his valuable assistance.
Encouraged by Sir W. Thomson, we determined to proceed with the
measurements, inasmuch as no disturbance, due to the rotation of the coil
with circuit open, could be detected until higher speeds were approached
than it was at all necessary to use.
One of the first points submitted to examination was the influence of
currents induced in the frame. Without altering the speed or making any
other change, readings were taken alternately with the contact-pieces in and
out. Observations made on several days agreed in showing a small effect,
due to the currents in the frame, in the direction of a diminished deflection.
The whole deflection being 516 divisions of the scale, the mean diminution
on making the top contacts was '86 division. When the coil was at rest no
difference in the zero could be detected on moving the contact-pieces.
In these preliminary experiments very consistent results were obtained at
constant speeds, whether the rotation was in one direction or the other ; but
when deflections at various speeds were compared, we were startled to find
the larger deflections falling very considerably short of proportionality to the
speeds. There are only two corrections which tend to disturb this pro-
portionality (1) the correction for scale-reading, (2) the correction for self-
induction. The effect of the first is to make the readings too high, and of
the second to make the readings too low at the greater speeds. According
to the figures given by the Committee (Report, p. 106), the aggregate effect
is to increase the readings, on account of the preponderance of (1) over
(2), whereas our results were consistently of the opposite character. Every-
thing that could be thought of as a possible explanation was examined
theoretically and experimentally, but without success. The coil was dis-
mounted and the wire unwound, in order to see whether there was any false
79] IN ABSOLUTE MEASURE. 11
contact which might be supposed to vary with the speed and so account for
the discrepancy. After much vexation and delay, it was discovered that the
error was in the statement in the Report, the effect of self-induction being
given at nearly ten times less than its real value. The correction for scale-
reading, instead of preponderating over the correction for self-induction, is in
reality quite a small part of the whole.
At this stage, as time was running short, we determined to proceed at
once to a complete series of readings at sufficiently varied speeds, postponing
the measurement of the coil to the end. The wire had been rewound
without extreme care to secure the utmost attainable evenness, and the
condition of the groove was such that a thoroughly satisfactory coil could not
have been obtained, even with extreme care. It appeared, however, on
examination that irregularities of this sort were not likely to affect the
final result more than one or two parts in a thousand, if so much ; and
many points of interest could be decided altogether independently of this
measurement.
The details of the experiments and reductions are given below by
Dr Schuster, who took all the readings of the principal magnetometer.
Mrs Sidgwick observed the auxiliary magnetometer; while the regulation
of the speed by stroboscopic observation fell to myself. Dr Schuster also
undertook the labour of the reductions and the final comparisons of our
arbitrary German silver coil with the standard ohms.
The observations were very satisfactory, and at constant speeds agreed
better than we had expected. The only irregularity that we met with was a
slight disturbance of the zero, due to convection currents in the air sur-
rounding the mirror, the effect of which, however, almost entirely disappears
in the means. This disturbance could be magnified by bringing a paraffin
lamp into the neighbourhood of the mirror. After about half a minute,
apparently the time occupied in conduction through the box and in starting
the current, the readings began to move off. Complete recovery would
occupy twenty or thirty minutes. In future experiments this kind of dis-
turbance will be very much reduced by increasing the moment of the magnet
five or six times, and by diminishing the size of the mirror, both of which
may be done without objection.
The comparison of the results at various speeds requires a knowledge
of the coefficient of self-induction L. Nothing is said in the Report as
to the value of L for the second year's experiments, but the missing in-
formation is supplied in Maxwell's paper on the " Electro- magnetic Field*/'
together with an indication of the process followed in calculating it. The
first approximation to the value of L, in which the dimensions of the
section are neglected in comparison with the radius of the coil, is 437,440
PAi/. Trait*. 1865.
12 ON THE DETERMINATION OF THE OHM [B. A. UNIT] [79
metres, but this is reduced by corrections to 430,165. The value which best
satisfies the observations is considerably greater, viz., 456,748. A rough
experiment with the electric balance gave 410,000; but Professor Maxwell
remarks that the value calculated from the dimensions of the coil is probably
much the more accurate, and was used in the actual reductions. I had
supposed at one time that the discrepancy between the results at various
speeds and the calculated value of L was due to the omission of the term in
tan 4 <, given above, whicli would have the same general effect as an under-
estimate of L ; but, as has been already mentioned, this term was in fact
included in the reductions made by Mr Hockin, in conjunction, moreover,
with the value L = 437,440.
A rough preliminary reduction of our observations showed at once that
they could not be satisfied by any such value of L as 437,000, but pointed
rather to 454,000, and we began to suspect that the influence of self-induction
had been seriously under-estimated by the Committee. Preliminary trials
by Maxwell's method with the electric balance giving promise of results
trustworthy within one per cent., we proceeded to apply it with care to the
determination of L, but the galvanometer at our command a single needle
Thomson of 2,000 ohms resistance was not specially suitable for ballistic
work. As this method is not explained in any of the usual text-books,
it may be convenient here to give a statement of it.
The arrangement is identical with that adopted to measure the resistance
of the coil in the usual way by the bridge. If P be the resistance of the
copper coil, Q, R, 8, nearly inductionless resistances from resistance-boxes,
balance is obtained at the galvanometer when PS= QR. This is a resistance
balance, and to observe it the influence of induction must be eliminated by
making the battery contact a second or two before making the galvanometer
contact. Let us now suppose that P is altered to P + 8P. The effect
of this change would be annulled by the operation of an electromotive force
in branch P of magnitude SP . x, where x denotes the magnitude of the
current in this branch before the change. Since electromotive forces act
independently, the effect upon the galvanometer of the change from P
to P + 8P is the same as would be caused by 8P . x acting in branch P,
if there be no E.M.F. in the battery branch at all*.
Returning now to resistance P, let us make the galvanometer contact
before making the battery contact. There is no permanent current through
the galvanometer (6r), but, at the moment of make, self-induction opposes an
obstacle to the development of the current in P, which causes a transient
current through 0, showing itself by a throw of the needle. The integral
* [18911. A slight error should here be corrected. The electromotive force should be
reckoned as 5P . x', where x' is the actual current flowing through SP. The ratio of x' to x is
very near unity in practice. See Phil. Trans, vol. CLXXIII. p. 677, 1882; Art. 80 below.]
79] ABSOLUTE MEASURE. 13
magnitude of this opposing E.M.F. is simply Lx, and it produces the same
effect upon G as if it acted by itself. We have now to compare the effects
of a transient and of a permanent E.M.F. upon G. This is merely a question
of galvanometry. If T be the time of half a complete vibration of the needle,
6 the permanent deflection due to the steady E.M.F., a the throw due to
the transient E.M.F., then the ratio of the electromotive forces, or of the
currents, is
T 2 sjn a
v tantf "
If, instead of the permanent deflection 0, we observe the first throw
(/8) of the galvanometer needle, this becomes
T
tan
In the present case, the ratio in question is, by what has been shown
above, SP.x : Lx, or &P : L; so that
L
a formula which exhibits the time-constant of the coil P in terms of the
period of the galvanometer needle. Further to deduce the value of L in
absolute measure from the formula requires a knowledge of resistances in
absolute measure.
In carrying out the experiment the principal difficulty arose from want
of permanence of the resistance balance, due to changes of temperature
in the copper coil. The error from this source was, however, diminished by
protecting the coil with flannel, and was in great measure eliminated in the
reductions. The result was L= 455,000 metres. This is on the supposition
that the ohm is correct. If, as we consider more probable, the ohm is one
per cent, too small, the result would be L = 450,000.
Without attributing too great importance to this determination, there
were now three independent arguments pointing to the higher value of L :
first, from the experiments of the Committee ; secondly, and more distinctly,
from our experiments; and thirdly, from the special determination; and
I entertained little doubt that a direct calculation from the dimensions
of the coil would lead to a similar conclusion.
This direct calculation proved no very easy matter. Mr W. D. Niven
(whom I was fortunately able to interest in the question ) and myself had
no difficulty in verifying independently the formula? given in Maxwell's
Electricity and Magnetism, 692, 705, from which the self-induction of a
simple coil of rectangular section can be found, on the supposition that the
dimensions of the section are very small in comparison with the radius. In
14 ON THE DETERMINATION OF THE OHM [B. A. UNIT] [79
the notation of the paper on the electro-magnetic field, if r be the diagonal
of the section, and 6 the angle between it and the plane of the coil,
L = 47rn 2 loge + T V - s ( 6 ~ i 7 "-) cot 2e ~ &* cosec 28
- % cot 2 6 log c cos - % tan 2 log e sin .......
(10)
In the paper itself, probably by a misprint, cos 20 appears, instead of
cosec20, in (10). The expression is, as it evidently ought to be, unchanged
when \TT-6 is written for 0. By an ingenious process, explained in the
paper, the formula is applied to calculate the self-induction of a double
coil*.
The whole self-induction of the double coil is found by adding together
twice the self-induction of each part and twice the mutual induction of the
two parts. The self-induction of each part is found (to this approximation)
by a simple application of (10). For twice this quantity Mr Niven found
301,802, and I found 301,920 metres. For twice the mutual induction
of the two parts I found, by Maxwell's method, 145,820 metres. Adding
301,920 and 145,820, we get 447,740 metres as the value of the whole
self-induction, on the supposition that the curvature may be neglected.
This corresponds to the value 437,440 given in the paper.
As to the origin of the discrepancy I am not able to offer any satisfactory
explanation. It should be noticed, however, that owing to his peculiar use
of the words " depth " and " breadth " as applied to coils, Maxwell has inter-
changed what, to avoid any possible ambiguity, I have called the axial
anil radial dimensions of the section. Thus the depth, i.e., in his use of the
word, the axial dimension, is given as '01008, but this is really the radial
dimension, as appears clearly enough from the Report of the Committee, as
well as from our recent measurements. The real value of the axial dimension
is '01841 metre. But I do not think that this interchange will explain the
difference in the results of the calculation.
When we proceed to apply corrections for the finite size of the section,
further discrepancies develope themselves. The second term in the expression
for L given in the paper (p. 508) does not appear to be correct, and the final
numerical correction for curvature (- 7,345 metres) differs in sign from that
which we obtain. Mr Niven has attacked the problem of determining the
correction for curvature in the general case of a single coil of rectangular
section, and (subject to a certain difficulty of interpretation) has obtained a
solutionf. The application of the result to the actual case of a double coil
* The following misprints maybe noticed : Page 509, line 11, for B read C; line 13, for
L(AC) read M (AC); line 13, for L (B) read L (C). Attention must be directed to the peculiar
meaning attached to depth.
t [1899. On this subject see Stefan (Wied. Ann. xxn. p. 107, 1884).]
79] IN ARSOLUTE MEASURE. 15
would, however, be a very troublesome matter. For the two particular cases
in which only one of the two dimensions of the section of a simple coil
is considered to be finite, Mr Niven and myself have independently obtained
tolerably simple results. Thus, if the axial dimension be zero (6 = 0),
(11)
and if the radial dimension be zero [c = 0],
Again, for a circular section of radius p,
In all these cases we see that the correction increases the value of L,
and there can be no doubt that the same is true for the double coil.
I have applied (13) to estimate the correction for curvature in the self-
induction of each part of the double coil. For reasons which it would take
too long to explain, I arrived at the conclusion that the value of the small
term must be very nearly the same for a circular section as for a square
section of the same area, and the actual section is nearly enough square to
allow of the use of this principle. The necessary addition to the originally
calculated self-induction of each part, in order to take account of curvature,
comes out 119'5 metres; so that the final value of L for the double coil will
on this account be increased 239 metres. This is a small quantity, but
a much larger correction for curvature must be expected in the mutual
induction of the two parts. By a sufficiently approximate method I find as
the correction to twice the mutual induction 3,469 metres, giving altogether
for twice the mutual induction 149,289 metres. This added to 302,159
(= 301,920 + 239) metres gives as the final calculated value of L for the
double coil, L = 451,448 metres. This result is confirmed by calculation of
the mutual induction by means of a table founded on elliptic functions.
In this way, and with a suitable formula for quadrature, we find, 2J/= 149,394
metres, agreeing nearly enough with the value found by Maxwell's method,
viz., 149,289 metres*. When all the evidence is taken into consideration,
there can remain, I suppose, Little doubt that the value 451,000 is sub-
stantially correct, and that the reductions of the Committee are affected by
a serious under- estimate.
* The arithmetical calculations were made from the data given by the Committee (Reprint,
p. 115), a =-153194, 2b'= -03851, 6 ='01841 (not -1841), c= -01608, all in metres. * = 313. The
whole number of turns (313) was supposed to be equally divided between the two parts.
16 ON THE DETERMINATION OF THE OHM [B. A. UNIT] [79
Professor Rowland, in ignorance apparently of Maxwell's previous calcu-
lation, has shown that if in the original experiments we assume an unknown
cause of error proportional to the square of the speed, and eliminate it,
we shall arrive at a value of the ohm differing very appreciably from that
adopted by the Committee. In this way he finds that
1.1 nnoc earth quadrant
1 ohm [B.A. unit] = '9926 - ^ -. -- .
Rowland is himself disposed to attribute the error to currents induced in the
frame. Our experiments prove these currents had not much effect, though
they may explain the difference between the value of L which best satisfies
our experiments (where the currents could not exist), i.e., 451,000, and the
higher value 457,000 calculated by Maxwell as most in harmony with
the original experiments. The process adopted by Rowland is evidently
equivalent to determining the coefficient of self-induction from the de-
flections themselves, and his result, rather than that given by the Committee,
must be regarded as the one supported by the evidence of the original
experiments.
Rowland's own determination, by a wholly distinct method, gives
-.LI rvn., earth quadrant
1 ohm TB.A. unit] = '9911 - - ? - ;
second
and according to our experiments the ohm is even smaller
1 ohm [B.A. unit] = -9893 earth q uadrant .
second
The question, therefore, arises whether any further explanation can be given
of the different result obtained by the Committee. The value of GK
employed in calculating the experiments according to (4) was GK = 299,775
metres. For the principal term in GK, as given by (7), we require the
values of n, a, and a. From p. 115 of the Reprint we find a = '158194 metre,
n = 313. The angle a must be recalculated, as the value of log sin 3 a
(1-9624955) is evidently incorrect. From 26' = '03851 metre, by means of
sin a = a/V(a + 6" 2 ), we find log sin 3 a = T-99043. From these data the final
value is GK= 299,290 metres, differing appreciably from that used by the
Committee. The further discussion of the question is a matter of difficulty
at this distance of time. There may have been some reason for the value
adopted, which it is now impossible to trace, so that I desire to be under-
stood as merely throwing out a suggestion with all reserve. But I think it
right to point out a possible explanation, depending upon the interchange of
the axial and radial dimensions in the paper on the Electro-magnetic Field.
The data there given are the mean radius, the two dimensions of the sections,
and the distance between the coils ("02010). This distance is correct, being
79] IN ABSOLUTE MEASURE.
17
equal to 26' -6, that is, to -03851 - '01841. The distance between the
mean planes of the coils is not given, but could, of course, be calculated by
addition of "02010 and '01841. If, however, the radial dimension '01608
were substituted for the axial dimension '01841, an erroneous value would
be obtained for 26', that is, -03618 instead of '03851. Using -03618 to
calculate a, I find GK= 2<J9,860 metres, agreeing much more nearly with
the value used in the reductions.
If it be thought probable that the value of GK was really 299,290,
a still further reduction of nearly two parts in a thousand must be made in
the number which expresses the ohm in absolute measure, and we should
get
1 ohm [BJL unit] = -9910 --
second
coinciding practically with the value obtained by Rowland from his own
experiments.
In the course of our experiments various doubts suggested themselves,
and were subjected to examination. It may be well to say a few words
about some of these, though the results are for the most part negative.
The energy of the currents circulating in the coil is expended in heating
the copper, and a rise of temperature affects the resistance. Calculation
shows that the disturbance from this cause is utterly insensible. If at the
highest speeds of rotation all the heat were retained, the rise of temperature
would be only at the rate of 3'2 x 10-" C. per second.
Much more heating may be looked for during the operation of taking the
resistance. Under the actual circumstances a rise of resistance of about one
part in 30,000 might be expected as the effect of the battery current in
one minute. The aggregate duration of the battery contact in each of the
resistance measurements was probably less than a minute.
Another question related to the possible effect of a want of rigidity
in the magnetism of the needle. It is known that galvanometers will some-
times, when it is certain that there is no average current passing through
the coils, show a powerful effect as a consequence of fluctuating magnetism
corresponding to the fluctuating magnetic field. In the present experiment
the magnetic field is fluctuating, and the magnet is expected to integrate
the effect as if its own magnetism were constant. It is unlikely that any
appreciable error arises in this way, as I find by calculation that a theoretically
soft iron needle would point in the same direction as a theoretically hard
needle when placed at the centre of the revolving coil.
From the details given the reader will be in a position to judge for
himself as to the accuracy of our experiments. If, as we believe, the
principal error to be feared is in the measurement of the coil, there is
2
18 ON THE DETERMINATION OF THE OHM [B. A. UNIT] [79
little to be gained by further experimenting with the present apparatus.
Accordingly a new apparatus has been ordered, from which superior results
may be expected. In designing this several questions presented them-
selves for solution.
All corrections being omitted, the effect
tan <f> oc n 2 ao)/R ;
and, if a denote the section of the wire, and S (= no-), the aggregate section
of the coil
R oc na/a- oc ri*a/S ;
so that if S be given, tan <f> is independent both of the number of turns
n and of the mean radius a. If < be given, the correction for self-induction
depends upon LjGK, while both L and G K vary approximately as if a. So
far, therefore, there is nothing to help us in determining n and a. The
following considerations, however, tell in favour of a rather large radius :
(1) Easier measurement of coil.
(2) Smaller correction for moment of suspended magnet.
(3) Smaller errors from maladjustment to centre, and from size of
magnet.
The question of insulation is important. During the rotation the electro-
motive force acts independently in every turn, and there is no strain upon
the insulation ; but in taking the resistance, when a battery is employed, the
circumstances are materially different. Any leakage from one turn to
another would, therefore, be a direct source of error. It is proposed to
use triply covered wire.
In order to obtain room for the tube encasing the fibre, it is necessary to
use a double coil. In the new apparatus there will be opportunity for a
much larger diameter, by which it is hoped to obtain an advantage in respect
of stiffness ; but the further question presents itself, whether the interval
between the coils should be increased so as to obtain a very uniform field, as
in Helmholtz's arrangement of galvanometer. The advantages of this plan
would be considerable in several respects, but on the whole I decided against
it, mainly on the ground that it would magnify the errors due to imperfect
measurement. If we call the effect (so far as it depends upon the quantities
now uuder consideration) u, we have, in previous notation,
u = a sin 3 a = a 4 (a 2 + b" 2 ) - ?,
so that
du _ 4, da ada + b'db'
u ~ a ~ a 2 + b'- '
79] IN ABSOLUTE MEASURE. 19
If b' = 0, duju = da a ;
bnt if, as in Helmholtz's arrangement, 6' = |r,
_
u 5 a 5 a
The increase of b' from to fa not only introduces a new source of
error in the measurement of V, but also magnifies the effect of an error
in the measurement of a. If V = ^a, we have nearly
du = da_3 L <W
u ~ a 10 a '
showing that an absolute error in V has about of the importance of an
equal absolute error in a.
As will be evident from what has been said already, the treatment of the
correction for self-induction is a very important matter. It is probable that
L may be best determined from the deflections themselves with the use
of sufficiently varied speeds. If L be arrived at by calculation, or by
independent experiments, it is important to keep down the amount of the
correction. We have seen, however, that L GK is almost independent of
, a, and S, so that if we regard tan <f> as given, the magnitude of the
correction cannot be controlled so long as a single pair of coils is used.
An improvement in this respect would result from the employment of two
pairs of coils in perpendicular planes, giving two distinct and independent
circuits. In virtue of the conjugate character, the currents in each double
coil would be the same as if the other did not exist, and the effects of both
would conspire in deflecting the suspended magnet. This doubled deflection
would be obtained without increase of the correction for self-induction, such
as would arise if the same deflection were arrived at by increasing the speed
of rotation with a single pair of coils. A second advantage of this arrange-
ment is to be found in the production of a field of force uniform with respect
to time.
However the correction for self-induction be treated, it is important
to obtain trustworthy observations at low speeds. In order to get a zero
sufficiently independent of air currents, it will be advantageous largely to
increase the moment of the suspended magnet. Preliminary experiments
have, however, shown that there is some difficulty in getting the necessary
moment in a very small space, in consequence of the interference with each
other of neighbouring magnets, and thus the question presents itself as
to the most advantageous arrangement for a compound magnet.
A sphere of steel, as used by the Committee, has the advantage that
if uniformly magnetised it exercises the same action as an infinitely small
magnet at its centre. But the weight of such a sphere is considerable in
2 2
20 ON THE DETERMINATION OF THE OHM [B. A. UNIT] [79
proportion to its moment, and it is probable that a combination of detached
magnets is preferable. It is possible so to choose the proportions as to
imitate pretty closely the action of an infinitely small magnet. Thus, if the
magnet consist of a piece of sheet steel bent into a cylinder and uniformly
magnetised parallel to the axis, the length of the cylinder should be to the
diameter as \/3 to V2- In this case the action is the same as of an infinitely
small magnet as far as the fourth term inclusive of the harmonic expansion.
Without loss of this property the cylinder may be replaced by four equal line
magnets, coinciding with four symmetrically situated generating lines. Thus,
if we make a compound magnet by placing four equal thin magnets along
the parallel edges of a cube, the length of the magnets should be v/3
times the side of the cube. This is on the supposition that the thin
magnets are uniformly magnetised, as is never the case in practice. To
allow for the distance between the poles and the ends of the bars, we may
take the length of the bars 2'3 times the side of the cube.
With the new apparatus, and with the precautions pointed out by
experience, we hope to arrive at very accurate results, competing on at
least equal terms with those obtained by other methods. Most of the
determinations hitherto made depend upon the use of a ballistic gal-
vanometer, and the element of time is introduced as the time of swing
of the galvanometer needle. There is no reason to doubt that very good
results may be thus obtained; but it is, to say the least, satisfactory to
have them confirmed by a method in which the element of time enters in a
wholly different manner.
Part II. By ARTHUR SCHUSTER.
Adjustment of the Instruments and Determination of Constants.
The only adjustments to be made consist in
1. The levelling of the coil.
2. The suspension of the magnet in the centre of the coil.
3. The proper disposition of the scale and telescope by means of which
the angles of deflection are read off.
Level. The first of these presents no difficulty, and any small error
can be easily taken account of in the calculations. It was found that the
upper end of the axis of rotation was inclined towards the north by an angle
of -0003 circular measure. Hence, as has already been explained [p. 3], we
must in the reductions write throughout G (1 + '0003 tan /) or T0008
for 0. This correction is small, but a little uncertain, as the coil was not
very steadily fixed in its bearings, and small variations in the inclination
79] IN ABSOLUTE MEASURE. 21
of the axis could be produced by slightly pressing on one side or the other of
the coil. When left to itself the coil seemed, however, very nearly to return
to the same position.
The Magnet. The magnet, which was suspended iu the centre of the
coil, consisted of four separate magnetised needles, each about 0'5 centiin.
long. These were mounted on four parallel edges of a small cube of cork.
A needle attached to the back of the mirror went through a small hole
in the cork, and was kept in its place by means of shellac, to prevent any
slipping between the magnets and the mirror. The proper suspension of the
magnet is a point of some delicacy and importance. As regards the vertical
adjustment, the distance of the cube of magnets from the highest and
lowest points of the circular frame was measured, and the magnet raised
or lowered until the distances became equal. A pointer was next fixed
to the frame, reaching very nearly to the centre of the coil. As the coil was
rotated, the pointer described a small circle round the axis of revolution, and
the position of the magnet could be easily altered until it occupied the
centre of the small circle. It is supposed that this adjustment was made
to within less than 1 millim., and could, therefore, for all practical purposes,
be considered as perfect. The magnetic moment of the magnet was
measured in the usual way. Two closely agreeing sets of measurements
showed that at a distance of 1 foot it deflected a suspended needle through
an angle, the tangent of which was '000298. Hence at the mean distance
of the coil (15-85 centims.) the deflection would have been '0021. This
number is equal to MKfGH, and will be referred to as tan /* in the
discussion of the calculations. The magnetic moment was determined a
few days after the last spinnings had been taken ; but on each day on which
experiments were made, the time of vibration of the magnet was determined,
and we thus assured ourselves that no appreciable change in the magnetic
moment had taken place while the experiments were going on. The time of
one complete vibration was 14'6 seconds.
Adjustment of Scale and Telescope. The telescope which served to read
the angle of deflection rested on a small table to which it could be clamped.
In front of the table and below the telescope, the scale could be raised
or lowered and fixed when the proper position had been found. It was
levelled by deflecting the magnet successively towards both sides, and
observing the point of the scale at which the cross wires of the telescope
seemed to cut the scale. If in both positions of the mirror the scale
was intersected at the same height, it was considered to be sufficiently
levelled. It remained to place the scale at right angles to the line joining
its centre to the mirror. This was done by measuring the distance of both
ends to the mirror by means of a deal rod, with metallic adjustable pointers
(presently to be described), and altering the position until these distances
22 ON THE DETERMINATION OF THE OHM [B. A. UNIT] [79
were equal. It is supposed that considerable accuracy was thus obtained.
A small remaining error would be eliminated by observing deflections on
both sides of the zero. To adjust the telescope we had now only to point
it to the centre of the mirror, and at the same time to place it in such a
position that its optic axis passed vertically over the centre of the scale. By
suspending a plumb-line from the telescope so as to divide its objective into
two equal parts, and focussing alternately on the mirror and on the image
of the scale, both points could be simultaneously attended to.
To measure the distance of the scale from the mirror the deal rod used
for the adjustment of the scale was cut down so as to have nearly the
required length. The two brass pointers attached to the two ends made an
angle of about 45 with the rod. One of the pointers was fixed, but the
other could be moved round a fixed point in the rod by means of a screw.
As it moved, the distance of the two points changed, and by properly
supporting the rod and leaning one point against the centre of the scale
at a known height from the ground, while the moveable point was made to
touch the centre of the mirror, the distance could be accurately found.
It was compared with the scale itself, in order that the calculation of
the angles of deflection should be independent of the absolute length
of a scale division. The length required is the shortest line between the
centre of the mirror and the plane of the scale, and this can be calculated
if the difference in height of the centre of the mirror and the point
to which the distance was measured, is known. These heights were de-
termined by means of a cathetometer. The height of the centre of the
objective was measured at the same time; so that all data required to find
the inclination of the normal of the mirror to the horizontal are known.
The following numbers were obtained; each division of the scale is for
simplicity supposed to be equal to 1 millim., which is very nearly correct,
but as has been said, its absolute value is of no importance.
Distance of mirror from scale in centims 252'28
As the position of the magnet was always read off
through a glass plate, a small correction equal to the
thickness of the glass (3'2 millims.) multiplied into
0- ~ M" 1 )* where /* is the refractive index, has to be
applied. This correction is subtractive and equal to Oil
Hence, D= 25217
It was also found that the mirror pointed downwards, and made an
angle of "004 with the horizontal. A small correction due to this cause
will be discussed in another place.
Torsion. The torsion was as much as possible taken out of the silk fibre,
which was about 4 feet long, before the magnet was attached to the mirror.
The coefficient of torsion was determined by turning the magnet through
79] IN ABSOLUTE MEASURE. 23
five whole revolutions and observing the displacement of the magnet. It
was calculated from the numbers obtained that one revolution shifted the
position of rest through 5'6 scale divisions, corresponding to an angle of
001107.
Another experiment in which the magnet was turned in the opposite
direction gave '001117.
Hence r = -00111/2^ = "00018.
The correction due to torsion is best applied to the value of G at the
same time as the correction for level by writing everywhere
Constants of the Coil. The accurate determination of the constants
of the coil forms the most difficult part of the measurements. Unwinding
the coil, the outer circumference of the successive layers was measured by
means of a steel tape. Each measurement was repeated several times, and
the agreement was satisfactory. The thickness of the wire was found to be
1*37 millims., which, on the circumference of the successive layers, should
produce a constant difference of 2'74 tr or 8'62 millims. Owing, however, to
defective winding, each layer enters a little into the grooves of the subjacent
layer, and the differences in circumference of successive layers were therefore
always smaller than they ought to have been. The differences varied
between 77 millims. and 8'6 millims., but generally were about 8'1 millims.
The wire was a little too thick, and as it had been taken off the coil
and rewound, small irregularities were formed which prevented a satisfactory
winding. Each coil had 156*5 windings. Of these 156 were in one coil
regularly distributed over twelve layers of thirteen windings each ; while
half a turn was outside. In the second coil the twelve lay ere only contained
155 windings, and one turn and a half was lying outside. In the calculation
for mean radius it was assumed that each complete layer contained the
same number of turns. Let S be the sum of all measurements for one
coil, also C the circumference of the layer containing the loose extra turns ;
then we find the mean circumference of the first coil,
99-680
.. ..
lOD'O
and for the second,
(13-1/12)3 + 1-5(7 99 . 651
156-5
Or as the circumference of the outside of the mean turn ... = 99*666
From this is to be subtracted a correction equal tr x thickness
of tape .............................................................. = -031
99-635
24 ON THE DETERMINATION OF THE OHM [B. A. UNIT] [79
To obtain the circumference of the axis of the mean winding we
have to subtract ir x thickness of wire ..................... =
Hence the final value of the mean circumference ............ /3 =
Or for the mean radius ............................................... a = 15 ' 789
The grooves of the coils and their distance was also measured,
and it was found that
6 = axial dimension of coil ................................ =
b' = distance of mean plane from axis of motion .... = T918
All measurements are given in centimetres.
We calculate
a = angle subtended at axis by mean radius = cot" 1 (b'fa) ..... = 83 04'
And logsin'o ............................................................ = 1'990458
The principal term in the expansion of GK is Trtfft sin 3 a . . . = 29,809,300
To this is to be added a small correction given on [p. 4]... = 100
The final value of GK being ........................................... 29,869,200
Applying the corrections for level and torsion to GK as explained, and
writing <!R38t for the value so corrected, we find
= 29,887,600.
The Observations.
The observations consisted of two parts : the comparison of the resistance
of the rotating copper coil with that of a standard German silver coil, and
the observation of the deflections during the spinnings. The comparison
of resistance was made by a resistance balance devised by Mr Fleming*,
to whom we are indebted for advice and assistance in all questions con-
cerning the comparison of resistances. In this balance, which only differs
by a more convenient arrangement from an ordinary Wheatstone's bridge,
Professor Carey Foster's method of comparing resistances is used. The
method consists in interchanging the resistances in the two arms of the
balance which contain the graduated wire, and thus finding the difference
between these two resistances in terms of that of a certain length of the
bridge wire. The balance was placed on a table near the rotating coil, and
could be electrically connected with it by means of two stout copper rods.
The German silver coil which served as the standard of comparison was
prepared so as to have a resistance nearly equal to that of the copper
coil, that is about 4'6 ohms. Any error due to thermo-electric currents,
which have sometimes been found to be generated at the moveable contact
of the galvanometer circuit with the bridge wire, is eliminated in Foster's
* Phil. Mag. vol. ix. p. 109, 1880.
79] IN ABSOLUTE MEASURE. 25
method, but to ensure greater accuracy and safety all measurements were
repeated with reversed battery current. The whole comparison seldom
occupied more than five minutes; and the readings obtained with the
battery current in different directions closely agreed with each other.
The spinnings were always taken in sets of four at the same speed, and
the comparison of resistance was made at the beginning and end of each set.
During the time of spinning the resistance was found to have altered owing
to a rise of temperature which always took place during the time of experi-
mentation. The corrections for the change of resistance and the possible
errors introduced owing to the uncertainty of this correction will be described
further on.
After the resistance of the coil had been measured, it was disconnected
from the balance and set into rotation with open circuit, so that no current
could pass. While the water supply was adjusted so as to give approximately
the required speed, the magnet in the centre of the coil, which had been
strongly disturbed during the measurement of resistance, was brought to rest
either by means of an outside magnet or more often by means of a small coil
and LeClanche cell, which was always placed in the neighbourhood of the
rotating coil. A key within reach of the observer served to make and break
contact at the proper time. When the speed had been approximately regu-
lated and the magnet was vibrating through a small arc only, its position of
rest was determined, while at the same time the auxiliary magnetometer
placed in the adjoining room was observed. The two ends of the rotating
coil were now connected together, by means of a stout piece of copper, the
well amalgamated ends of which were pressed into cups containing a little
mercury, into which they tightly fitted.
As the coil was set into rotation the magnet slowly moved towards one
side, and a proper use of the damping coil brought it quickly to approximate
rest near its new position of equilibrium. When the swings extended
through no more than ten or twenty divisions of the scale, the coil was kept,
as nearly as possible, at the proper speed, by the observer at the tuning-fork
(Lord Rayleigh, see p. 8). Readings of the successive elongations were
taken for about two minutes, and a signal given at the beginning and end of
each set of readings enabled the observer at the auxiliary magnetometer
(Mrs Sidgwick) to note its position as well as any changes in the direction of
the earth's magnetic force during the time of observation. The direction of
rotation was now reversed, and the deflection observed in the same manner;
the whole process being twice repeated, so that four sets of readings were
obtained When all the observations for the given speed had been com-
pleted, the position of rest of the magnet, when no current passed through
the coil, was again determined and compared with the auxiliary magneto-
meter. A recomparison of resistance with the standard completed the set.
26 ON THE DETERMINATION OF THE OHM [fi. A. UNIT] [79
The magnet in the centre of the coil should, when no current is passing
through the coil, always go through the same changes as the magnet of the
auxiliary magnetometer. If this could be ensured, the two might be com-
pared once for all, or the comparison might even be omitted altogether, for
the difference between the deflections of positive and negative rotations,
when corrected for changes in the earth's magnetism, would give the double
deflection independently of the actual zero position. Unfortunately, however,
and this was our greatest trouble, the comparison between the magnet and
the auxiliary magnetometer showed that we had to deal with a disturbing
cause, which rendered a frequent comparison between the two instruments
necessary. This disturbing cause, which we traced to air currents circulating
in the box containing the magnet, will be discussed presently.
The observations were taken on three different evenings and one after-
noon. The evenings (8h. P.M. to llh. P.M.) were chosen on account of the
absence of disturbances, which, during the usual working hours, are almost
unavoidable in a laboratory. We may give, as an example for the regularity
with which the magnet vibrated round its position of rest, a set of readings
which were taken while the coil revolved about four times in one second, the
circuit being closed.
T = 9" 36 m . = 13-OC.
Eotation. Negative.
374-4 362-1
373-3 362-8
372-2 362-0
373-9 361-4
372-8 362-0
372-8 362-0
372-4 363-8
371-8 364-0
371-1 364-0
370-5
Mean.... 372'52 362-68
Position of rest, 367*60.
T=9 h 38 m -5. =130 C.
The number of readings taken was not always the same, but varied
generally between sixteen and twenty.
We used, in the course of our experiments, four different speeds. The
method of obtaining and regulating these has been explained by Lord
Rayleigh. For simplicity we generally denoted the speed by means of the
number of teeth on the circle which seemed stationary when looked at
79] IN ABSOLUTE MEASURE. 27
through the tuning-fork ; thus we spoke of a speed 24 teeth, 32 teeth, and
60 teeth. To obtain the lowest speed the circle containing 60 teeth was
looked at over the top of the tuning-fork, so that only one view for each
complete vibration was obtained; this was equivalent to a circle of 120 teeth
in the ordinary arrangement, which allowed a view for each half vibration,
and, consequently, the lowest speed was called 120 teeth. The velocity of
rotation depends, of course, on the frequency of the fork, which varied only
within narrow limits, and was always very near 63'69. If f denote this
frequency and N the number of teeth on the stationary card, the velocity of
rotation &> is given by the equation to = 4>7rf/X. In the " British Association
Report" the speed is always indicated by means of the time occupied by
100 revolutions. If T is this time, we find T = oOX/f. The following table
gives the comparison of &>, T, and N, on the supposition that the frequency of
the fork was always the same and equal to 63'69.
N.
120
tt,
6-670
T.
94-206
Number of turns
per second.
1-06
60
13-339
47-103
2-12
32
25-011
25-122
3-98
24
33-348
18-841
5-31
The last column gives the number of turns per second.
Three speeds were taken on each of the three nights, and one set of
observations with the lowest speed was secured in the course of one after-
noon. We obtained in this way three sets for each of the two intermediate
speeds and two sets for the lowest and highest speeds. A comparison with
the Report of the British Association Committee shows that we do not go
up quite to their highest speeds, but that on the other hand our lowest
speed was considerably below the one used by them. In the Report for the
year 1863, it is mentioned that the forced vibration of the magnet, depending
on the rotation of the coil, could not be noticed, and it is calculated that the
amplitude of this vibration was less than ^ of a millimetre on the scale.
No mention is made of this forced vibration in the Report for 1864, although
much lower speeds were used during that year. In our lowest speed a slight
shake of the needle, due to the varying action of the currents in the coil,
was distinctly seen ; but as calculation showed that the amplitude was only
the eighth part of a millimetre on the scale, no appreciable error is supposed
to have been introduced by it. The moment of inertia of the suspended
parts was higher in the experiments made by the British Association, and
this, no doubt, is partly the reason why this forced vibration escaped their
notice.
28 ON THE DETERMINATION OF THE OHM [l3. A. UNIT] [79
Air Currents.
It has already been noticed that air currents in the box containing the
magnet effected its position to some extent, and we had to investigate in
how far our final results might be affected by this disturbance. During the
first night (December 2) our attention had not been drawn so much as it was
afterwards to the effect of these air currents. We had previously ascertained,
by a series of careful measurements, that the rotation of the coil with open
circuit did not sensibly affect the zero position of the magnet, and we con-
sidered it sufficient to note the zero as short a time as possible before each
set of four spinnings. The comparison of these zeros with the auxiliary
magnetometer showed that during the two hours of experimenting, the
needle had kept its zero within two divisions of the scale, so that the changes
during two successive spinnings (generally about five minutes) must have
been very small. On the second, night (December 6), however, the zeros
were taken at the beginning and end of each set of four spinnings, and the
disturbance due to air currents was found to be of more importance. The
following table reveals the fact that during a set of spinnings the magnet
seems to have moved in one direction, but that during the time the coil was
at rest and the comparison of resistance was made, it went in the opposite
direction. The numbers given are corrected for changes in the direction of
the earth's magnetic force as shown by the auxiliary magnetometer.
December 6.
Number of teeth on Time. Position Approximate
stationary circle. h. m. of rest. deflection.
60 8 53 763-60 218
9 12 766-35
32 9 31 7.64-88 397
9 56 765-78
24 10 9 762-67 514
10 38 766-48
Here, then, we have a gradual rise in the zero from one to over three
divisions during a set of four spinnings. The approximate deflection is given
in order to give an idea what amount of error the uncertainty of the zero
might introduce.
Special experiments were now made, and it was found that by placing a
lamp about a foot and a half from the magnet box, changes amounting to
eighteen divisions of the scale would be observed ; greater precautions were
taken, in consequence of the experience thus gained, to secure the box from
the radiation of the lamp and gas-jets, which could not be dispensed with in
the course of the experiments. The magnet box was covered with gold-leaf
so as to reflect the heat as much as possible. On the last night of work
79] IN ARSOLUTE MEASURE. 29
frequent determinations of the position of rest were made, but in spite of all
precautions an unknown cause produced a sudden displacement of five scale
divisions. The exact time at which this change took place could not be de-
termined, and two spinnings were therefore rejected. During the remainder
of the evening the magnet gradually came back to its original position.
With the exception of the two spinnings just mentioned we have not rejected
any observations.
When we come to inquire into the amount of uncertainty to which our
results are liable, owing to the effects of these air currents, we find that it
cannot be greater than the more dangerous, because less evident, errors to
which the determination of our constants (mean radius and distance of mirror
from scale) are subject. As long as the changes of the position of rest take
place irregularly, the error would tend to disappear in the mean, and the
probable error deduced from our experiments would give a fair idea of the
uncertainty due to this cause. This probable error, as we shall see, is very
small. A regular displacement of zero in one direction would, however,
produce a constant error which would not disappear in the final mean. We
have some evidence that such a regular displacement has to some extent
taken place. The comparison of zeros on December 6, as quoted above, for
instance, shows the position of rest in the course of the spinnings shifted
towards increasing numbers. Such a shift, if not taken into account, would
tend to make the deflections towards increasing numbers (positive rotation)
appear larger than those towards decreasing numbers. This, indeed, was
observed. Supposing the shift takes place regularly during the time of
spinnings we might have taken it into account. But the correction which
we should have had to apply is so small and uncertain that it is doubtful
whether we should have improved our final result, and it would certainly not
have altered it within the limits within which we consider it accurate ; for
we find that reducing the deflections on the supposition, 1st, that the zero
has kept constant ; and 2nd, that it has changed uniformly during each set
of spinnings ; the two results agree to within about one and a-half tenths of
a division, which, even at the lowest speeds, would only make a difference of
about 1 in 750, and on the highest speeds four times less. The fact that a
regular shift in the zero position of the magnet affects the mean of four
spinnings is due to the arrangement of experiments, adopted during the first
two nights, in which four rotations succeeded each other in alternate direc-
tions. If, after a rotation in the positive direction, two negative rotations,
followed again by a positive one, had been taken, a regular displacement
such as that we are discussing would not have affected the mean. This
latter plan was adopted on the last night. In the measurements undertaken
by the British Association Committee, the deflections in one direction were
generally greater than in the other, and the difference was ascribed to a
considerable torsion in the fibre of suspension, which, in order to explain the
30 ON THE DETERMINATION OF THE OHM [fi. A. UNIT] [79
discrepancy, must, as pointed out by Rowland, have displaced the magnetic
axis considerably out of the meridian. The differences in the readings taken
when the coil was spinning in opposite directions were, on the average,
about 3 per cent., and amounted in one case to 8 per cent. They show no
regularity dependent on the speed of rotation. We also observed some slight
differences of the same nature ; but they are very small, and always remain
within such limits that they may easily have been produced by a displace-
ment due to air currents. On the last night, when more frequent zero
readings were taken, the differences were, it is true, not much reduced in
amount, but their sign was reversed. It is, perhaps, worth remarking that,
owing to the absence of any controlling instrument equivalent to our
auxiliary magnetometer, the Committee of the British Association had no
opportunity of discovering the presence of these air currents, as any changes
in the zero position would naturally have been ascribed by them to a casual
change in the direction of the earth's magnetic force. Owing to the different
shape and material of the box containing the mirror, it seems possible that
the effect of air currents may have been considerably larger than it has been
in our experiments.
Reduction of Observations.
Scale Corrections. The first step in reducing the observations consists
in calculating the value of 2 tan <J) from the observed deflection. The cor-
rection to be applied to the reading in order to obtain numbers proportional
to the tangents of deflection, if the position of rest of the magnet is at the
centre of the scale, would be d 3 {4<D 3 ; d being the observed reading, and D
the distance of the mirror from the scale. If the zero, however, is at a point
8 of the scale, the correction becomes (d S) (d + S) 2 /4Z) 2 , where S is to be
reckoned positive when in the same direction as d. For the higher speeds a
second correction, to +d 5 /8D 4 , was applied, which comes just within the
limits of accuracy aimed at in the actual readings. The corrected deflections
so obtained are equal to 2Z) tan <. Small errors, due to a faulty division of
the scale, ought also to be applied. It is difficult to obtain a proper scale in
one piece of sufficient length to be used in these experiments ; and the one
in actual use consisted of three parts, cemented with caoutchouc cement to a
thick piece of deal. No appreciable error was introduced by a very slight
warping of the wood, and the scales were found to be very accurately divided,
but the small errors existing were corrected ; small corrections had also
to be introduced, which are due to the imperfect joining of the different
pieces. The combined correction never amounted to more than '15 of a
division. Each division, as has already been stated, being very nearly equal
to 1 millim.
It has already been noticed that the normal to the mirror pointed slightly
downwards. The correction due to this want of adjustment seems to have
79] IN ABSOLUTE MEASURE. 31
been generally neglected, yet it is not altogether unimportant. If p is the
vertical distance between the centre of the objective and that point of the
scale where it is cut by the normal to the mirror ; also if a is the inclination
between the normal to the mirror and the horizontal, the correction to be
applied to a deflection d is dpa/D, where D is the distance of the mirror from
the scale. In our experiments the correction amounted to d x 0*00014,
although the angle between the normal and the horizontal was only about
14 minutes of arc. The correction is positive only if the normal lies between
the horizontal through the mirror and the line joining the mirror to the
cross wires of the telescope.
Correction for Temperature. We have now to discuss a series of cor-
rections which have to be applied in order to make a comparison of the
results obtained in different spinnings possible. It has already been noticed
that four spinnings at the same speed were always taken into one set. The
comparison of resistance at the beginning and end of each set showed that
during the time of spinning the temperature had altered ; before combining
the mean within each set we had, therefore, to correct for the change of
temperature. We endeavoured to keep the room as much as possible at a
constant temperature during the experiments; the lamps used were always
lighted nearly two hours before beginning, but, in spite of all precautions, the
temperature always rose after we had entered the room and begun to work.
The thermometer rose at first pretty rapidly through about 1 degree, and
then rose slowly until at the end of the evening it stood generally nearly
'2 degrees higher than at first. When the first set of spinnings commenced,
the rapid rise, as shown by the thermometer in the room, had already subsided ;
but, as was to be expected, the temperature of the coil was lagging somewhat
behind that of the room, and its resistance still rose appreciably. Thus,
during the first night, the resistance of the copper coil rose almost '4 per cent,
during the course of the first set of four spinnings. If the curve of tempera-
ture of the coil is known, there is of course no difficulty in applying the
proper correction. This curve can be obtained approximately by plotting
down the measured resistances as ordinates with the time as abscissae. This
was done for all observations made on December 2 ; but during the succeeding
nights it was found that the curve could not be sufficiently well determined
by the observations, and that the assumption of a uniform rise in resistance
during the time elapsing between two successive measurements would give
as good results as the experiments themselves would allow us to obtain.
The proper determination of this correction is a subject to which we shall
have to give some attention in the more accurate measurements which we
have in view. At present it will suffice to point out that, as far as we can
judge, the error due to uncertainty of temperature is not more than "05 per
cent, during the first set of spinnings on each night. It is much smaller in
the succeeding sets. It may increase the clearness of this account if at this
32
ON THE DETERMINATION OF THE OHM [B. A. UNIT]
[70
point we give a specimen, worked out in detail, of one set of deflections.
Let the resistance of our standard German silver coil, which we always
have assumed to be at the temperature of the air, be called S, and the
resistance of the rotating coil C. A comparison by means of the balance
shows that
C = 8 + a,
where a is the resistance of a certain length of the bridge wire, differing
slightly at the beginning and end of the experiment.
December 6.
Number of teeth on stationary circle, 32.
Comparison of resistance, C= +'0225. Time = 9 1 ' 17 m .
Position of rest 766'48. Time = 9 h 32 m .
Auxiliary magnetometer 26 '9.
Time= 9 h 37 m ... 9 h 42 in .
t= 13'0 ... 13-0 .
Rotation negative... positive.
Deflected reading 367'60 ... 1166'40.
Auxiliary magnetometer 27*55 ... 28'24.
Auxiliary magnetometer 27 -2.
Position of rest 767'08.
Comparison of resistance, <7=S + *0272. Time = 10 h O m .
From the comparison of zeros with the auxiliary magnetometer at the
beginning and end of the experiments, we find for the corresponding
readings during the experiments, 76678 and 27*05. Considering that
increased readings, if the magnet in the coil correspond to decreased
readings in the auxiliary magnetometer, we find the following numbers for
the positions of rest during the experiments :
765*53
1166-09
+ 400-56
2-94
0-35
.. 9 h 47 m .
.. 13-0
.. negative
.. 366'23
.. 28 - 50
Time = 9
. 9" 53 m
. 13-0
. positive
. 1166-09
28-30
57 m .
Position of rest
766-28
765-59
765-33
Deflected reading
367*60
1166-40
366-23
Deflections
- 398-61
+
400-81
- 399-10
Scale correction .
+ 2-08
2*93
+ 2-08
Reduction of temperature
to Time=9 h 37 m
+ 0-05
+
0*05
- 0-21
Corrected deflection ...
Mean
- 396-55
deflection.
+
397*93
397*42.
- 397-23
397-97
S+ 0-0248.
79] IN ABSOLUTE MEASURE. 33
When all the spinnings had been reduced in this way, the final cal-
culations for the actual resistance were made. The determination of all
quantities involved has been explained, with the exception of the measure-
ment of the absolute pitch of the tuning-fork.
Rate of Vibration of Tuning-fork. As has already been explained, the
tuning-fork which was used to regulate the speed was on every night
compared with a standard fork, and our determinations, therefore, all depend
on the absolute pitch of this standard fork. The method used to determine
that pitch has been described by Lord Rayleigh*.
A fork, vibrating about 32 times a second, maintained by means of an
electric current, and tl riving a second fork of fourfold frequency, was
compared directly with the clock. The vibrations of the driven fork were
simultaneously compared with the standard by counting the number of
beats in a given time. A few experiments have to be made in order to see
whether the fork gains on the clock, or vice versa, and also whether the
standard vibrates quicker or slower than the driven fork. This can be done
by gradually shifting weights on the driver. The difference in the time of
vibration of the clock and driving fork was generally such as to give one
cycle in between 20 or 30 seconds. The driven fork gave at the same time
from 5 to 11 beats per minute.
The experiments agreed well with each other, and both the rate of
vibration and the temperature variation are in close agreement with the
determinations made by Professor McLeod and Mr G. S. Clarke-f- of other
tuning-forks which, like ours, were made by Konig.
The following series of determinations was made at a temperature of
about 13 C. :
128-179 128-181 128-174 128-189
128-180 128-179 128180 128185
The small discrepancies would very likely be still further reduced if
greater care was taken to ascertain the exact temperature of the fork. As a
mean of different sets we find
Number of vibrations in 1 second = 128-180 for t = 13-0 C.
128-161 t=U-QC.
From these data and the number of beats counted during each course of
experiments we can, with the necessary accuracy, determine the number of
vibrations of the fork, which served to regulate the velocity of the revolving
coil
Nature, vol. xvn. p. 12, 1877. [Art. 49, vol. i. p. 331.]
t Phil. Traiu., voL CLXXI. p. 1, 1880.
34 ON THE DETERMINATION OF THE OHM [B. A. UNIT] [79
Calculation of Results. For accurate calculation, the expansion given in
the Report of the British Association is not sufficient. Instead of taking
into account a greater number of terms, we may with equal facility have
recourse to the original quadratic equation for the resistance. We find
R = [ (1 + tan p, sec <) + \/ (1 + tan fi sec <) 2 - U tan 2 </>].
In this equation,/, as before, is written for the frequency of the electrically
maintained fork, and N for the number of the teeth on the apparently
stationary circle.
U is written for
The equation is correct if the torsion and deviation from level are taken
into account in the value of GK as has been explained. The only approxi-
mation used in the equation is that of writing tan^, for KM/GH.
Results.
The results of the calculation are collected in the following table. The
first column contains the date on which the experiments were made ;
the second, the speed in terms of the number of teeth on the stationary
card ; the third column gives the deflection corrected for all scale errors and
variations of temperature during each set ; the fourth column shows the
value of resistance in absolute measure as obtained by calculation on the
assumption that the coefficient of self-induction of the coil is 4'51 x 10 7
centims. This absolute resistance refers to the German silver coil, and a
small length of the bridge wire at a given temperature. As both the
temperature and this length of bridge wire varied in different experiments,
the different results cannot be directly compared, but we can easily apply a
correction which shall reduce the numbers to the absolute resistance of the
German silver coil at a fixed temperature. The temperature chosen was
H'5 C., which was approximately the lowest temperature observed in
the course of the experiments. The fifth column contains the corrected
values, which now can be compared together, and give the absolute resistance
of the standard coil as observed on different occasions, and with different
speeds. In the last column the mean value for the different speeds is given.
In taking these, as well as the final mean, it must be observed that the
set of observations made on December 10 with speed 60 teeth contained
only two spins, or half the usual number.
79]
IN ABSOLUTE MEASURE.
Date
Speed.
No. of teeth
OB stationary
card
Deflection
,^
' fixlO-* !
corrected
Mean
Dec. 7.
10...
120
11042
110-22
4-5486
4-5568
4-5419
4-5309
4-5364
Dec. 2 ...
60
213-61
i.-'-s
4-5487
6... ;
218-30
4--5620
4-5471
4-5467
10...
218-72
4-5531
45422
Dec. 2...
32
3&7-7S
- -
45417
6...
10 ... j
397-39
397-26
4-5672
4-5687
4-5415
4-5448
4-5427
Dec. 2 ...
6...
34
513-73
513-58
4-5719
4-5734
4-5446
4-5438
4-5442
The mean of all the observations is
B- 4-5427 <**h qoadrant.
second
The value of the self-induction which was adopted in these calculations is
slightly smaller than the values calculated by Lord Rayleigh and Mr Niven.
A comparison of the results obtained with different speeds shows that
the value must be very nearly correct, for there is no decided difference
between the results. Nevertheless, it seemed of interest to calculate the
value of the self-induction which best agreed with the experiments, and to
see whether that value gave an appreciably different result for K
We may, in feet, treat both R and L as unknown quantities, and employ
the methods of least squares to find out the most probable values. We
use for this purpose the approximate values already found, and find the most
probable corrections to them. Neglecting the small corrections for torsion,
magnetic moment, and level, and writing P = 2LR GKa>, we find for the
quadratic which determines R
r=0,
where U as before is written for
ZL /2Z, \
GK(GK~ I )'
We find approximately by differentiation, remembering that
dPjP = dRjR,
dR f\ 3U\ dU
32
36 ON THE DETERMINATION OF THE OHM [B. A. UNIT] [79
We may consider dR/R and dU to be the unknown quantities to be
determined. The coefficients with which they are multiplied are known
with sufficient accuracy, d tan <f> is found for each observation by putting
dU=0 and dR equal to the difference between the value of R calculated
by means of that observation, and the value of R provisionally adopted.
The usual methods to form the normal equations were employed. We
find in this way
R = 10 9 x (4-5433 0'0019),
Z = 10 7 x(4-5130-0110).
It is satisfactory to note that the final value of R derr ed with the aid of
the theory of probability is practically identical with the mean value directly
calculated from our experiments with 4'51 x 10 7 as coefficient of self-induction.
A remarkable agreement is shown between the value of this coefficient of
self-induction best fitting our experiments 4'5130 x 10 7
and the value calculated from the dimensions of
the coil 4-5145 x 1() 7 .
The large probable error, however, shows that the agreement is partly
accidental.
To give an idea of the accuracy with which R has been determined by
means of our experiments independently of constant errors, it may be
mentioned that the probability of our value being wrong by one in a
thousand is only one in ten, while the experiments made by the British
Association give an even chance for the same deviation.
It only remains to determine the resistance of the German silver standard
in ohms [B.A. units] at a temperature of 11'5 C.
We had at our disposal the original standards prepared by the Committee
of the British Association. These are very nearly equal at the temperature
at which they are supposed to be correct, and the ohm as determined by the
Committee is, of course, uncertain within the limits within which the
standards differ, but for our present purpose these may be considered
identical. The coils were carefully compared by Mr Fleming, who also
determined their temperature coefficients. One coil in a flat case (hence
called the " flat coil "), which had the smallest temperature coefficient, and
supposed to be right at 14'8 C., was taken at that temperature as the true
ohm. Six of the standards were so arranged and joined together by means
of mercury cups, that four were in a row, and the remaining two in double
circuit, the whole system of coils being thus equivalent to about 4*5 ohms.
Our standard German silver was nearly 4'6 ohms. As the difference was too
great to allow a direct comparison by means of Mr Fleming's bridge, a piece
of German silver wire was prepared so as to have a resistance of -1 ohm ;
79] IN ABSOLUTE MEASURE. 37
this could easily be done within the required limits of accuracy by means of
a set of resistance coils belonging to the Laboratory. Having thus a set
of resistances very nearly equal to the one to be measured, a series of
experiments was made on two successive days. Knowing all the temperature
coefficients, we could easily reduce the measurements to ohms. Four different
experiments gave for the German silver standard at 8'o C.
4-5902, 4-5896, 4'5869, 4'5879, 4-5890. Mean = 4'5887.
Assuming the German silver to vary 4'4 per cent, for 100 C., we find
for our standard at ll c- 5 C. 4 - 595 ohms [B.A. units]. We have hitherto
neglected to take account of the resistance of the two stout copper rods
which connected the rotating coil with the resistance bridge. This resistance
was determined by Mr Fleming to be '003 ohm. To make matters equal, we
ought to have added that resistance to the British Association standards in
comparing them with the standard used by us, and we should then have
found that the absolute resistance found by us to be equal to
earth quadrant
second
was equal to 4'592 ohms [B.A. units].
From this we calculate that the ohm [B.A. unit] as fixed by the
Committee of the British Association is
9893 e^h quadrant
second
this being the final result of our experiments.
108U77
80.
EXPERIMENTS TO DETERMINE THE VALUE OF THE BRITISH
ASSOCIATION UNIT OF RESISTANCE IN ABSOLUTE
MEASURE.
[Phil Trans. CLXXIII. pp. 661697, 1882.]
THE present paper relates to the same subject as that entitled " On the
Determination of the Ohm in Absolute Measure," communicated to the
Society by Dr Schuster and myself, and published in the Proceedings for
April, 1881 [Art. 79] referred to in the sequel as the former paper. The
title has been altered to bring it into agreement with the resolutions of
the Paris Electrical Congress, who decided that the ohm was to mean in
future the absolute unit (10 9 c. G. s.), and not, as has usually been the in-
tention, the unit issued by the Committee of the British Association, called
for brevity the B.A. unit. Much that was said in the former paper applies
equally to the present experiments, and will not in general be repeated,
except for correction or additional emphasis.
The new apparatus [fig. 0] was constructed by Messrs Elliott on the same
general plan as that employed by the original Committee, the principal
difference being an enlargement of the linear dimensions in the ratio of
about 3 : 2. The frame by which the revolving parts are supported is
provided with insulating pieces to prevent the formation of induced electric
currents, and more space is allowed than before between the frame and those
parts of the ring which most nearly approach it during the revolution. It is
supported on three levelling screws, and is clamped by bolts and nuts to the
stone table upon which it stands. The ring is firmly fastened by nuts
to two gun-metal pieces which penetrate it at the ends of the vertical
diameter, and which form the shaft on which it rotates. The lower end
of the bottom piece is rounded, and bears upon a plate of agate, on which
the weight of the revolving parts is taken. A little above this comes
the driving pulley (9 inches in diameter), and above this again the screw arid
80]
OX THE VALCE OF THE BRITISH ASSOCIATION UNIT, &C.
30
not by which the divided card is held. The top piece is hollow, forming
a tube with an aperture of 1 J inches, and is held by a well-fitting brass collar
attached to the upper part of the frame. On this bearing the force is very
small, so that the considerable relative velocity of the sliding surfaces
has no ill effect Notwithstanding its great weight, the ring ran very
lightly, and the principal resistance to be overcome was that due to setting
air in motion.
Fig. .
In the original apparatus the ring is very light, in fact scarcely strong
enough to stand the forces to which it is subjected in winding on the wire.
In order to avoid this defect, and also on account of its larger size, the
new ring was made very massive. Cast solid, with lugs at the ends of what
was to be in use the horizontal diameter, it was cut into two equal parts
along a horizontal plane. The two parts were then insulated from one
another by a layer of ebonite, and firmly joined together again at the lugs by
bolts and nuts, after which the grooves, &c., were carefully turned. As
it was intended to use two coils of wire in perpendicular planes, two
rings were prepared. The smaller ring fitted into the larger, the end
pieces passing through holes along the vertical diameters of both. But
for a reason that will presently be given, only the larger ring was used
in the present experiments.
40 ON THE VALUE OF THE BRITISH ASSOCIATION [80
In the spring of 1881 the larger ring was wound in Messrs Elliott's shop
under the superintendence of Dr Schuster and myself, and the necessary
measurements were taken. On mounting the apparatus a few days later
in the magnetical room of the Cavendish Laboratory, and making pre-
liminary trials, we were annoyed by finding a very perceptible effect upon
the suspended magnet even when the wire circuit was open. The currents
thus indicated might have been due to a short circuit in the wire, or more
probably (considering that the wire was triple covered, and that the winding
had been carefully done) had their seat in the ring itself. Experiment
showed that the insulation between the two parts of the ring as well as
between the wire and each part, was very good, so that no currents could
travel round the entire circumference ; but on consideration it appeared
not unlikely that currents of sufficient intensity might be generated in
those parts of the ring which lie nearest to the ebonite layer. The width of
the ring (in the direction of its axis) was 4 inches, and the least thickness
that at the bottom of the grooves about f inch, so that the operative parts
may be compared to four vertical plates f inch thick, 4 inches broad, and
(say) 6 inches high. In these plates currents will be developed during
the rotation, whose plane is perpendicular to that of the currents in the
The unwished-for currents could doubtless have been diminished by saw
cuts in a vertical plane extending a few inches upwards and downwards
from the insulating layer, but it appeared scarcely safe to assume that
the ring would retain its shape under such treatment. It would have
been wiser to have tried the effect of spinning the ring alone before winding
on the wire, but we were off our guard from the fact that the old ring gave
no perceptible disturbance.
Theory having shown that these currents, if really formed in the manner
supposed, could be satisfactorily allowed for, we decided to proceed with the
experiment. At the worst, the differential effect between wire circuit
closed and wire circuit open could only be in error by a quantity depending
upon the square of the speed, and therefore capable of elimination upon the
evidence of the spinnings themselves ; while if the view were correct that
the disturbing currents were principally in a plane perpendicular to that of
the wire, even the correction for induction would not be much affected.
A special experiment, in which the ring (with wire circuit open) was
oscillated backwards and forwards through a small angle in time with the
natural vibrations of the magnet, allowed us to verify the plane of the
currents. A marked effect was produced when the plane of the ring was
east and west, but nothing could be detected with certainty when it was
north and south the opposite of what would happen with the wire circuit
closed. After this, no doubt could remain but that most of the disturbance
80] UNIT OF RESISTANCE IX ABSOLUTE MEASURE. 41
was due to currents in the ring, and subsequent spinnings after the removal
of the wire have proved that no sensible part of it was caused by leakage
through the silk insulation. The existence of this disturbance, however,
so far modified our original plan as to induce us to omit the second ring as
giving rise to too great a complication.
The suspended magnet was made of four pieces of steel attached to the
edges of a cube of pith and of such length (about inch) as to be equivalent
in their action to an infinitely small magnet at the centre of the cube.
Before the pieces were put together the approximate equality of their
magnetic moments was ascertained. The resultant moment was between
six and seven times as great as that used in our former experiments.
In virtue of the greater radius of the coil, this important advantage was
obtained without undue increase of the correction for magnetic moment,
which amounted to about "004, only twice as great as before. The effect of
mechanical disturbances, such as air currents, was still further reduced by
diminishing the size of the mirror, particularly in its horizontal dimension.
On both accounts the influence of air currents was probably lessened about
15 times, and, in fact, no marked disturbance was now caused by the
proximity of a lamp to the magnet box*. In consequence of these changes,
however, it was found necessary to introduce an inertia ring in order to bring
the time of vibration up to the amount (about 5^ seconds from rest to rest)
necessary for convenient observation. The diameter of the ring was about
| inch, and the whole weight of the suspended parts was not too great to be
borne easily by a single fibre of silk. A brass wire passing between the
spokes of the ring prevented the needle from making a complete revo-
lution.
The enlarged scale of the apparatus allowed us to introduce a great
improvement into the arrangement of the case necessary for screening the
suspended parts from the mechanical disturbance of the air caused by
the revolution of the coiL A brass tube of an inch in diameter was not
too large to pass freely through the hollow axis. At its lower extremity
(fig. 1) it was provided with an outside screw, to which the magnet box was
attached air-tight By unscrewing the box, whose aperture was large enough
to allow the inertia ring to pass, the suspended parts could be exposed
to view, and by drawing up the brass tube they could be removed altogether,
so as to allow the coil to be dismounted, without breaking the fibre. The
upper end of the fibre was attached to a brass rod sliding in a socket at the
upper end of the tube, by which the height of the magnet could readily
be adjusted. The whole was supported on three screws passing through
the corners of a brass triangle attached to the tube not far above the place
* See pp. 115, 132 [ol. n. pp. 11, 28] of the former paper.
42
ON THE VALUE OF THE BRITISH ASSOCIATION
[80
where it emerged from the hollow axis. The points of the screws rested
upon the same overhanging stand as in the former experiments [p. 9]*.
The larger diameter of the tube made the system so rigid that no
mechanical disturbance of the kind formerly met with was to be detected
at the highest speed to which we could drive the coil. Even a tap with
the finger-nail upon the magnet box produced but a small disturbance.
Fig. l.
No change was required in the arrangements for regulating and de-
termining the speed of the coil, which worked, if possible, more perfectly
than before, in consequence of the greater inertia of the revolving parts.
The divided card was, however, on an enlarged scale, and the numbers of
the teeth in the various circles were so arranged that each circle was
available for a distinct pair of speeds according as it was observed through
the slits in the plates carried by the electric fork or over the top of the upper
plate. The speeds actually used corresponded to 80, 60, 45, 35, and 30 teeth,
seen through the slits, i.e., about 127 times per second.
The greater resistance of the copper coil (23 instead of 4'6) rendered
necessary a modification in the method of making the comparisons with
the standard. The whole value of the divided platinum-iridium wire on
Fleming's bridge being only ^ ohm, a change of temperature in the copper
of not much more than a degree would exhaust the range of the instrument.
To meet this difficulty it was only necessary to add resistances to the copper
circuit so as to compensate approximately the temperature variations, for it
is evident that it can make no difference whether the change of resistance of
the entire revolving circuit is due to a rise of temperature, or to the insertion
of an additional piece. The platinum-silver standard was therefore prepared
so as to have a resistance (about 24 ohms) greater than any which we were
likely to meet with in the copper, and the additional pieces were relied upon
to bring the total within distance. As at first arranged, the additional
resistance was inserted at the mercury cups, instead of a contact piece of
* June, 1882. The general disposition of the apparatus is shown in fig. 2.
80]
UNIT OF RESISTANCE IN ABSOLUTE MEASURE.
no appreciable resistance. During the comparison with the standard it was
transferred to another part of the circuit.
In the course of May, 1881, a complete series of spinnings were taken,
the arrangements and adjustments being (except as above-mentioned) in
all respects the same as with the old apparatus. Five different speeds were
used, and each of them on three different evenings. The work of observing
was also distributed as before, Dr Schuster taking the readings of the
principal magnetometer, and Mrs Sidgwick the simultaneous readings of
the auxiliary magnetometer, while I observed the divided card and regulated
the speed. At each speed on each evening four readings were taken with
Kg. 2.
Stand for suspended parts.
Frame of revclying coil.
Driving cord.
Electro-magnetic fork and telescope.
Water engine.
Principal telescope and scale.
H. Fleming's bridge.
J. Platinnm-silTer standard.
-/. Bridge galvanometer.
K. Telescope and scale of auxiliary magneto-
meter.
L. Auxiliary magnetometer needle and
wire circuit closed, two with positive and two with negative rotation, and in
like manner four readings were taken with the wire circuit open. Observa-
tions on the zero with the coil at rest were for the most part dispensed with,
as it was thought that the time could be better employed otherwise ; in fact,
the mean of the two not very different positions of equilibrium obtained with
positive and negative rotation when the wire circuit was open, gires all
that is wanted in this respect In the actual reductions we only require
the difference of readings with positive and negative rotations.
It was hoped that these observations would have been sufficient, but
on the introduction by Dr Schuster of the various corrections for tempe-
rature, for the beats between the two forks, and for the outstanding bridge-
wire divisions, the necessity for which disguises the significance of the
44 ON THE VALUE OF THE BRITISH ASSOCIATION [80
numbers first obtained, it was found that the agreement of the results
corresponding to a given speed was by no means so good as we had expected
in view of the precautions taken and the accuracy of the readings. What
was worse, there was evidence of a decided progression, as if the absolute
resistance of the standard had gradually diminished during the time occupied
by the spinnings.
It is not impossible that there really was some change in the standard
which had then been newly prepared; but the discrepancies were not, as
according to this view they ought to have been, proportional to the speeds
of rotation. I am inclined rather to attribute them to shiftings of the paper
scales. The principal magnetometer scale was composed of three lengths
of 50 centims. each, cemented with indiarubber to a strip of deal. The
compound scale thus formed was examined by Dr Schuster in March, 1881.
Between the graduations of the first and of the middle piece there was
a gap of about \ millim., and another of nearly the same magnitude between
the middle and the third piece. When I re-examined the scale in July,
the gap at 500 divisions had increased to T % millim., and that at 1000 to
millim. Curiously enough, there were no observable errors in the equality
of the divisions of the three parts taken separately; but the changes above-
mentioned are sufficient to throw considerable doubts upon the value of the
first series of spinnings. They have, however, been reduced by Dr Schuster,
and the result is given below for the sake of comparison.
To be free for the future from uncertainties of this kind, I replaced
the paper scale by a long glass thermometer tube by Casella, graduated
into millimetres. The divisions were fine and accurately placed, but the
imperfect straightness of the tube has rendered necessary certain small
corrections in the final results. Probably a straight strip of flat opal would
have been an improvement.
The second series of spinnings was made in August, 1881, and this,
it was fondly hoped, would be final. To guard against possible change in
the platinum-silver coil a careful comparison with the standard units was
previously instituted by Mrs Sidgwick, of which the details are given later.
As we had unfortunately lost the advantage of Dr Schuster's assistance, the
observations at the principal magnetometer devolved upon Mrs Sidgwick.
The much easier post at the auxiliary magnetometer was usually occupied
by Lady Rayleigh ; occasional assistance has been rendered by Mr A. Mallock
and by Mr J. J. Thomson.
In the conduct of the second series one or two minor changes were
introduced. In order to know the temperature of the standard tuning-fork
more accurately, a thermometer was placed between its prongs and read
at the same time as the number of beats was taken. The insertion of the
80] UNIT OF RESISTANCE IN ABSOLUTE MEASURE. 45
small resistances necessary to bring the copper coil within range of the
standard was also arranged in a different manner. Some trouble had been
experienced in getting a sufficiently good fit between the contact pieces used
in the first series and the mercury cups. It is necessary that the stout
copper terminals should press down closely upon the bottoms of the cups,
and also that the mercury should not be liable to escape at high speeds from
the effect of centrifugal force. Bits of indiarubber tubing were placed
round the copper legs, by which a fair fit with the sides of the cups
was effected ; but I thought that it would be an improvement to revert
to a single contact piece for the mercury cups of no sensible resistance,
whose fit could be carefully adjusted, and to insert the extra resistances at
the connexion of the other (outer) ends of the component coils. For this
purpose binding screws were employed, pressing firmly together the flat
copper terminals of the copper wire and of the German-silver resistance
pieces. It is almost unnecessary to say that these short lengths of German-
silver wire were doubled upon themselves before being coiled, and that the
pieces were not touched between a spinning and the associated resistance
comparisons. Used in this way the screwed up contacts seemed unobjection-
able, even though the surfaces were not amalgamated.
On each night and for each speed a set of twelve spinnings was made,
six with wire circuit open, and six with wire circuit closed. It was usual to
take, first, two of the former (one with positive and one with negative
rotation) ; secondly, to compare the resistances of the revolving circuit
and the standard ; thirdly, after inserting the contact piece and adjusting
the indiarubber strap by which it was held down, to make the six closed
contact spinnings ; fourthly, to compare the resistances again ; and lastly, to
complete the open contact readings. Each spinning, it will be under-
stood, involved the reading of several elongations (about six for the open
contact and ten for the closed), from which the position of equilibrium
was deduced.
Table II. [p. 70] gives all the results of the second series, except one
for 35 teeth on August 27th, which was rejected on the ground that it
exhibited such large internal discrepancies, as to force us to the conclusion
that the contact piece had been inserted improperly. It will be seen
that the agreement is good except on August 29th, in which case the
deflections are as much as four or five tenths of a millimetre too small.
These discrepancies, though not very important in themselves, gave me
a good deal of anxiety, as they were much too large to be attributed to
mere errors of reading, and seemed to indicate a source of disturbance
against which we were not on our guard.
The least unlikely explanations seemed to be (1) a change in the distance
of the mirror from the scale, which unfortunately had not been remeasured
46 ON THE VALUE OF THE BRITISH ASSOCIATION [80
at the close of the spinnings, though this would require to reach 3 millims. ;
(2) imperfect action of the contact piece from displacement of mercury
or otherwise ; (3) a change of level in the axis of rotation. The anomalous
result of August 27th seemed to favour (2), while on behalf of (1) it must be
said that the stand of the telescope and scale as well as the support for
the suspended parts of the principal magnetometer were of wood. It was
just conceivable that under the influence of heat or moisture some bending
might have occurred.
On my return to Cambridge in October we proceeded to investigate
these questions with the closest attention. As repeated direct measure-
ments of the distance of the mirror and scale were inconvenient, measuring
rods (like beam compasses) were provided to check the relative positions of
the telescope stand and of the upper end of the suspending fibre with
regard to fixed points on the walls of the room. But no changes com-
parable with 3 millims. were detected, even under much greater provocation
than could have existed during the August spinnings. The next step
was to examine the action of the contact piece. For this purpose the
coil was balanced against the standard as usual, except that the contact
piece was inserted and connexion with the bridge made at the other ends
of the double coil. It was presently found that the resistance did depend
upon the manner in which the contact piece was pressed, and that to an
extent sufficient to account for the August discrepancies. Eventually it was
discovered that one of the legs of the contact piece, which by a mistake
had been merely rivetted and not soldered in, was shaky.
After this there could be no reasonable doubt that the faulty contact
piece was the cause of our troubles. In all probability the leg became
loose on the 27th, in which case the earlier results would be correct.
Moreover, the final means are not very different, whether the spinnings
of August 29th are retained or not. This being the case, we might perhaps
have been content to let the matter rest here ; but in view of the importance
of the determination, and the desirability as far as possible of convincing
others as well as ourselves, we thought that it would be more satisfactory
to make a third and completely independent series of spinnings.
In this series the faulty composite contact piece was replaced by a
horse shoe of continuous copper, and a check was instituted upon the
distance between mirror and scale. The opportunity was also taken to make
a minor improvement in connexion with the auxiliary magnetometer. The
somewhat unsteady table on which the telescope and scale had stood was
replaced by one of stone, and the arrangements for illumination were im-
proved by throwing an image of a gas flame on the part of the scale under
observation. The same number of readings were made as in the second
series, but we found it more expeditious to take the six open contact
80] UNIT OF RESISTANCE IN ABSOLUTE MEASUBE. 47
spinnings together. At the beginning of the evening it was desirable to
commence with these open contact spinnings in order to give more time
for the coil to acquire the temperature of the room, which always rose
somewhat, although the lamps and gas were lit a couple of hours beforehand.
Later in the evening we sometimes took the closed contact readings for
two speeds consecutively, in order that the intermediate resistance com-
parison might serve for both. In other respects the arrangements were
unaltered.
Full details of the observations and reductions are given below. It
will be sufficient here to mention that the maximum discrepancy between
any two deflections at the same speed amounts only to y^ of a millimetre,
so that the agreement on different nights is more perfect than could have
reasonably been expected. At the lowest speed the above-mentioned dis-
crepancy is less than one part in 3000, and at the highest speed less
than one part in 6000. No spinnings in the third series were rejected,
except on one or two occasions when it appeared at the time of observation,
from the behaviour of the auxiliary magnetometer, that there was too much
earth disturbance. The spinnings were then suspended, and the observa-
tions already obtained were not reduced.
At the close of the spinnings, Mrs Sidgwick made a further comparison
of our platinum-silver coil with the standard units.
The value arrived at for the B.A. unit ("9865 ohm) differs nearly three
parts in a thousand from that which we obtained with the original apparatus.
This difference is not very great, and may possibly be accounted for by errors
in the measurement of the coil (see [p. 11] of former paper). If a coil be
imperfectly wound, the mean radius, as determined by a tape, is liable to be
too great At any rate, this discrepancy sinks into insignificance in com-
parison with that which exists between either of these determinations and
that of Professor Kohlrausch*, according to whom the B.A. unit would be as
much as 1*0196 ohms. With respect to the method employed by Kohlrausch,
I agree with Rowland^ in thinking it difficult, and unlikely to give the
highest accuracy ; but how in the hands of a skilful experimenter it could
lead to a result 3 per cent, in error, is difficult to understand. The only
suggestion I have to make is that possibly sufficient care was not taken in
levelling the earth-inductor. Although estimates are given of the probable
errors due to uncertainties in the various data, nothing is said upon this
subject. In consequence, however, of the occurrence of the horizontal
intensity as a square in the final formula, in conjunction with the largeness
of the angle of dip, the method is especially sensitive to a maladjustment of
* Pogg. A**., Erginanngtand TI. PhiL Mag., April, 1874.
t America* Journal, April, 1878.
48 ON THE VALUE OF THE BRITISH ASSOCIATION [80
this kind. I calculate that a deviation of the axis of rotation from the
vertical through 21' in the plane of the meridian, would alter the final result
by 3 per cent.*
According to Rowland's determination, the value of the B.A. unit is
'9912 ohm. The method consists essentially in comparing the integral
current in a secondary circuit, due to the reversal of the battery in a primary
circuit, with the magnitude of the primary current itself. The determination
of the secondary current involves the use of a ballistic galvanometer, whose
damping is small, and whose time of vibration can be ascertained with full
accuracy ; and it is here, I think, that the weakest point in the method is to
be found. The logarithmic decrement is obtained by observation of a long
series of vibrations, and it is assumed that the value so arrived at is appli-
cable to the correction of the observed throw. I am not aware whether the
origin of damping in galvanometers has ever been fully investigated, but the
effect is usually supposed to be represented by a term in the differential
equation of motion proportional to the momentary velocity. This mode of
representation is no doubt applicable to that part of the damping which
depends upon the induction of currents in the galvanometer coil under the
influence of the swinging magnet. If this were all, a correction for damping
would be accurately effected on the basis of a determination of the loga-
rithmic decrement, made with the galvanometer circuit closed in the same
manner as when the throw is taken. In all galvanometers, however, a very
sensible damping remains in operation even when the circuit is open, of
which the greatest part is doubtless due to aerial viscosity ; and it is certain
that the retarding force arising from viscosity is not simply proportional to
the velocity at the moment, without regard to the state of things imme-
diately preceding.
In particular, the force acting upon the suspended parts as they start
suddenly from rest in the observation of the throw, must be immensely
greater than in subsequent passages through the position of equilibrium,
when the vibrations have assumed their ultimate character. I calculate that
in the first quarter vibration (i.e., from the position of equilibrium to the first
elongation) of a disc vibrating in its own plane and started impulsively from
rest, the loss of energy from aerial viscosity would be 1'373 times that under-
gone in subsequent motion between the same phases. From this it might at
first appear that in this ideal case the logarithmic decrement observed in the
usual manner would need to be increased by more than a third part in order
to make it applicable to the correction of a throw from rest ; but in order to
carry out this view consistently we should have to employ in the formula the
time in which the needle would vibrate if the aerial forces were non-existent,
instead of the actually observed time of vibration. Now since the action of
* See p. [63].
80] UNIT OF RESISTANCE IX ABSOLUTE MEASURE. 49
viscosity is to increase the time of vibration, the second effect is antagonistic
to the first, so that probably the error arising from the complete neglect of
these considerations is very small.
There is another point in which it appears to me that the theory of the
ballistic galvanometer is incomplete. It is assumed that the magnetism of
the needle in the direction of its axis is the same at the moment of the
impulse as during regular vibrations. Can we be sure of this ? The impulse
is due to a momentary but very intense magnetic force in the perpendicular
direction, and it seems not impossible that there may be in consequence a
temporary loss of magnetism along the axis. If this were so, the actual
impulse and subsequent elongation would be less than is supposed in the
calculation, and too high a value would be obtained of the resistance of the
secondary circuit in absolute measure. In making these remarks I desire
merely to elicit discussion, and not to imply that Rowland's value is certainly
four parts in a thousand too high.
Determinations of the absolute unit have been made also by H. Weber*,
whose results indicate that the B.A. unit is substantially correct. In the
absence of sufficient detail it is difficult to compare this determination with
others, so as to assign their relative weights.
The value of the B.A. unit in absolute measure is involved in the two
series of experiments executed by Joule on the mechanical equivalent of heat-f-.
The result from the agitation of water is 24868, while that derived from the
passage of a known absolute current through a resistance compared with the
B.A. unit was 25187. The latter result is on the supposition that the B.A.
unit is really 10 9 C.G.S. If we inquire what value of the B.A. unit will
reconcile the two results, we find
1 B.A. unit = '9873 ohm,
in very close agreement with the measurement described in the present
paper. It should be remarked that in the comparison of the two thermal
results some of the principal causes of error are eliminated ; and it is not
improbable that an experiment in which heat should be simultaneously
developed in one calorimeter by friction, and in a second similar calorimeter
by electric currents, would lead to a very accurate determination of resist-
ance, more especially if care were taken so to adjust matters that the rise of
temperature in the two vessels was nearly the same, and a watch were kept
upon the resistance of the wire while the development of heat was in
progress.
[June, 1882. Since this paper was sent to the Society, Mr Glazebrook
has worked out the results of a determination of the B.A. unit in absolute
* Phil. Mag., Jan., Feb., March, 1878.
t Phil. Trans., Part n., 1878. Brit. Ass. Rep., 1867; Reprint, p. 175.
50 ON THE VALUE OF THE BRITISH ASSOCIATION [80
measure by a method not essentially different from that adopted by Rowland.
The final number is practically identical with that of the present paper ; and
the agreement tends to show that the difference between ourselves and
Rowland is not to be attributed to the use of a ballistic galvanometer.
Reference should have been made to the results of Lorenz*. He finds as
the value of the mercury unit defined by Siemens
^OOH. earth quadrant
1 mercury unit = '9337 - .
second
The corresponding number calculated from the results of the present
paper with use of the value of the specific resistance of mercury lately found
(Proc. Roy. Soc., May 4, 1882) is '9413. If we invert the calculation, we find
that according to Lorenz the value of the B.A. unit wonld be '9786 absolute
measure. The method of Lorenz is ingenious, and apparently capable with
good apparatus of giving a result to much within 1 per cent. Mrs Sidgwick
and myself are at present making a trial of it.]
It will be desirable here to consider briefly some of the criticisms of
Kohlrausch and Rowland upon the method of the original British Associa-
tion Committee, which has been adopted in the present investigation without
fundamental alteration. The difficulty, remarked upon by Kohlrausch, of
obtaining a rapid and uniform rotation, has not been found serious, and I
believe that no appreciable error can be due either to irregularity of rotation
or to faulty determination of its rapidity. It has also been brought as an
objection to the method that a correction is necessary on account of the
magnetic influence of the suspended magnet upon the revolving circuit.
The theory of this action is, however, perfectly simple, and the application
of the correction requires only a knowledge of the ratio of the magnetic
moment to the earth's horizontal force. If the magnetic moment is very
small, the correction is unimportant ; if larger, it can on that very account
be determined with the greater ease and accuracy. It is probable that in
the original experiments too feeble a magnetic moment was used, and that
in consequence the suspended parts were too easily disturbed by non-
magnetic causes ; but this might have been remedied without increasing
objectionably the correction in question. At any rate the larger coil of the
new apparatus allows the use of any reasonable magnetic moment.
Perhaps the least advantageous feature in the method is the necessity
for creating a violent aerial disturbance in the immediate neighbourhood of
a delicately suspended magnet and mirror. If, however, any deflection occurs
in this way, very little error can remain when the open contact effect is
subtracted from the closed contact effect. The difficulty of avoiding a
* Fogg. Ann., 1873.
80] UNIT OF RESISTANCE IN ABSOLUTE MEASURE. 51
sensible deflection, due to currents in the ring, when the wire circuit is
open, is connected with a special advantage i.e., the possibility of assuring
ourselves that there is no leakage from turn to turn of the coil. In the
method followed by Rowland, for instance, such a leakage would lead to
error, and could not be submitted to any direct test.
The correction for self-induction cannot be made very small without a
disadvantageous reduction of the whole angular deflection ; but so far as the
wire is concerned it can be calculated a priori, or determined by independent
experiment, with the necessary accuracy. There is reason, however, to think
that the best method of treatment is to determine this correction from the
spinnings themselves, combining the results of widely different speeds so as
to obtain what would have been observed at a small speed. At small speeds
it is certain that all effects of self-induction and of mutual induction between
the wire circuit and other circuits in the ring will disappear.
Measurements of coil.
The mean radius of the coil, being the fundamental linear measurement
of the investigation, must be found with full accuracy. There has been some
difference of opinion as to the best method of effecting this. The greatest
accuracy is probably attained by the use of the cathetometer. The measure-
ment of the circumference of every layer by a steel tape has the advantage
that the subject of measurement is three times as large, and is much less
troublesome. The disadvantage is that if a layer be not quite even, there is
danger of measuring the maximum rather than the mean outside circum-
ference. In the present investigation the coil was so large that the tape
could be employed without fear*.
Each of the component coils marked A and B had 18 x 16 = 288 windings,
but in consequence of variations in the thickness of the triple silk covering,
there was a difficulty in getting exactly 18 turns into each layer. In the
eleventh layer of A it was necessary to be content with 17 turns, and to
place an extra turn on the outside, so as to form the commencement of a
seventeenth layer a circumstance which of course was taken into account in
calculating the mean. The number thus arrived at, after correction for the
thickness of tape, is the mean outside circumference. What we require is
the mean circumference of the axis of the wire ; it may be derived from the
first by subtraction of half the difference between the tape readings for the
first layer, and for the bottom of the gun-metal groove.
The original Committee also employed the tape method. Their measurement of the length
of the wire when unwound was not in order to find the mean radius, as Siemens and Kohlrausch
suppose, but to verify the number of turns.
52 ON THE VALUE OF THE BRITISH ASSOCIATION [80
The results obtained by Dr Schuster and myself when the coils were
wound are :
Coil A. Coil B.
Mean of readings in millims. . . 1489'3 1487'5
Correction for tape . . . . '. '6 "6
Mean outside circumference . . . 1488'7 1486'9
Correction for thickness of wire 3 '4 3'4
Mean circumference .... 1485'3 1483'5
Mean radius . ". . , ,-*.-''-. 236'39 23611
Mean circumference of A and B 1484'4
Mean radius of A and B (a) . . . 236'25
Axial dimension of section in millims. . 19*9 19'9
Radial 15'9 15'4
Distance of mean planes (26') . . 65'95
Two or three readings were taken of the circumference of every layer,
and to prevent mistakes in the number of turns, the plan described by
Maxwell*, of simultaneously winding string on wooden rods, was followed.
Without some such device, there is great risk of confusion.
In estimating the degree of accuracy obtainable, we must remember that
the circumference of each layer is measured before the outer layers are
wound on ; any change produced by the pressure of these outer layers is a
source of error. We had already observed a tendency in the measurements
to be less during the unwinding of a coil than during the winding, and we
fully intended to remeasure the coil after the spinnings were completed.
This was done on December 6, 1881, by Mrs Sidgwick and myself. As we
expected, somewhat smaller readings (by about f millim.) were obtained for
the circumference of the middle layers. The results were :
Coil A. Coil B.
Mean radius . .. 236'31 236'02
Mean of both . . 23616
or nearly one part in 2000 less than before. Of the two values, it would
appear that the latter is more likely to represent the actual condition of the
coil during the spinnings, and is therefore entitled to greater weight. If we
give weights in the proportion of two to one, we get
Mean radius = 23'619 centims.f
* Electricity and Magnetism, n. 708.
t August, 1882. At the time of use the tape was compared with a measuring rod, which
again has been compared with a standard metre verified by the Standards Department of the
Board of Trade. For the purposes of this investigation the differences observed are altogether
negligible. I may add that the clock with which the standard tuning-fork was compared (see
p. [33] of former paper) was rated from astronomical observations.
80] UNIT OP RESISTANCE IN ABSOLUTE MEASURE. 53
Calculation of GK.
have
GK = ^-n^n-a sin'a \ 1 + A ^ + i ^ s\n-a cos'ec - i sin'al
( 6 a 1 a* 8 a 1
We have
G
in which
a = mean radius = 23'625 (1st measurement)
b = axial dimension of section = T990
c = radial dimension of section = 1'565
n = total number of turns = 576
26' = distance of mean planes = 6*595
sin a = a + >J(a? + 6'*).
From these data we find
log 2ir s n 2 = 6-81617
log a = 1-37337
log sin 3 a = 1*98744
log{...} =1-99995
logGJ-f =8-17693
But if we substitute the adopted value of a, i.e., 23'619 centims., we have
by subtraction of '00011
loGjK- =8-17682.
Calculation of L.
We may write
L = 16 s x 18 s (A + L, +
where L 1} L* are the coefficients of self-induction of the two parts, and J/ the
coefficient of mutual induction without regard to the number of turns. L^
and L 2 may be calculated from the formula
L = *7ra pog (8a/r) + & - 1 (0 - TT) cot 2^ - ITT cosec 2(9
- 1 cot 2 log,, cos 6 - % tan 2 log sin 0],
in which r is the diagonal of the section, and 9 the angle between it and the
plane of the coiL With this formula and with the dimensions as measured
when the coil was wound, we get
L, (for A) = 1029-3 centims. L 2 (for B) = 1031 '9 centims.
It would not be difficult to calculate an approximate correction for the
curvature of the coil, but this is scarcely necessary. (See p. [15] of former
paper.) Adding the above, we have
206 1-2 centims.
54 ON THE VALUE OF THE BRITISH ASSOCIATION [80
The value of M was found from the tables given as Appendix I. to 706
of the new edition of Maxwell's Electricity. If we suppose each coil con-
densed into the centre of its section, we find M= 4nr x 33'061. A more exact
calculation by the formula of interpolation explained in Appendix II. gives
Jlf=47rx33-140, so that
2M = 832-88 centims.
The final result is accordingly
L = 1G 2 x 18' x 28941 = 24004 x 10 8 centims.
These calculations of the coefficients of induction have been made
independently by Mr Niven and myself, and are so far reliable ; but we must
not forget that the accuracy of the result depends upon the accuracy of the
data, and that in the present case the diagonal of the section (r) on which
the most important part of L depends is an element subject to considerable
relative uncertainty. It is probable that the effective axial dimension of
the section is somewhat less than the width of the groove, and therefore that
the real value of L may be a little greater than would appear from the
preceding calculation.
Theory of the ring currents.
If the circuits are conjugate, the currents in the wire and in the ring are
formed in complete independence of one another, a circumstance which
simplifies the theory very materially. In the same notation as was used in
the former paper (p. 105) [p. 2], and with dashed letters for the ring circuit,
we have as the equation determining the angle of deflection (<) when the
wire circuit is closed,
\G'K'<*
+ ^2 i i'z m z V* + *** tan $ + R' tan /* s ?c 0j.
When the wire circuit is open, the equation determining the angle of
deflection (</> ) is
tan </> + T . = {R + L'a tan + R' tan /, sec
Since T is an extremely small quantity it is unnecessary to keep up the
distinction between r</>/cos tf> and rtan</>. By subtraction
(1 + T) (tan </> - tan < )
= p 2 'j- a - 2 {R + La) tan </> + R tan /j, sec </>}
n ^ ~ tan ^ + R tan ^ ^ sec ^ ~ sec ^'
80] UNIT OF RESISTANCE IX ABSOLUTE MEASURE. 55
The last term is small, and we may neglect (sec </> sec <^) in combina-
tion with # tan /z.
Moreover
R- + L*a)- ~ R' + L'a> tan fa '
so that
If now we write (<?#) for "/(! + T), we get
The effect of L' would therefore be to increase disproportionately the
deflections at high speeds, i.e., contrary to the effect of L. It appears,
however, that in these experiments it could not have been sensible. At the
highest speed tan< was about 3^, and <a about 26 per second, so that
< tan fa would be about ^. The value of L' R' is difficult to estimate with
any accuracy. But the value of L'R for the wire circuit is about - 01 second,
and that for the ring circuit must be much less, so that the terms involving
L' may safely be omitted.
The quadratic in R then becomes
Lw _ Cg)ZM , ..._ =o,
tan <f> tan fa tan <f> - tan fa
whence
where
2L
(GK) \(GK) tan </> - tan 0,1 '
L by direct experiment*.
Although the calculated value of L was the result of two independent
computations, I considered that it would be satisfactory still further to
* In consequence of the necessity which ultimately appeared of introducing an arbitrary
correction proportional to the square of the speed of rotation, the result of the present section
does not influence the final number expressing the B.A. unit in absolute measure. The method,
however, is of some interest, and (it is believed) has not been carried out before with the pre-
cautions necessary to secure a satisfactory result
56 ON THE VALUE OF THE BRITISH ASSOCIATION [80
verify it by an experiment with Wheatstone's balance. The statement
of this method and the final formula, as given on pp. [12, 13] of the
former paper, being approximate only, it will be convenient here to repeat
them with the necessary corrections.
The four resistances in the balance are two equal resistances (10 units
each), that of the copper coil P, and a fourth resistance Q (nearly equal
to P) taken from resistance boxes, of which P is the only one associated
with sensible self-induction. When P and Q are equal, there is no per-
manent current through the galvanometer ; but if the galvanometer circuit
be first closed and then the battery current be made, broken, or reversed,
the needle receives an impulse, whose magnitude depends upon L.
If x denote the change of current in the branch P, the action of self-
induction is the same as that of an electromotive impulse in that branch
of magnitude Lx, and the effect upon the galvanometer is that due to
this electromotive impulse acting independently of the electromotive force
in the battery branch.
In order now to get a second quantity with which to compare the
induction throw, the resistance balance is upset in a known manner. If
while Q remains unaltered, P be increased to P + 8P, there is a steady
current through the galvanometer, which we may regard as due to an
electromotive force SP. x' in the branch P + SP, x being the current
through the branch. If be the deflection of the needle under the
action of the steady current, a the angular throw, and T the time of
swing from rest to rest, we have by the theory of the ballistic galvano-
meter as the ratio of the instantaneous to the steady electromotive force
TT tan0 '
subject to a correction for damping; so that this expression represents the
ratio of Lx : SP. x'. If the induction throw be due to the make or break
of the battery circuit, x represents simply the current in the branch P.
In the case where the battery current is reversed, we may write 2Lx for
Lx, understanding by x the same as before. As this method was the one
actually adopted, we will write the result in the appropriate form
Tsm$a
TT tan0 '
In the formula as originally given by Maxwell, and as stated in the
former paper, the distinction between x and x (the currents before and
after the resistance balance is upset) was neglected. This step is legiti-
mate if SP be taken small enough, to which course however there are
experimental objections. In order that tan 6 might be of suitable magni-
80] UNIT OF RESISTANCE IN ABSOLUTE MEASURE. 57
tude, it was found necessary to make the ratio of &P : P equal to about
xfa, a fraction too large to be neglected.
In carrying out the experiment it was found more convenient to insert
the additional resistance in the branch Q, leaving P unaltered. By the
symmetry of the arrangement it is evident that this alteration is imma-
terial, and that we may take the formula in the form
L = L = ly.i;' Jsinja
Q~ P~ Q.x TT tan0 '
JT being the current in the branch Q when the resistance balance is perfect,
x the diminished current when the additional resistance SQ is inserted.
The principal difficulties in carrying out the experiment arose from
variation in the battery and in the resistance balance. From these causes
the results of two days' experiments were rejected, as unlikely to repay
the trouble of reduction. On the last day (December 3, 1881) the first
difficulty was overcome by using three large Daniell cells (charged with
zinc sulphate) in multiple arc. As precautions against rapid change of
temperature the copper coils were wrapped thickly round with strips of
blanket aud deposited in a closed box. The delicacy of our arrangement*
was such that about -^^ of a degree centigrade would manifest itself, so
that it was hopeless to try to maintain the resistance balance absolutely
undisturbed. The mode of applying a suitable correction will presently
be explained. On December 3, partly by good luck, the necessary correc-
tion remained small throughout. In order to avoid a direct action of the
current upon the galvanometer needle, the coil was placed at a consider-
able distance, at the same level, and with its plane horizontal Any
outstanding effect of the kind would, however, be eliminated from the
final result by the reversals practised
The induction throws were always taken by reversal ot the battery
current. A reversal has two advantages over a simple make or break.
In the first place the effect is doubled and is therefore more easily mea-
sured; and in the second the battery is more likely to work in a uniform
manner, the circuit being always closed except for a fraction of a second
at the moment of reversal. The key was of the usual rocker and mercury
cup pattern.
The galvanometer was one belonging to the laboratory of about 80 ohms
resistance. It was set up by Mr Glazebrook for his experiments by an
allied method, and with its appurtenances was ready for use at the time
that this determination of self-induction was undertaken. The scale was
divided into millimetres, and was placed at a distance of 218 centims.
from the galvanometer mirror. The instrument was adapted for ballistic
58 ON THE VALUE OF THE BRITISH ASSOCIATION [80
work, as the vibrations were subject to a logarithmic decrement of only
about -0142.
The electric balance was provided for by a resistance box from Messrs
Elliotts. The battery current after leaving the reversing key divides itself
on entering the box, each part traversing 10 ohms. At the ends of these
resistances come the galvanometer electrodes. The first part of the current
now traverses the copper coil, and the second part other resistances, after
which the two parts reunite and pass back to the battery. In the use
of the "other resistances," a special arrangement was adopted which I
must now explain. The resistance of the copper coil being somewhat
under 24 ohms, the most obvious way to obtain a balance was to add to
it a piece of adjustable wire until the whole would balance 24 ohms from
the box. The objection to this plan is that the smallest known disturb-
ance which we can then introduce, i.e., by the addition or subtraction of
a single unit, is much too great for the purpose.
The difficulty thus arising is completely met by the use of high resist-
ances, taken from a second box, in multiple arc with the 24 ohms.
In order to balance the copper coil and its leading wires at the actual
temperature (about 14), 753 ohms were required in multiple arc with
the 24. To calculate the resultant resistance we have
= -041666667 + "001328021 = '042994688 = .!. ,
so that the resistance of the copper coil in terms of the units of the box
is 23-25869. A suitable deflection 6 was obtained by the substitution of
853 for 753 in the auxiliary box. In this case
^ + ufo = '041666667 + "001172333 = "042839000 = 9a . a \ a ^ ;
so that the additional resistance was
SQ = -08453 unit.
It may be remarked that if the copper coil had been about 1 warmer,
its resistance would have been greater by ^th part, and the balance would
have required 853 instead of 753 in multiple arc with the 24.
On account of the progressive changes already mentioned, it was advis-
able to alternate the observations of a and 6 as rapidly as possible, and to
occupy no more time than was really necessary in taking the readings.
A good deal of time may be saved by working the key suitably, and by
opening and closing the galvanometer branch (at a mercury cup provided
for the purpose) so as to avoid producing unnecessary swings, and to stop
those due to induction when done with ; but it is unnecessary to go into
detail in this part of the subject. After a little practice two induction
80] UNIT OF RESISTANCE IX ABSOLUTE MEASURE. 59
throws, starting with opposite directions of the current, and two observa-
tions of steady deflection, one in each position of the reversing key, could
be made in about seven minutes. The vibrations of the galvanometer
needle were damped by the operation of a current in a neighbouring
coil, the current being excited by a Leclanche cell and controlled by a
key within reach of the observer at the telescope. The readings were
taken by Mrs Sidgwick, while I reversed the battery current, shifted the
resistances, and recorded the results.
In the simple theory of the method the induction throw is supposed
to be taken when the needle is at rest and when the resistance balance
is perfect. Instead of waiting to reduce the free swing to insignificance,
it was much better to observe its actual amount and to allow for it.
The first step is, therefore, to read two successive elongations, and this
should be taken as soon as the needle is fairly quiet. The battery
current is then reversed, to a signal, as the needle passes the position
of equilibrium, and a note made whether the free swing is in the same
or in the opposite direction to the induction throw. We have also to
bear in mind that the zero about which the vibrations take place is
different after reversal from what it was before reversal, in consequence
of imperfection in the resistance balance. At the moment after reversal
we are therefore to regard the needle as displaced from its position of
equilibrium, and as affected with a velocity due jointly to the induction
impulse and to the free swing previously existing. If the arc of vibration
(i,e., the difference of successive elongations) be o, before reversal, the arc-
due to induction be a, and if 6 be the difference of zeros, the subsequent
vibration is expressed by
| (a t) sin nt + 6 cos nt,
in which t is measured from the moment of reversal, and the damping
is for the present neglected. The actually observed arc of vibration is
therefore
or with sufficient approximation
a + a, + 26 i /a,
so that
a = observed arc + a, 26 s a.
In most cases the correction depending upon 6 was very small, if not
insensible. The "observed arc" was the difference of the readings at the
two elongations immediately following the reversal. As a check against
mistakes the two next elongations also were observed, but were not used
further in the reductions. The needle was then brought nearly to rest,
and two elongations observed in the now reversed position of the key,
60 ON THE VALUE OF THE BKITISH ASSOCIATION [80
giving with -the former ones the data for determining the imperfection
of the resistance balance. As the needle next passed the position of
equilibrium, it was acted upon by the induction impulse (in the oppo-
site direction to that observed before), and the four following elongations
were read.
These observations of the throw were followed as quickly as possible
by observations of the effect of substituting 853 for 753 units in the
auxiliary arc. As soon as the vibrations could be reduced to a moderate
amplitude, readings of three or four consecutive elongations were taken.
The galvanometer contact was then broken, and the battery key reversed.
When the needle had swung over to the other side, the galvanometer
contact was renewed, and four elongations were observed. The difference
between the two positions of equilibrium represented the disturbance of
the resistance balance.
The whole of this disturbance, however, was not due to the additional
100 introduced, but required correction for the corresponding effect ob-
served even with 753 units in the auxiliary arc. For this purpose it was
only necessary to add or subtract the difference between the equilibrium
positions of the needle with the key in the two positions, as deduced
from the observations immediately preceding the induction throws ; and
in order to eliminate the influence of the progressive change, the mean
of these differences as found before and after the insertion of the extra
100 units was employed. This result was compared with the mean of the
four induction throws contiguous to it, two preceding and two following,
and in this way a ratio obtained which was independent of the gradual
but unavoidable changes in the battery current and in the copper resist-
ance. After about half the readings had been taken the galvanometer
connexions were reversed.
A specimen set of observations will now be given.
3 h 36 m [753] L 264'4
Induction 246'6
3 h 38 m R 262-5
3 h 38 m Induction 245'9
3 h 40 m [853] R 182-3
3 h 41 m L 344-7
3 h 44 m [753] L 264-4
3 h 44 m Induction 245'7
3 h 45 m R 2631
Induction 245*6
80] UXTT OF RESISTANCE Df ABSOLUTE MEASURE. 61
At 3 k 36" with 753 units in the auxiliary arc and with, battery key
to the left, the position of equilibrium, as deduced from two elongations,
was 264'4 on the galvanometer scale. The arc of vibration due to induc-
tion consequent on shifting the key from left to right, corrected for the
free swing, but uncorrected for damping, was 246-6. In like manner with
key to the right, the equilibrium position at 3 k 38" was 262*5 and the
arc due to induction was 245 "9. The difference 1*9 between 264'4 and
262*5 represented the defect of balance. In the second set of induction
throws the corresponding difference is 1-3. showing that the changes of
temperature in progress were (at this stage) improving the balance of
resistances. The difference between the readings R and L with 853 units
is 162*4, the reading L being the higher. Since the reading L is also higher
with 753 units, we have to subtract from 162-4 the mean of 1*9 and 1-3.
ijg., 1'6. The corrected value is thus 160*8. With this we have to compare
the mean of 246*6, 245-9, 245'7, 245-6, iLe., 245-9, and we thus obtain as
the ratio of the two effects
245-9/160-8, or 1-529.
The numbers obtained in this way were 1-535, 1-532, 1'529, 1'528, mean
1-5310; and with galvanometer reversed 1-534, 1-529, 1-530, 1'530, 1-532.
mean 1*5310. The reversal of the galvanometer appears to have made
no difference, and we have as the mean of all 1*5310. The comparison
of the partial results shows that during the hour and a half over which
the readings extended the battery current fell slowly about one part in 1 20,
and that the resistance of the copper gradually increased, until the balance-
was perfect, and afterwards became too great, the whole change being about
one part in 6000, which would correspond to about oiie-twentieth of a
degree centigrade,
A small correction is required in identifying the above determined ratio
with 2 sin fa tan 0. If A be the induction arc and B be difference of equi-
librium positions with 853 units when the commutator is reversed.
tan 2a = AJD, tan = \BD,
where D distance of mirror from scale = 218 centims.
From these we get
2 sin fr _ A 1 - &A* 4D*
tan ~ B 1 -
or in the present case with ^1 = 24*5, 5=16*0,
and
A 5=1*5310.
62 ON THE VALUE OF THE BRITISH ASSOCIATION [80
So far we have omitted to consider the effect of damping, which must
necessarily cause the observed value of A to be too small. If X be the
logarithmic decrement, the correcting factor is (1 + X). The throw from
zero to the first elongation is diminished by the fraction ^\, and the dis-
tance from zero to the second elongation is too small by the fraction |X.
Observations made in the usual manner after the other readings were
concluded gave with considerable accuracy
X=-0142.
The time of vibration was taken simultaneously. It appeared that
T =11-693 seconds.
A sufficient approximation to the ratio of currents x' : x can be obtained
by neglecting in both cases the current through the galvanometer, whose
resistance (80 units) was considerable in comparison with the other resist-
ances. On account of the small resistance of the battery, the difference of
potentials at the battery electrodes may be regarded as given. On these
suppositions we get at once
at 10 + 23-25869 , . . Tnnoni
*T 10 + 23-34322' whence ^ (^) = 1^891.
A more elaborate calculation, in which the finite conductivity of the
galvanometer was taken into account, gave a practically identical result,
log (x'jx) = 1-99886.
We may now enter the numbers in the formula
L = 8Q - . ~ . 4 (-99925) (1 + X),
x 'ZTT J)
in which we must remember that 8Q is to be expressed in absolute measure.
Now the value given before, viz. SQ = -08453, is expressed in B.A. units.
What this would be in absolute units involves the entire question to whose
solution this paper is directed. We will suppose that
1 B.A. unit = -987 ohm,
SQ log -08453 x 10 9 =7-92701
Correction to absolute units log '987 = 1 "99432
A:B log 1-5310 = _'18498
Correction for finite arcs log '99925 =1-99967
Correction for damping log 1'0142 = "00612
Time of vibration log 11 "693 = T06793
Ratio of currents log (xjx) = 1-99886
9-17889
log 2?r = -79818
\o S L =8-38071
80] UNIT OF RESISTANCE IN ABSOLUTE MEASURE. 63
whence
L = 24028 x lO'centims.
The value by a priori calculation is
L = 2-400 x 10 9 centims.
about one part in a thousand lower*.
Correction for lend.
If the axis of rotation deviate from the vertical in the plane of the
meridian a corresponding correction is required. If / be the angle of dip,
and ft the deviation of the axis from the vertical towards the north, the
electromotive forces are increased in the ratio (1 + ft tan /) : 1, in which
proportion we must suppose GK increased. (See pp. 106, 124 [pp. 3, 20] of
former paper.) The angle of dip at Greenwich for 1881 is about 67" 30', so
that
tan 7= 2 414.
The correction for an error in level is thus of the first order, and is
magnified by the largeness of the angle of dip in these latitudes. If the
experiments were made at the magnetic equator, we should not only reduce
the correction for level to the second order, but also obtain the advantage of
a nearly doubled horizontal force.
Observations on the level were made by Dr Schuster on June 1, by
myself on August 30, and by Mrs Sidgwick on October 13, and on November
11 and 23. The August observations gave ft = '26': the October observations
gave ft = $&: and the November observations gave ft = '25'. The position of
the axis is necessarily to a slight extent indefinite, and the differences are
probably accidental. The same level was used throughout, and the value of
its graduations was tested. We may take
0= + -27' = + -000079 circular measure
and
1 + ft tan / = 1-00019.
* A farther small correction is called for by the fact that at actual temperature of the room
(about 14=) the resistances given by the boxes were not exactly multiples of the B.A. unit. The
difference in the case of the principal box. which is marked as correct at 14 2. may be n*glnrtod.
but the resistances taken from the auxiliary box (marked 13 : -3 mnst have been smaller than
their nominal value, to the extent of a little over one part in a thousand. By the same fraction
SQ, and consequently L, must be greater than is supposed in the above calculation. The
corrected value of I. will be
L = 2-4052 x IV.
It is about ftro parts in a thousand greater than that found from the measured dimensions, and
is, in my opinion, quite as likely to be comet. -
64 ON THE VALUE OF THE BRITISH ASSOCIATION [80
Correction for torsion.
To determine T, about five complete turns in either direction were given
to the upper end of the fibre. The difference of reading for one turn was
found to be in June 2'58, and in August 2'45. If we take as the mean 2 '51,
we get
Value of GK corrected for level and torsion.
Calling the corrected value <ffir3Bt, we have
gJ5T(l+0tan/) 1-00019 CR
sm= I + T = roooo?5^'
so that
log CBrlt + 8-17686.
The corrections are in fact almost insensible.
Calculation of U.
In this we take for (fir1& the value just found. For L we take the mean
of the values found by a priori calculation and by direct experiment, i.e.,
L = 2-4026 x 10 8 .
Thus
For the values of tan <f> and tan we must anticipate a little. The
ratio is itself in some degree a function of the speed, but it will suffice to
take the values applicable to the highest speed, for which
tan <o : tan <j> = 7'81 : 439*41.
Thus
_^Lf =1-0181,
tan tan </>
and
log U= -84325.
Measurement of tan p.
This is the tangent of the angle through which a suspended magnetic
needle would be turned when the principal magnet is presented to it at a
distance <J(a z + b'*) to the east or west, the axis of the principal magnet lying
east and west. Actual measurements with the aid of the auxiliary magneto-
meter were made in April, June, and November ; and as a check upon the
80] UNIT OF RESISTANCE IN ABSOLUTE MEASURE. 65
constancy of the magnetic moment frequent observations were taken of the
time of vibration.
To explain the procedure it will be sufficient to take the data of the
November measurement. Two positions were chosen for the principal
magnet, nearly equidistant from the suspended magnet, to the east and
west. The length of the line joining the two positions was 695 millims., and
it passed horizontally about 36 millims. below the suspended magnet. In
each position the magnet was reversed backwards and forwards several
times and readings taken. When the principal magnet was to the east, the
mean difference of readings due to reversal was 13*55 divisions on the
millimetre scale. When the principal magnet was in the westerly position,
the corresponding difference of readings was 14'61. We are to take the mean
of these, i.e., 14-08, as the difference of readings due to reversal at a distance
of 347 5 millims. The half of this, or 7-04, corresponds to the simple
presentation or removal of the magnet. The distance from mirror to scale
was 2670 millims., so that the tangent of the angle of deflection was
7*04
9 ocTn- This result has to be adjusted to correspond with the distance
(n* + &'), in place of 347 5. Hence
In this calculation the error due to the principal magnet having been
necessarily placed at a different level from that of the suspended magnet is
ignored. As a matter of fact a relatively considerable correction is required.
If be the altitude of one magnet as seen from the other, the observed effect
is too small in the ratio (1 30 s ) : 1. The above written value of tan p
requires to be increased about 3 per cent. : so that we take
tan p = -00420.
Measurement of D.
For the first and second series of spinnings the distance from mirror to
scale was measured exactly as described by Dr Schuster (p. [22] of former
paper). The value adopted for the second series, after correction for the
thickness of the glass window in the magnet box, was
D = 2669-0 millims.
The same method of measurement was applied at the beginning of the
third set, and a watch was kept by means of the measuring rods already
spoken of [p. 46]. Slight movements were in fact observed, principally
of the nature of a recovery of the telescope stand from the rather violent
5
66 ON THE VALUE OF THE BRITISH ASSOCIATION [80
treatment to which it had been subjected as a test. Minute corrections
are accordingly introduced into the tabular statement [p. 72], so as to make
the results of different days comparable. At the close of the spinnings the
direct measurement was repeated, when there appeared a slight discrepancy
between the results obtained by Mrs Sidgwick and myself. It is in fact
rather a difficult matter to say exactly when the pointer has advanced to the
equilibrium position of the centre of a suspended mirror, which cannot be
prevented from swinging. Although the amount in question was not im-
portant, I thought it might be more satisfactory to check the result by
another method, and therefore arranged a travelling microscope focussed
alternately upon the centre of the mirror and upon a scratch on the window
of the magnet box, by which the distance between these two points was
determined. The remaining distance between the scratch and the scale was
easily measured with the rod. The result tended to confirm the smaller
value previously found. The value adopted for the spinnings of the third
series before November 5 is
2668-8 millims.
and for November 5 and subsequent nights
2669-4 millims.
From these numbers we have to subtract I'l millim., as a correction for
the thickness of the glass window ; so that
D before November 5 = 2667'7 millims.
D November 5 and after = 2668'3 millims.
These distances are expressed in terms of the divisions of the scale, whose
exact agreement with millimetres is of no consequence.
Reduction of Results.
In order to give a clear idea of the results and of the manner in which
they have been reduced, it will be advisable to quote from the note-book the
details of one set of spinnings. I have chosen at random one of the third
series made on October 31, 1881, with a speed of "45 teeth."
The first column gives the number of the spinning, the first six being
made with wire circuit open, and the last six with the wire circuit closed.
In spinnings I., III., V., VIII., X., XII., the rotation was in the direction
reckoned negative, and in the remaining ones positive. The second column
gives the time, the third the reading of the auxiliary magnetometer, the fourth
the reading of the principal magnetometer, the fifth the result of correcting
the latter by the former, the sixth and seventh the approximately constant
sums and differences of consecutive pairs of numbers in the fifth column, and
80]
UNIT OF RESISTANCE IX ABSOLUTE MEASURE.
67
|c
5-1
o
2
1 .
3 I
m
0*
rt
1
|1
jiii
>
s s
lo
S2*l
Sis!
s
si
II
11H
9
+
If
H
g 0.95 u3 K
li
11
2
1
2 90-81 mK
a
'S *
5 i
2 2
-1
11
e 1
gOOI ^^H o
I s
lltll
l fe =|l
P
ea ?~
5J gc - a^H g
+ 1
14
.s |
a
|j
OS O
?" . H 3
5 I C4
il
11
1
sg^gpg r ? ? ? ?
OOOOO ' OOiO^^ [
1 |
2 1
I
sssss sss
SSSaTS : Sg
|:
o
2 x
11 14*
^52^ ^S^??S.
J 1
> -S
32-^5*
51 ii S
liliil 'llllli
l|
1 14; i
S25Sc? sg^32Sf.
Is
1=1 ill
111111 -IS1111
S
II
Hi
cp^woa)?! .^-p-*"??^ ;
SSSSS -SSSSSao
Ii
s I
-3*
ii
S2 .8385858 3. 88.
cd ad ad ad ad ad ad CD oo eo oo ao
2
M
- ri 3 > > 5: p 3 H ^ S g
s i
* 3
nado ?iuoo P 980 ! 3 ^^O
52
68 ON THE VALUE OF THE BRITISH ASSOCIATION [80
the eighth gives the mean deflection from zero, i.e., 5'29 for the open contacts,
and 302*56 for the closed contacts.
The ninth column shows the results of the resistance comparisons between
the platinum-silver standard and the revolving copper coil before and after
the closed contact spinnings. The first number (+ 212) means that at
8 h 36 m the resistance of the standard exceeded that of the copper by 212
bridge-wire divisions, each of which represents ^^Q of an ohm. It will be
seen that during the spinnings the resistance of the copper increased, which
accounts for the gradual fall observable in the seventh column. The mean
of the comparisons before and after spinning is taken to correspond with the
mean deflection 302'56. The three following columns show respectively the
temperatures of the water in which the standard was immersed, of the air in
the neighbourhood of the copper coil, and of the standard tuning-fork, while
the thirteenth column gives the number of beats per minute between the
electrically maintained and the standard fork.
For the sake of more convenient comparison of the results obtained at the
same speed on different nights, small corrections are calculated to reduce the
actually observed deflections in the eighth column to what they would have
been in a standard condition of the resistance and of the speed. Under each
of these heads we have two corrections to consider. In the first place the
copper circuit differed in resistance from the standard coil by the outstanding
( 52) divisions of the bridge wire. The resistance of the whole being about
24 ohms, each division of the wire corresponds to one part in 480,000, so that
in the present case the correction is additive and equal to 52 parts in
480,000, i.e., is equal to + '03 division of the scale. This is given in the
fourteenth column. Secondly, the resistance of the standard itself depends
upon a variable temperature. The mean temperature of the standard in
this series was about 13, to which all the observations are reduced. In the
present case the temperature was below the normal, so that the resistances
were too small and the deflections too large. Accordingly the correction is
negative. To estimate its amount the change of resistance with temperature
is taken at three parts in 10,000 per degree ; so that in the present case we
are to subtract 2'8 parts in 3000 of the whole deflection, i.e., '27, as entered
in the fifteenth column. With use of these corrections we obtain the deflec-
tion as it would have been observed had the resistance of the revolving
circuit (together with the long copper bars by which it was connected with
the bridge) been on every occasion exactly that of the standard at 13.
In like manner two other very small corrections have to be introduced to
make the results correspond exactly to a normal speed of rotation. The
standard number of beats is taken at 59, and the standard temperature of
the fork at 17. In the specimen set the number of beats is 2 per minute
too small, which means that the octave of the electrically maintained fork
80] UNIT OF RESISTANCE IN ABSOLUTE MEASURE. 69
made (relatively to the other fork) 2^ complete vibrations per minute too
many. The whole number of vibrations per minute being 60 x 127, the
speed was too great by 2 parts in 60 x 127, by which fraction the observed
deflection must be reduced. The correction is thus 10. But besides this
the standard fork at 13'05 vibrated faster than its normal rate at 17, by
about one part in 10,000 for each degree of difference. The correction for
this is --12.
In addition to the corrections already mentioned the observations of
November 5 and after were subjected to another small correction for the
observed change in D.
The accompanying Table (II.) exhibits the results of the second series in
a manner which after what has been said will not require much explanation.
Column VIII. gives in each case what the deflection would have been if the
revolving circuit and the copper connecting bars had exactly balanced the
platinum-silver standard at 16, the electric fork vibrating at such a speed as
to give 59 beats per minute with the standard fork at 17, and thus allows
us to test th'e agreement or otherwise of the results obtained on various
occasions at the same speed. From this point onwards the means only need
be considered ; but as there is reason (as already explained) to distrust the
observations of August 29, I have added a second mean from which the
distrusted elements are excluded. The deflection (d) thus arrived at is equal
to D tan 20, whereas what we require is 2D tan <f>. The connexion between
the two quantities is obtained in a moment from the formula
2 tan ^ = tan 20 (1 - tan 2 0),
by successive approximation. Thus
2 tan = tan 20 (1 - tan 2 20 + | tan 4 20],
or
W tan = d - i $\I> + 1 &!&.
Column X. gives the value of 2D tan 0, XI. that of W (tan - tan )
in the notation of p. [54], and XIII. that of log (tan tan ).
For the further calculation we require the value of to. If / be the
frequency of vibration of the electrically maintained fork, F that of the
standard at 17, N the number of teeth,
and when the number of beats is 59 per minute,
For F at 17 we take 128-130 (see p. [33] of former paper), so that
/= 63-574.
70
ON THE VALUE OF THE BRITISH ASSOCIATION
[80
it
I7i|
111!
Hm
0-6996 = (7 0-699S=OT
SOIT|I10I
0-699S=(T 0-6995= a
inn
V
T
0000
CO CO
cp
oO
CO
(MO
-* o
li
QO <M tO CO -^ ^< O O
os o o 5 o oo o o
rt ' II rH rH CO ' II CO CO
00OCO
y'? 5 ? 5
T-HT-HT t
000
COCOCO
oOO
ooo
999
+ I +
O-HO
000
CO <N rH
999
+ + I
*s
+ I I
o?
I I +
lMIM
s coe
80] UNIT OF RESISTANCE IN ABSOLUTE MEASURE.
Thus
log (2-7T ./. lEt) = 10-77832 = log (10 10 x 6'0023),
and
in which
+ '00422 sec <) 2 -
log ^=-843 25.
- tan
71
TABLK III. Second Series.
Number of teeth
60
45
35
30
R by preceding formula in ohms .
Resistance of standard at 16 . .
23-639
23-642
23-655
23-658
23-660
23-651
23-663
23-654
23-670
23-659
23-673
23-662
Resistance of standard at 13 . .
23-621
23-637
23-642
23-633
23-652
23-641
The immediate result of the formula is the resistance in absolute measure
of the revolving circuit, on the supposition that with the connecting bars it
exactly balances the standard at 16. The resistance of the standard itself is
therefore given by addition of the resistance of the bars, i.e.. "003 ohm. In
the last line the results are reduced to the temperature of 13 for comparison
with the third series.
72
ON THE VALUE OF THE BRITISH ASSOCIATION
[80
I
rife
111!
iili is
Ilij
I
Correction
to middle
of bridge
wire
t-ocsi
rH i-H 1 1 i 1
COCOCOCO
"?'<?>
?> *
I + ' +
Til O5 O>
Cp CO (N
+++
O5 IN 00
f ?
CO CO CO
$z .g
CO IM <M
IM i-l t- O5St-O
<NCO a IH p
..> *:>>>
a a
80] UNIT OF RESISTANCE IN ABSOLUTE MEASURE. 73
TABLE V. Third Series.
Number of teeth
60 45 35 30
R by formula 23-616 23-618 23-627 23-635
Resistance of standard at 13 . 23-619 23-621 23-630 23-638
If we compare the results of the second and third series at the same
speed, we find the agreement satisfactory (with a partial exception at the
speed corresponding to 45 teeth), especially if we take the means from which
the observations of August 29 in the second series are excluded. Adding
together all the results of each series we should obtain from the second
series 23'638, or with exclusion of August 29, 23'633, and from the third
series 23'627, between which the extreme difference is less than one part in
2000. When, however, we compare the values obtained from observations at
different speeds, we see from both series, but more especially from the third,
evident signs of a tendency to rise with the speed, as if the self-induction of
the revolving circuit had been underestimated. In view of the remarkable
concordance of the results obtained at the same speed on different nights, it
is impossible to attribute these discrepancies to errors of observation, and it
is important to consider what cause of systematic disturbance can have
remained unallowed for. The first question which presents itself is whether
it is possible to admit an error in the adopted value of L sufficient to explain
the progression. The proportional correction for self-induction is approxi-
mately U tan 3 <, or for the speed of 30 teeth '0457. For the speed of
60 teeth the correction will be only one-fourth of this. To bring the results
for the two speeds into agreement it would be necessary to increase the value
of U by nearly 3 per cent., which would correspond to an increase of about
one per cent, in L. It is difficult to believe that the value of L adopted for
the wire circuit can be in error to this extent.
Another direction in which an explanation might be looked for would be
the influence of air disturbance, or from tremor. The accompanying table,
however, shows such an extraordinary agreement of the open contact deflec-
tions, both among themselves and with numbers proportional to the speeds
of rotation, as to prevent us from supposing that this cause of disturbance
can have operated.
On the whole, it would appear to be the most probable explanation that
there were currents in the ring flowing in circuits not conjugate to the wire
circuit, and therefore influencing the induction phenomena. But whatever
view we may take on this matter, there is no reason to doubt that the true
74 ON THE VALUE OF THE BRITISH ASSOCIATION [80
TABLE VI. Deflections with wire circuit open.
Number of teeth
60
45
35
30
3-89
3-93
3-93
3-94
5-22
5-24
5-24
5-23
5-25
6-74
6-77
C-76
6-76
6-76
7-85
7-82
7-92
7-88
After the wire had been removed, December 7
Numbers proportional to speed
value will be obtained by introducing such a correction proportional to the
square of the speed as will harmonise the several results, a course equivalent
to determining the coefficient of self-induction from the spinnings themselves.
In this way the numbers corresponding to any two speeds may be made
arbitrarily to agree, but the numbers for the two remaining speeds will
afford a test of the admissibility of this procedure. The only hypothesis
upon which the simple mean of the numbers already obtained for the various
speeds should be taken as final would appear to be one that would attribute
to the discrepancies an accidental character, and seems quite out of the
question.
The simplest way to carry out the correction will be to determine the
amount of the coefficient from the two extreme speeds. The squares of the
speeds are as
l:tf :W = 4J
so that the difference of the numbers for the two extreme speeds, 23'638
23'619, i.e., '019, is three times the quantity by which the lowest is to be
reduced. We are accordingly to subtract respectively
0063, J x -0063, J % 4 - x >0063 > 4 x ' 0063 >
with the following results.
TABLE VII. Third series.
Number of teeth
Mean
60
45
35
30
Resistance of standard at 13, uncorrected
Correction proportional to square of speed
Resistance of standard at 13, corrected .
23-619
006
23-613
23-621
Oil
23-610
23-630
018
23-612
23-638
025
23-613
23-627
23-612
80] UNIT OF RESISTANCE IN ABSOLUTE MEASURE. 75
It will be seen that the agreement is practically perfect, the coefficient
given by the extreme speeds suiting also the requirements of the inter-
mediate speeds. The maximum difference corresponds to about y^ths of a
millimetre only in the deflections of the principal magnetometer. The
number 23*612 x 10 9 is therefore to be regarded as the resistance in absolute
C.G.S. measure of the platinum-silver standard at 13. If, however, the
correction be rejected, the result will be different by decidedly less than one
part in a thousand.
Although the experiments of the second series will not bear comparison
with those of the third, it may be well to mention that they lead to sub-
stantially the same conclusion. The simple mean (taken with exclusion of
August 29) of all the values is 23*633, and after introduction of the correc-
tion proportional to the square of the speed, 23 618.
The results 9f the first series of spinnings are given in Table VIII.
They have been reduced by Dr Schuster, so as to show the value of the
platiimm-silver standard in absolute measure from the observations of each
night at each speed. The mean radius of the coil was taken from the first
measurements, and a somewhat higher value of U was employed than the
subsequent calculation of the ring currents seemed to justify.
TABLE VIII. First series.
Teeth. Resistance in absolute measure of standard at 13~.
80 23*651, 23-632, 23*628 ... Mean 23637
60 23*646, 23*629, 23*601 ... Mean 23*625
45 23*678, 23*691, 23*686 ... Mean 23*685
35 23*608, 23615., 23*632, 23*665 Mean 23*630
30 23*644, 23*639, 23*628 ... Mean 23*637
23-643
Comparison with the standard B.A. units.
Four distinct sets of comparisons between the platinum-silver standard
and the ultimate B.A. units have been effected in the course of these investi-
gations, and two distinct methods have been followed. In the first method
two coils of about five units, called for brevity [5]'s, were compared separately
with five standard units combined in series with mercury cups. Secondly,
the two [5]'s in series were compared with a [10]. Thirdly, the [10], the two
[5]'s, and four singles were combined in series and compared with the
platinum-silver standard [24]. The differences in every case were expressed
in divisions of the wire of Fleming's bridge, whose value in terms of the
76 ON THE VALUE OF THE BRITISH ASSOCIATION [80
B.A. unit is known. This method is simple enough in principle, but the
arrangement of so many mercury connexions is troublesome, and the calcula-
tion of the innumerable temperature corrections necessary is tedious. The
labour would have been greater still had we not been able to avail ourselves
of the previous work of Professor Fleming, who had carefully compared the
various standard units, and had drawn up a chart on which is exhibited the
comparative resistances of the coils over a considerable range of temperature.
The mean B.A. unit, in terms of which our results are expressed, was denned
by him, but the difference between the single standards is scarcely of im-
portance for our purpose. In calculating the temperature corrections for
the two [5]'s, the [10], and the [24], which were all of platinum-silver wire,
the coefficient '0003 per degree has been used. The temperatures were
those of the water in which the coils were immersed. They never differed
much from the temperature of the room, and were referred to a Kew
standard. The results of three comparisons, executed by Mrs Sidgwick, are
as follows :
Resistance in mean B.A. units of platinum-silver standard at 13.
July, 1881 .................................... 23-9326
September, 1881 .............................. 23'9341
November, 1881 .............................. 23'9348
In February, 1882, a fourth determination was executed by myself, in
which a different method was employed. Five coils approximately equal to
each other and to five units were arranged in a closed case upon a tube of
brass. The ten copper terminals emerged below from the ebonite bottom of
the case, and rested in mercury cups upon an ebonite base-board, which was
so arranged that by a single motion the terminals could be transferred from
one set of cups which combined the coils in series to another set which
combined them in multiple arc*. In this way resistances are obtained in the
proportion of 25 : 1, independently of any exact equality between the single
coils ; for it is obvious that if the resistance in series is given, the resistance
in multiple arc is a maximum in the case of equality, and therefore varies
little, even if the equality be not exact f. By the aid of this apparatus the
[24] was compared with a standard unit, without the assistance of other coils.
* I believe that Professor Rowland has used a contrivance of this sort.
t [1899. If there be n coils of mean resistance r and of actual resistances equal to r(l + a),
r(l + /3), &c., so that
the resistance of the coils combined in series is nr. Thus
Resistance in series _ n n _ , ( a 2 +/3 2 +...|
Resis~tanceln parafieT = ITa + 1+/3 + ' ' ' = " p * n j
approximately.]
80] UNIT OF RESISTANCE IN ABSOLUTE MEASURE. 77
In the first place [24] + [1] was compared with the five coils in series, and in
the second place the [1] was compared with the five coils in multiple arc.
The only precaution necessary is to effect the second comparison so quickly
after the first that the five coils have no time to change their temperature.
Two determinations by this method on different days gave as the resistance
of [24] at 13
23-9350, 23-9358 mean, 23*9354.
It would seem not impossible that the resistance of [24] has gradually
increased, but the changes are unimportant. We will take as the resistance
with which the absolute measurement is to be combined, that found in
November, 23*9348; so that
23-9348 B.A. units = 23'612 x 10- C.G.S. = 23-G12 earth ^-^ .
second
Hence, as the result of the investigation, we conclude that
, earth quadrant
1 B.A. unit= -986ol -
second
[1899. Further experimental work by Mrs Sidgwick and myself upon
this subject is recorded in Phil. Trans. 174, p. 295, 1882, Art. 94 below.]
81.
ON THE SPECIFIC RESISTANCE OF MERCURY.
By Lord RAYLEIGH and Mrs H. SIDGWICK.
[Phil. Trans. Vol. 174, pp. 173185, 1882.]
OUR experiments on the determination of the British Association unit of
electrical resistance in absolute measure are detailed in two memoirs com-
municated to the Society*. The conclusion to which they led us is that
non-> earth quadrant
1 B.A. umt= -986o
second
but this result differs considerably from that obtained by some other experi-
menters, the original Committee included. Although in the present state of
the question it is not desirable that the B.A. unit should fall into disuse,
there can be no question as to the importance of connecting it with the
mercury unit introduced now more than twenty years ago by Siemens. It
will then be possible, as recommended by the Paris Conference, to express
our absolute measurements in terms of mercury, by stating what length of a
column of mercury at of 1 square millimetre section has a resistance of
1 ohm. Accordingly the experiments about to be described relate to the
expression in terms of the B.A. unit of the resistances of known columns
of mercury at 0.
This investigation was the more necessary, as the principal authorities on
the subject, Dr Werner Siemens and Dr Matthiessen, had obtained results
differing by as much as *8 per cent.
The earlier determinations of Siemens were vitiated by the assumption
of an erroneous value (13'557) for the specific gravity of mercury, a constant
* Proceedings, April 12, 1881; Phil. Trans. 1882, Part II. [Arts. 79, 80].
81] ON THE SPECIFIC RESISTANCE OF MERCURY. 79
which it is necessary to know in order to infer the mean section of a tube
from the weight of contained mercury. The error, pointed out by Matthiessen,
was afterwards* admitted by Siemens, who gives as the corrected expression
of the relation between the two units,
1 mercury unit = '9536 B.A. unit.
On the other hand, the independent measurements of the resistance of
mercury by Matthiessen and Hockinf gave
1 mercury unit = *9619 B.A. unit,
the mercury unit being defined as the resistance at of a column of mercury
1 metre long and 1 square millimetre in section.
Our own experiments lead us to a value not differing much from that of
Siemens. We find
1 mercury unit = -95418 BJL unit
If we assume that the B-A. unit is "98651 ohm (in accordance with our
determination), we find
1 mercury unit = -94130 ohm,
the ohm being 10 9 C.G.s. The same result may be expressed in another way
by saying that the ohm is the resistance of a column of mercury at : ,
1 square millimetre in section, and 1062'4 millims. in length.
Through the kindness of Dr C. W. Siemens we have had an opportunity
of comparing with the B~A. units a standard mercury unit (No. 2513) issued
ty Messrs Siemens and Halske. At the proper temperature (16"7) we find
that its resistance is
-95365 R.A. unit,
agreeing very closely with previous comparisons of Siemens' mercury
measurements with the B.A. unit
The determination of the specific resistance of mercury is simple enough
in principle, though the execution is somewhat tedious, and the calculation
of the results is complicated in practice by the necessity of introducing
various temperature corrections. In a first sketch of the method it will be
convenient to omit these corrections, which is tantamount to supposing that
all the measurements are made at zero. If L be the length and s the section
of the column of mercury, R its resistance, r the specific resistance of the
metal,
fjj -. 8
R = , or r = /fcy.
s lj
* PJW. Mag. voL KM. 1866.
t Reprint of British Association Reports, p. 114.
80 ON THE SPECIFIC RESISTANCE OF MERCURY. [81
The length L can be measured directly, but s can only be found with the
necessary accuracy from the contents. Thus if p be the specific gravity of
mercury, and W the weight of the whole column in grammes, pLs = W,
whence s = W/pL, and
RW
Apart from the temperature corrections already referred to, the simplicity
of the formula is disturbed by the inevitable departure from the truly
cylindrical form of the glass tubes used to contain the mercury. It is true
indeed that to a first order of approximation the formula stands unaltered,
as we may see if we understand by s the mean section of the tube. The
volume is still truly expressed by sL, and the resistance is approximately
expressed by rL/s. If, however, the squares of the variations of section
cannot be neglected, the actual resistance is greater than the formula would
lead us to suppose, as is evident if we imagine the section to become at one
place very small.
In general we must regard s as a function of the position (#) along the tube
at which it is taken. For the purposes of the present paper we may assume
with sufficient approximation (see Lord Rayleigh's Theory of Sound, 308)
The necessary data with respect to s are obtained by a calibration of the
tube. " If a small quantity of mercury is introduced into the tube and
occupies a length X of the tube, the middle point of which is distant x from
one end of the tube, then the area s of the section near this point will
be s= Gj\, where C is some constant. The weight of mercury which fills the
whole tube is
where n is the number of points at equal distances along the tube, where X
has been measured, and p is the mass of unit of volume.
" The resistance of the whole tube is
R=[ r =-Z(\)-
" Hence
and
WR
P L* S(X).2(X- 1 )
gives the specific resistance of unit of volume" (Maxwell's Electricity, 362).
81] ON THE SPECIFIC RESISTANCE OF MERCURY. 81
In the sequel
71 s
is denoted by /* ; it is a numerical quantity a little greater than unity.
Another correction is required in our method of working to take account
of the resistance offered by that part of the mercury in the terminal cups,
which is situated just beyond the ends of the tube. The question is identical
with that of the correction necessary in calculations of pitch for the open
ends of organ pipes (see Theory of Sound, 307, and Appendix A), and
it scarcely admits of absolutely definite solution. We cannot, however, be
far wrong in adding to the actual length of the tube "82 of its diameter,
which corresponds to the supposition that the diameter of the mercury
column suddenly becomes infinite. Since, in our experiments, the whole
correction only amounts to about a thousandth part, even a ten per cent.
error iu our estimate would scarcely be material.
Let ? = resistance of a column of mercury 1 metre long and 1 square
millimetre in section, at 0, expressed in B.A. units.
R = resistance of the tube full of mercury at in B.A. units.
L = length of the tube at t' in centimetres as measured with brass
rod.
I = length of a thread of mercury of nearly the length of the tube at
t as measured with brass rod.
W= weight of the same thread in grammes.
p = coefficient correcting for conicality of tube.
SL = correction to L on account of the connecting rods not being close
up to the ends of the tube = '82 x diameter of tube.
p = specific gravity of mercury at = 13'595.
7 = cubic expansion of mercury per degree = '0001795.
q = glass = '000025.
y o
b = linear expansion of brass = "000018.
t = temperature of brass measuring rod to which the lengths are
corrected = 17'2.
Then the volume of the thread at = W/p.
r=
P
Mean section of the tube at t = ^M +b(t-to)\ '
Mean section at O c = ,j { i + & (f _ <,)} {l + fofj '
82
ON THE SPECIFIC RESISTANCE OF MERCURY.
Length of the tube at - (*>
[81
R = 10~ 4 .r./jt.
(L + BL) {I + b(t'-
- t )}
The value of p is that used by the Committee of the British Association
in reducing Dr Matthiessen's experiments (see reprint of Reports on Electrical
Standards, p. 114), and stated to be the mean of the values given by Kopp,
Regnault, and Balfour Stewart. The values of g, 7, and 6 are taken from
Everett's Units and Physical Constants 7 being Regnault's value for the
expansion of mercury. The measurements of the other quantities, which
depend on the particular tube used, are given in the following table, together
with the resulting value of r. The description of the means employed to
obtain these data follows.
Number
of
observa-
tion
Date of
observation,
1882
Number
of tube
R
Tempera-
ture of
coilF
Tempera-
ture of
second coil
L
in centi-
metres
ft
1
Feb. 23 & 24
I.
79912
J12-2
|11 -5
12-1 1
11-oi
87-771
1-00314
2
25 . . ! I.
79912
12-7
12-5
,,
3 21 . .
I.
M
,,
,,
4 March 18 .
I.
79920
13-7 13-75 : ,,
,,
5
Weighed )
Feb. 14 j
I.
79912
:;,
-
"
6
Feb. 24 . .
II.
99088
12-0
96-400
1-00007
7
21 to 23
11.
99081
13-2
!
,,
8
March 7 .
II.
99081
11-5
,,
,,
9
8 .
II.
99079
12-2
(|
n
10
.. 30 .
II.
99085
13-25
11
,, 6 .
III.
99711
11-2
123-566
1-00046
12
10 . III.
99725
12-9
M
13
13 . III.
99720
12-7
H
14
,, 14 .
III.
99725
13-4
,,
,.
15
22 .
IV.
50783
13-0
12-9
194-137
1-000838
16
24 . IV.
50774
12-7
12-7
81]
ON THE SPECIFIC RESISTANCE OF MERCURY.
Number
of
observa-
tion .
1
in centi-
metres
W
t
t'
aB5x ' L 'rt(fif at)
L
r
Mean
values of
r from
each tube
1
-7--.M4
12-442
16-5
16-5
00103 + -00002
95386
2
87-310
12-4545
17-2
17-2
ooooo
95412
3
87-035
12-4185
18-4
,,
-00002
95424
95416
4
87-558
12-486
20-6
,,
-00006
95436
5
87-771
12-523
16-5
16-5
+ -00002
95421
6
96-054
12-096
16-7
16-4
000883 + -00002
95389
,
7
95-452
12-0245
16-4
"
+ -00003 -95414
8
9-5-831
12-074
17-1
+ -00002 -95437
95419
9
9B-151 12-113
18-0
18-0 - -00003
-95436
10
--
11
122-218
13-620
16-2
18-7
000778 - -00001
95424
1-2
123-288 19-780
18-5
-00005
95418
13
123-221 19-7665
18-4
' - -00005 -95399
j- -95416
14 123-058 19-745
18-3 .. - -00005 -95425
1
15 193-410 95859
14-5 14-5
000869 ! + -00009 -95440 ,
16 192-576 95-402
16-8
+ -00005 -95415 J '
Mean of all the above values of r in B.A. nnits -95418.
The mercury used for all the measurements except 10 and 14 was
distilled in -vacua with an apparatus fitted up by Mr Shaw. In order to see
whether a different result might not be obtained with other mercury, some
was procured from the chemical laboratory for measurements 10 and 14.
For the latter a portion of this mercury was treated with nitric acid and
distilled at atmospheric pressure. For measurement 10 it was treated with
nitric acid, but not distilled. An accident occurred in carrying out this
measurement, so that only the resistance of the column was ascertained ;
but this agrees so well with the resistances found with the same tube
for the other mercury, that there is no reason to suppose that any discrepancy
would have appeared in proceeding with the measurement further.
The glass tubes used were supplied by Cassella, and were selected for
uniformity of bore, so that the correction for conicality should be small.
They were slender and easily broken, which made the manipulation of them
difficult, and it was in fact owing to a breakage that the tube called No. I.
was used so short. The measurements taken with it, at first intended to be
preliminary, were, however, made with the same care as in the case of the
other tubes, and the difference of length and resistance adds some variety to
the data. Tubes II. and III. were cut so that their resistance should be as
nearly as possible one B.A. unit. The section of tubes L, II., and III., was
62
84 ON THE SPECIFIC RESISTANCE OF MERCURY. [81
approximately 1 square millimetre. Tube IV. was a much larger one,
introduced with a view of varying the data as much as could conveniently be
done. The diameter of its bore was about 2 millims., and its length was
nearly 2 metres. It was cut so as to give a resistance of about half a
B.A. unit.
The ends of the tubes were ground into a convex form with emery
powder on a lathe, in order that the length (L) of the bore might be
measured accurately. This measurement was effected by setting two
microscopes, which could be adjusted longitudinally to the exact position
required by micrometer-screws graduated to Tuir&u inch, so that their cross-
wires should coincide with the ends of the tube. Observations were made
in three or four different positions as the tube was turned round its axis, and
the mean taken. After removal of the tube, a brass measuring rod belonging
to the British Association was substituted for it, and the number of whole
divisions corresponding most nearly to the distance between the cross-wires
of the two microscopes was read off. The outstanding fraction of a millimetre
was then ascertained by screwing the microscope up to the whole division
and reading the difference on the screw-head. For the long tube the
measuring rod was too short, and a third microscope had to be used to
fix an intermediate point as a fresh departure for the scale. A thermometer
laid beside the tube during the measurement gave the temperature (If) at the
moment. The brass measuring rod was carefully examined, and its divisions
were found to agree among themselves.
The tubes were cleaned by passing through them in succession, by means
of a suction-pump, sulphuric acid, nitric acid, caustic potash, and distilled
water, followed by air dried with chloride of calcium. The process with
omission of the acids was in general repeated between each refilling with
mercury, but it was omitted in measurement 7, and there is no record of its
having been done in 1 , 3, and 6.
To calibrate the tubes a short thread of mercury was inserted and moved
to the various positions required, by blowing through a chloride of calcium
tube. In the case of tubes I. and II., the length, X, of the thread was
measured by adjusting microscopes to its two ends, with subsequent substi-
tution of an ivory scale divided in fiftieths of an inch. But this method was
troublesome ; and with tubes III. and IV. the scale was simply placed against
the thread and the length read off with a magnifying-glass, a procedure
which was found to give sufficiently accurate results, notwithstanding the
difficulty arising from parallax owing to the thickness of the glass. The
following table gives the different values of \ for each tube.
As a check upon the correction for conicality, two distinct values of /j,
were in some cases calculated from the alternate observations of \, and were
found to agree closely. It may not be superfluous to mention that in carrying
81]
ON THE SPECIFIC RESISTANCE OF MERCURY.
85
out the computations we must work to six or seven places, although the
observed values of X themselves may not be accurate beyond the third
place.
The lengths are in fiftieths of an inch
Tube I.
Tube II.
Tube UI.
Tube IV.
80-8
104-5
135-0
171-0
80-0
104-1
134-0
172-0
77-0
104-5
133-0 171-5
75-8
105-0
132-0
170-5
76-0
104-5
131-5
171-5
76-4
105-2
130-5
174-5
75-0
104-3
128-0 175-0
74-0
104-0
127-5 174-5
73-4
104-7
126-5 175-5
73-0
104-0
126-5 176-5
7'2-7
103-0
126-5 177-0
72-3
101-8
126-0 180-0
72-5
125-0
180-5
71-9
125-5
180-7
71-1
126-0
182-2
70-1
126-0
183-7
69-7
126-0
183-5
68-0
126-5
182-5
67-9
127-0
184-0
67-6
127-0
186-0
65-9
128-5
186-5
65-3
128-0
128-5
128-0
'
To find the mean section of the tubes we at first tried the method
adopted by Messrs Matthiessen and Hockin in their experiments for the
British Association. After aspirating the tube with dry air we placed it
in a wooden trough full of mercury, and filled it by suction. It was then
held down in the trough with iron weights till it was presumably of the
same temperature as the mercury in the trough, which was taken at three
places. It was then held by the fingers (previously cooled in other mercury),
pressed against its two ends, and taken out of the trough, the mercury
adhering to the outside was brushed off, and the contents of the tube were
emptied into a small porcelain crucible and weighed. But there was no
doubt that when the fingers holding the tube were bare they pressed a little
way how much it was difficult to determine into the tube, and when they
were covered with stiff leather, or other stiff material, it was difficult to get a
sufficiently good hold. However, in one case (No. 5) r was calculated from
the weight so obtained with leather on the fingers.
86 ON THE SPECIFIC RESISTANCE OF MERCURY. [81
The method, followed by Siemens and Sabine, of screwing an iron plate
up against the end of the tube, was attempted, but we did not succeed
in closing the orifice sufficiently tightly in this way. Ultimately we came
to the conclusion that the best results would be obtained by weighing a
thread of mercury nearly as long as the tube, of which we could ascertain
the actual length by direct measurement, We thought, also, that there
might be some advantage in ascertaining the volume of the mercury from
the same filling as that of which the resistance had been taken, as we could
not be sure that the closeness of contact between the mercury and the glass
was always the same, so that the same volume of mercury would always be
contained in the same length of tube, nor that the tube itself was in no
way altered by the action of the caustic potash used to clean it. The plan
adopted was, therefore, after measuring the resistance, to keep the tube
horizontal so as to retain in it most of the mercury while the terminals were
removed, and then with microscopes and divided rod to measure the thread
of mercury in the same way as the tubes were measured. The length so
obtained is called in the table 1. The greatest difference between I and L
(that in measurement 11) is scarcely over 1 per cent., and in most cases the
difference is considerably less, so that, considering how nearly cylindrical the
tubes were, the error in the mean section introduced by using a thread
of length I instead of L is quite inappreciable. It was another advantage of
our method that it avoided the necessity of filling the tube under mercury,
which it would have been difficult to do with a tube so long as IV.
The only difficulty in measuring the thread of mercury arose from the
convexity of its ends. This was overcome by pressing them flat with little
flat-ended vulcanite pins made to fit into the tube. The curvature of the
ends when free was not always the same ; but it was found that the length
of the mercury held with pins varied little from the number calculated on
the assumption that the ends were hemispherical, namely, the length of the
portion of the column of mercury which was in contact with the glass added
to two-thirds of the difference between this length and that between the
convex extremities. In some cases, where, owing to the pins not fitting very
well or other causes, there was a difficulty in flattening the ends properly,
the calculated value was used. A thermometer lay beside the tube during
the measurement, so as to give the temperature t. After the measurement,
the mercury was blown out into a small crucible and weighed. Care had to
be taken not to leave behind minute globules, which, owing probably to the
small portion of the tube unoccupied by mercury during the measuring
becoming damp from the air of the room or from the fingers, tended to
adhere to the glass near the ends.
In three cases (No. 5 as above mentioned and Nos. 3 and 9) the mercury
weighed and measured was not that of which the resistance was taken.
81] ON THE SPECIFIC RESISTANCE OF MERCURY. 87
No. 3 was done before it occurred to us that there might be an advantage
in carrying out both operations with the same filling, and in No. 9 about
one-tenth of the mercury was spilt accidentally and had to be replaced.
The equality of the arms of the balance used for the weighing was
tested. The weights were compared among themselves and found to be free
from appreciable error.
The terminals were composed of L-shaped pieces of ebonite, hollowed
out in the manner shown (about full-size) in the figure. Each end of the
tube was furnished with a short length of thick rubber tubing, by which the
aperture between the glass and the ebonite was closed air-tight. As a further
precaution, the space at cc beyond the rubber was filled up by pouring in
melted paraffme wax.
After the terminals were fitted the tube was again aspirated with dry air
through tubes in corks inserted at a a, and then filled with mercury, which
was poured in to one terminal and allowed to run slowly through to the
other till it stood at a considerable height, represented by dd, in both
terminals. The tube was then placed in a wooden trough and covered with
ice. Our reason for using vulcanite terminals rather than glass ones was the
fear that under the influence of the ice moisture would collect on the portion
of glass above the mercury and serve as a conductor. We certainly avoided
all difficulty of this kind by using vulcanite. On the other hand, we probably
increased a difficulty which would have existed in any case, namely, that of
getting the temperature of the portion of the tube which was within the
terminal down to 0. This portion of the tube was about 2 centims. at each
end, or about 5 per cent, of the length in the case of tube I., and about 2 per
cent, in the case of tube IV. What the exact temperature of this part of the
tube was it is impossible to say, but it was ascertained that the temperature
of the mercury in the terminals with the copper connecting rods in situ was
not higher than 5 or 6, depending in some degree on the extent to which
88 ON THE SPECIFIC RESISTANCE OF MERCURY. [81
the ice was piled up round the cup. The mean temperature of the parts of
the tubes not directly exposed to ice can hardly have been so high as 2.
Supposing it to have been 2, and taking the case of tube I., where the
largest proportion of the whole length was within the terminals, the effect
would be an overestimate of r by about '00008. In the case of tube IV. the
error in r would be less than the half of this.
The tubes were connected with the resistance balance by copper rods,
well amalgamated, of which one end stood on the bottom of the vulcanite
terminals, so that a considerable portion of the amalgamated copper surface
was in contact with the mercury. The rods were kept at a little distance
from the ends of the tubes. Dr Matthiessen brought flattened copper rods
up against the ends of his tubes, but this plan appeared open to objection,
since it would be very difficult to secure complete contact between the copper
and glass all round the edge of the orifice, especially under an opaque fluid
like mercury ; and any defect in such contact would render necessary an
unknown correction. We preferred, therefore, to let the ends of the tube
open without obstruction into the mercury cup, which may be regarded as of
infinite extent by comparison. The correction necessary to take account
of the resistance of the mercury beyond the ends of the tube has already
been considered.
The resistance of the rods used to connect I., II., and III. with the bridge
was about '00215 B.A. unit. With tube IV. an additional rod had to be
introduced to get the necessary length. This brought the resistance of the
rods up to '00291. The other end of the rods fitted into mercury cups on
the resistance balance.
The balance used was one designed by Professor Fleming (Phil. Mag. IX.
p. 109, 1880), in which Professor Carey Foster's method is employed of
interchanging the resistances in the two arms of the balance containing the
graduated wire, so that the difference between them is expressed in terms of
the wire. One thousand divisions of the graduated wire are stated by
Professor Fleming to equal '0498 B.A. unit, and experiments of our own
also showed it to be about '05. The wire is of platinum-iridium, and as it
has a high temperature coefficient compared with the platinum-silver of the
standard coils, we thought it undesirable to use much over 100 divisions of
it. In order to avoid this in the case of tubes I. and IV. it was necessary to
introduce coils from a resistance box in multiple arc. The resistance box
employed was one by Messrs Elliott Brothers. With tube I., 20 ohms from
the box were used in multiple arc with the standards against which the tube
was balanced, and in the case of tube IV. 24 ohms were used in multiple arc
with the tube itself. Tubes II. and III. were balanced against the standard
coil belonging to the British Association and deposited at the Cavendish
Laboratory, called F. For tube IV. another of their unit coils, called the
81] ON THE SPECIFIC RESISTANCE OF MERCURY. 89
Flat coil, was used in multiple arc with F. For tube I., F and a five-ohm
coil were used in multiple arc. The standard coils belonging to the British
Association have recently been carefully compared with each other by
Professor Fleming, who has drawn out a chart in which is recorded their
variation with temperature, together with their resistance in terms of the
mean of their resistances at the temperatures at which they were originally
considered to be correct The values of F and of the Flat coil both
platinum-silver coils were taken from this chart. The five-ohm coil had
been compared with the British Association standards by ourselves. It was
also of platinum-silver, and its temperature coefficient was assumed to be the
same as that of the others.
The standard coils were immersed in water whose temperature was
observed each time a resistance was measured. These temperatures are
given in the table. It may be worth remarking that the resistances were
taken in a different room from that in which the lengths were measured,
which accounts for the difference between t and the temperature of the
standards. The thermometer used to find all the temperatures was graduated
to fifths, and was corrected by one which had been verified at Kew.
When one coil only was used to balance the tube, its terminals fitted
directly into the mercury cups of the bridge, but when two were used
in multiple arc their terminals were put into larger mercury cups, which
were connected with the mercury cups of the bridge by short copper connect-
ing pieces of about '00017 ohm resistance.
All the measurements were repeated with reversed battery currents, in
order to eliminate thermoelectric disturbance. The readings with battery
current each way usually agreed very closely, and the mean of the two was
adopted.
It will be observed that the values of R for tube IV. differ by nearly two
parts in 10,000, and that there is a less proportional difference, but still an
appreciable one, for the other tubes. The greatest actual difference between
any two of the values in the table for the same tube is '00014 ohm. Some
small error is due to neglect of the change of resistance of the copper
connecting rods and of the bridge wire with temperature. A change of 4 in
the temperature of the rods would make a difference of about '00003 ohm.
There is further a probability of error in ascertaining the temperature of the
standard coil. A difference of ^ in this also introduces a difference of
00003 ohm in the resistance; and there is not only a probable error of
perhaps ^ in finding the temperature of the water in which the coil is
immersed, but there is no certainty that the coil follows the water exactly.
There is evidence, however, that the differences in R are partly due to a real
difference in the resistance of different fillings of the tube whether owing
90 ON THE SPECIFIC RESISTANCE OF MERCURY. [81
to microscopic bubbles or to a thin varying layer of air between the mercury
and the glass, or to what cause, we were unable to determine*.
We found some reason for thinking that the resistance tended to diminish
with time when the mercury remained long in the tube. To examine this
we filled tube II. on April 3rd, and found its resistance to be '99077. It was
then left standing full of mercury till April 18th, when the resistance was
99055. This difference can hardly be relied upon; and in any case the
experiments we have tabulated cannot well be affected by any change of this
kind, as the interval between the measurement of resistance and that of
volume was very short, except in cases 1 and 7. In case 7 the tube stood
full of mercury for two days after the resistance was taken. In case 1 the
resistance was measured on two successive days, and the mean of the two
values taken. The second was the lowest by '00020, possibly owing to an
error. The length was measured immediately after the last measurement of
resistance.
The variations in the values of r are, as we should expect, greater than
those in R, being affected by probable errors in the other data. The extreme
difference amounts to less than 6 in 10,000, and the greatest divergence from
the mean value is 3'3 in 10,000.
The mean value of r according to these experiments, '95418, lies between
that deduced from Dr Siemens' experiments for his 1864 standard, namely,
9534, and Dr Matthiessen's value, namely, "9619 (Phil. Mag. May, 1865),
but the difference between our value and Dr Matthiessen's, namely, '00772,
is nearly ten times as great as that between ours and Dr Siemens'. We are
unable to account satisfactorily for this large difference. One point, however,
is worth noting. Dr Matthiessen measured the resistance of the mercury in
his tubes, not at zero, but at temperatures between 18 and 19'l (Report of
British Association Committee for 1864). To deduce the specific resistance
at zero, therefore, he must have assumed the coefficient of variation with
temperature, and presumably though it is nowhere stated in the Report
he used that found from his own experiments (Phil. Trans. 1862), namely,
'074f per cent, per degree. Our own observations have led us to suspect
that this value is too small. We made three comparisons of the resistance of
tube III. in ice, and in water at approximately the temperature of the room,
and one similar comparison with tube IV. The results are given in the
following table. Our arrangements were not adapted for observing the
resistance at other temperatures, as the open trough afforded no means of
checking rapid change.
* A variation in the closeness of contact between mercury and glass amounting to less than
one-fifth of a wave-length of mean light would account for the difference of resistances in the two
fillings of tube IV.
t This is the value which results from the experiments made at and at about 20.
81] OX THE SPECIFIC RESISTANCE OF MRRCT7RT. 91
.
Date
Mean tempe- P.. . Mean of the
No. of ratureof Resistance Resistance /*?< >r foor values in
tube water in the in water at the last
trough tOP
March 13 .
in.
12-7
1-00814
-99720
000663
14.
in.
13-25
1-00874
99725
000670
38 - i
in.
12-8
1-00810
99720
000854 -000861*
24 .
IV. j
12-5
51318
50774
-000857
The above determined mean coincides with the value found by Schroder
van der Kolk-f% whose observations, however, related to a much greater range
of temperature. An observation by Werner Siemens* between the tempera-
ture 18 D '5 and D gives for the coefficient '00090.
The difference between the coefficients -00074 and "00086, as applied to the
reduction from 18 7 (the mean temperature of the tubes in Dr Matthiessen's
observations) to 0, would account for about one quarter of the difference
between his results and our own.
The remainder of the discrepancy may possibly be connected with the
manner in which Dr Matthiessen's tubes were calibrated. Although iu the
description of the process a small column of mercury is spoken of (Reprint,
p. 1 28), it is distinctly stated on the preceding page that the lengths of the
columns of mercury were 383, 291, 245 millims. respectively, i.e., nearly halt
the lengths of the tubes. It is possible that this may be a mistake ; but if
such lengths were really used, the correction for conicality would have been
much underestimated, so that the specific resistance of mercury would come
out too high. In the case of uniform conicality the true correction would be
four times as great as that obtained by applying the formula applicable to
short threads, to cases where the length is about half that of the tube.
[January, 1883. The measuring rod and the weights used in the above
investigation have been compared with standards verified by the Board of
Trade, and the errors have been found to be negligible. But since the value
of p employed relates to weighings in vacua, a corresponding correction is
called for here. On this account the final number, '95418, should be
reduced to
-95412.]
* It should be noticed that the resistances here compared are those of the contents of a
certain glass tube at various temperatures, so that the accompanying temperature variations
of length and section are determined by the properties of glass and not by the properties
of mercury. The results are therefore not quite comparable with those obtained in similar
experiments with solid metallic wires, which are free tc determine for themselves their length
and section.
t Pogg. Ann. vol. ex. 1860.
+ Ibid. voL cxra. 1861.
82.
THE USE OF TELESCOPES ON DARK NIGHTS.
[Proceedings of the Cambridge Philosophical Society, iv. pp. 197, 198, 1882.]
IN Sillimans Journal for 1881 Mr E. S. Hold en, after quoting observa-
tions to a like effect by Sir W. Herschel, gives details of some observations
recently made with a large telescope at the Washburn Observatory, from
which it appears that distant objects on a dark but clear night can be seen
with the telescope long after they have ceased to be visible with the naked
eye. He concludes, " It appears to me that this confirmation of Herschel's
experiments is important, and worth the attention of physicists. So far as I
know there is no satisfactory explanation of the action of the ordinary Night-
glass, nor of the similar effect when large apertures are used."
It is a well-known principle that no optical combination can increase
what is called the ' apparent brightness ' of a distant object, and indeed that
in consequence of the inevitable loss of light by absorption and reflection the
' apparent brightness ' is necessarily diminished by every form of telescope.
Having full confidence in this principle, I was precluded from seeking the
explanation of the advantage in any peculiar action of the telescope, and was
driven to the conclusion that the question was one of apparent magnitude
only, that a large area of given small ' apparent brightness' must be visible
against a dark ground when a small area would not be visible. The experi-
ment was tried in the simplest possible .manner by cutting crosses of various
sizes out of a piece of white paper and arranging them in a dark room against
a black back-ground. A feeble light proceeded from a nearly turned-out
gas-flame. The result proved that the visibility was a question of apparent
magnitude to a greater extent than I had believed possible. A distance was
readily found at which the larger crosses were plainly visible, while the
smaller were quite indistinguishable. To bring the latter into view it was
82] THE USE OF TELESCOPES ON DARK NIGHTS. 93
necessary either to increase the light considerably, to approach nearer, or
lastly to use a telescope. With sufficient illumination the smallest crosses
used were seen perfectly defined at the full distance.
There seems to be no doubt that the explanation is to be sought within
the domain of Physiological Optics. It has occurred to me as possible that
with the large aperture of the pupil called into play in a dark place, the
focussing may be very defective on account of aberration. The illumination
on the retina might then be really less in the image of a small than in the
image of a large object of equal ' apparent brightness.'
[1899. See Camb. Proc. iv. p. 324, 1883 ; Art. 96 below.]
83.
ON A NEW FORM OF GAS BATTERY.
[Proceedings of the Cambridge Philosophical Society, iv. p. 198, 1882.]
IN Grove's well-known gas battery it would seem that the only efficient
part of the platinum surface is where it meets both the gas and the liquid, or
at any rate meets the liquid and is very near the gas. In order to render
a larger area effective I have substituted for the usual platinum plates
platinum gauze resting upon the surface of the liquid in a large trough
in such a manner that the upper surface is damp but not immersed. One
piece is exposed to the oxygen of the air ; the other forms the bottom of an
enclosed space into which hydrogen is caused to flow. The area of each
piece is about 20 square inches.
To test the efficiency, the current was passed through an external
resistance of about 6 ohms, including a galvanometer. Under these cir-
cumstances the permanent current was about one-fourth of that obtained
when a large Daniell cell was substituted for the gas element. An inferior,
but still considerable, current was observed when coal gas was used instead
of hydrogen prepared from zinc.
84
ACOUSTICAL OBSERVATIONS. IV.
[Philosophical Magazine, xm. pp. 340347, 1882.]
ON THE PITCH OF ORGAN-PIPES. SLOW VERSUS QUICK BEATS FOR COM-
PARISON OF FREQUENCIES OF VIBRATION. ESTIMATION OF THE
DIRECTION OF SOUNDS WITH ONE EAR. A TELEPHONE-EXPERIMENT.
VERY HIGH NOTES. RAPID FATIGUE OF THE EAR. SENSITIVE FLAMES.
On the Pitch of Organ-Pipes.
IN the Philosophical Magazine for June 1877 [Art. 46, vol. i. p. 320]
I described some observations which proved that the note of an open organ-
pipe, when blown in the normal manner, was higher in pitch than the natural
note of the pipe considered as a resonator. The note of maximum resonance
was determined by putting the ear into communication with the interior of
the pipe, and estimating the intensity of sounds of varying pitch produced
externally.
A more accurate result may be obtained with the method used by
Blaikley*, in which the external sound remains constant and the adjustment
is effected by tuning the resonator to it. About two inches were cut off
from the upper end of a two-foot metal organ-pipe, and replaced by an
adjustable paper slider. At a moderate distance from the lower end of the
pipe a tuning-fork was mounted, and was maintained in regular vibration by
the attraction of an electromagnet situated on the further side, .into which
intermittent currents from an interrupter were passed. Neither the fork
nor the magnet were near enough to the end of the pipe to produce any
sensible obstruction. By comparison with a standard, the pitch of the fork
Phil. Mag. May 1879.
96 ACOUSTICAL OBSERVATIONS. [84
thus vibrating was found to be 255 of Konig's scale. The resonance of the
pipe was observed from a position not far from the upper end, where but
little of the sound of the fork could be heard independently ; and the paper
slider was adjusted to the position of maximum effect. This observation was
repeated many times, the distance between marks fixed on the pipe and on
the slider respectively being recorded. The following numbers give the
results, expressed in fiftieths of an inch [inch = 2'54 cm.] :
31 33 30 25 31
25 32 31 34 29
35 28 29 30
The extreme range being only one-fifth of an inch, shows that the observa-
tion is capable of considerable precision, corresponding as it does to only
about 2 vibrations per second out of a total of 255. Finally, the slider was
fixed at the mean of the above-determined positions, and the natural note of
the pipe was then considered to be 255. The error in length was probably
less than ^ inch, and the error in pitch less than half a vibration per second.
The pipe was then blown from a well-regulated bellows ; and the beats
were counted between its note and that of the standard fork above referred
to, the pressure being taken simultaneously with a water- manometer. Three
observers were found to be necessary for accurate working one to count the
beats, rising to the rate of ten per second, one to keep the bellows uniformly
supplied with wind, and one to observe the manometer. At pressures
between 4*2 inches and 1'53 inch the pitch of the pipe was very well defined
and considerably higher than the natural note. Below 1 inch the pitch
became somewhat unsteady, and distinct fluctuations in the frequency of the
beats were perceived, while no corresponding variation of pressure could be
detected. At about '8 inch the pitch of the pipe falls to unison with the
natural note, and with further diminishing pressures becomes the graver of
the two. Below '7 inch the unsteadiness is such as to preclude accurate
estimations of pitch.
The results are embodied in the accompanying table, which shows the
correspondence of pitch and pressure. Instead of the actual number of beats
counted, which involves a reference to the extraneous element of the pitch of
the standard fork, the number (greater by unity) is given which expresses
the excess in the frequency of vibration of the actual over that of the natural
note of the pipe. It will be seen that at practical pressures the pitch is
raised by the action of the wind, but that this rule is not universal.
84]
ACOUSTICAL OBSERVATIONS.
- Pressure, in
inches
Difference of
frequencies
Remarks
4-2
+ 11-0
2-72
9-3
2-26
8-4
1-86
7-1
1-53
5-6
1-32
4-2
1-06
2-1
88
1-5
82
+ -1
75
"5
68
1-2
64
2-3
57
3-9* About this point a discordant high note comes in alone-
53
46
40
3-7
3-l
- 4-0
side of the normal note.
Here the discordant note
* About this point the octave of the normal note is heard,
after which the normal note itself disappears.
The normal note reappears, the octave continuing.
The octave goes, and then the normal note, after which
there is silence.
Octave comes in again, and then the normal note, at
a pitch which falls from considerably above to a little
below the natural pitch. At the lowest pressures the
normal note is unaccompanied by the octave.
Slow versus quick Beats for comparison of Frequencies of Vibration.
Most of those who have had experience in counting beats have expressed
a preference for somewhat quick beats. Perhaps the favourite rapidity has
been four beats per second. There is no doubt that in the case of insuffici-
ently sustained sounds slow beats are embarrassing. The observer gets
confused between the fall of sound which is periodic and that which is due
to the dying away of the component vibrations, and loses his place, as it
were, in the cycle. But it is also possible, I think, to trace an impression
that, independently of the risk of confusion, quick beats can be counted with
greater accuracy than slow ones. It is indeed true that the number of beats
in a given time, such as a minute, can be determined with greater relative
accuracy when there are many than when there are few ; but it is also true,
as a little consideration will show, that in the comparison of frequencies we
98 ACOUSTICAL OBSERVATIONS. [84
are concerned not with the relative, but with the absolute number of beats
executed in the given time. If we miscount the beats in a minute by one, it
makes just the same error in the result whether the whole number of beats
is 60 or 240.
When the sounds are pure tones and are well maintained, it is advisable
to use beats much slower than four per second. By choosing a suitable
position we may make the intensities at the ear equal ; and then the phase
of silence, corresponding to antagonism of equal and opposite vibrations, is
extremely well marked. Taking advantage of this, we may determine slow
beats with very great accuracy by observing the time which elapses between
recurrences of silence. In favourable cases, the whole number of beats in the
period of observation may be fixed to within one-tenth or one-twentieth of a
single beat, a degree of accuracy which is of course out of the question when
the beats are quick.
In some experiments, conducted by Dr Schuster and myself*, to determine
the absolute pitch of a Konig standard fork, I had occasion to observe some
very slow beats. The beating sounds were of pitch 128. One of them was
steady, proceeding from an electrically maintained fork ; the other (from the
standard fork) gradually died away. In order to be more independent of
disturbing noises to which we were exposed, a resonator was used connected
with the ear by an india-rubber tube. The standard fork was mounted at
the end of a wooden stick, so that it might not be heated by the hand. As
the vibrations became less powerful, the prongs of the fork were caused
slightly to approach the mouth of the resonator, so as to maintain the
equality of the two component sounds. In this way it was possible to
obtain very definite silences, and to measure the interval of recurrence
with accuracy. In one observation, extending over about two minutes, the
beat occupied as much as twenty-four seconds, and there was no confusion.
I have little doubt that even slower beats might be observed satisfactorily if
both components were steadily maintained.
Estimation of the Direction of Sounds with one Ear.
In my former experiments (Phil. Mag. June 1877 [Art. 46, vol. I.
p. 315]) I found it difficult to obtain satisfactory observations with one ear
closed, although it was not doubtful that the power of estimating directions
was greatly curtailed. My desire to experiment upon an observer deaf on
one side has since been gratified by the kind assistance of Mr F. Galton. In
January 1881 experiments were tried with him similar to those on normal
hearers described in my former paper. It was found that Mr Galton made
mistakes which would be impossible for normal ears, confusing the situation
of voices and of clapping of hands when to his right or left, as well as when
* Proc. Roy. Soc. May 5, 1881, p. 137. [Art. 79.]
84] ACOUSTICAL OBSERVATIONS. 99
in front or behind him. Thus, when addressed loudly and at length by a
little boy standing a few yards in front of him, he was under the impression
that the voice was behind. In other cases, however, there seemed to be some
clue, whose nature we could not detect. Bad mistakes were made ; but the
estimates were more often right than mere chance would explain.
After this experience it seemed unlikely that there could be any success
in distinguishing whether pure tones came from right or left, and from in
front or behind. The experiment was tried, however, with in the main the
expected result. But when the sounds were close, there appeared to be some
slight power of distinguishing right and left, which may perhaps have been
due to incomplete deafness of the defective ear.
A Telephone-Experiment.
In Maxwell's Electricity and Magnetism, vol. n. 655, it is shown that a
perfectly conducting sheet acts as a barrier to the magnetic force : " If the
sheet forms a closed or infinite surface, no magnetic actions which may take
place on one side of the sheet will produce any magnetic effect on the other
side." In practice we cannot use a sheet of perfect conductivity; but the
above-described state of things may be approximated to in the case of
periodic magnetic changes, if the time-constants of the sheet circuits be large
in comparison with the periods of the changes.
The experiment is made by connecting up into a primary circuit a
battery, a microphone-clock, and a coil of insulated wire. The secondary
circuit includes a parallel coil and a telephone. Under these circumstances
the hissing sound is heard almost as well as if the telephone were inserted in
the primary circuit itself. But if a large and stout plate of copper be
interposed between the two coils, the sound is greatly enfeebled. By a
proper choice of battery and of the distance between the coils, it is not
difficult so to adjust the strength that the sound is conspicuous in the one
case and inaudible in the other.
Very High Notes. Rapid Fatigue of the Ear.
In former experiments with bird-calls I had often been struck with what
seemed to be the capricious behaviour of these sources of sound, but had
omitted to follow up the observation. In the spring of last year the apparent
caprice was traced to the ear, which very rapidly becomes deaf to sounds of
high pitch and moderate intensity. A bird-call was mounted in connexion
with a loaded gas-bag and a water-manometer, by which means the pressure
could be maintained constant for a considerable time. When the ear is
placed at a moderate distance from the instrument, a disagreeable sound
72
100 ACOUSTICAL OBSERVATIONS. [84
is heard at first, but after a short interval, usually not exceeding three or
four seconds, fades away and disappears altogether. A very short intermission
suffices for at any rate a partial recovery of the power of hearing. A pretty
rapid passage of the hand, screening the ear for a fraction of a second, allows
the sound to be heard again. During his visit to Cambridge in March 1881,
I had the pleasure of showing this experiment to Prof. Helmholtz.
The uniformity of the sound in the physical sense may be demonstrated
with a sensitive flame, which remains uniformly affected so long as the
pressure indicated by the manometer does not vary. The sensitive flame
may also be employed to determine the wave-length of the sound, in the
manner described in the Philosophical Magazine for March 1879, p. 154
[Art. 61, vol. I. p. 406]. In the case of two bird-calls blown with a
pressure of about 2 inches of water, the wave-lengths were found to be
respectively T304 inches and 1*28 inches [one inch = 2'54 cm.]. The method
was found to work easily and with considerable accuracy, almost identical
results being obtained from observations of the loops, where the flame is most
affected, and from the nodes, where it is least affected.
By modifying the pressures with pinch-cocks, the two notes could be
brought into unison. Although both bird-calls were blown from the same
gas-bag, it was not possible to keep the beats slow for more than a few
seconds at a time ; but that period was quite sufficient for the effects of the
beats to manifest themselves in a striking manner by the behaviour of the
flame. In repeating these experiments, it may be necessary to bear in mind
that many people cannot hear these high notes at all, even at first. With a
shorter wave-length of about ^ inch, as determined by the flame, I was myself
quite unable to hear any sound from the situation of the flame. A slight
hissing was perceived when the ear was brought up close to the source ;
but it is probable that this was not the part of the sound that agitated the
flame.
Sensitive Flames.
In the chapter devoted to this subject in Tyndall's Sound (third edition,
p. 231) the accomplished author remarks : " An essential condition to entire
success in these experiments disclosed itself in the following manner. I was
operating on two fishtail flames, one of which jumped to a whistle while the
other did not. The gas of the non-sensitive flame was turned off, additional
pressure being thereby thrown upon the other flame. It flared, and its cock
was turned so as to lower the flame ; but it now proved non-sensitive, how-
ever close it might be brought to the point of flaring. The narrow orifice
of the half-turned cock interfered with the action of the sound. When the
gas was fully turned on, the flame being lowered by opening the cock of the
84] ACOUSTICAL OBSERVATIONS. 101
other burner, it became again sensitive. Up to this time a great number
of burners had been tried, but with many of them the action was nil.
Acting, however, upon the hint conveyed by this observation, the cocks which
fed the flames were more widely opened, and our most refractory burners
thus rendered sensitive." In the abstract of a Royal-Institution lecture
(Phil. Mag. Feb. 1867) a rather more definite view is expressed: "Those
who wish to repeat these experiments would do well to bear in mind, as an
essential condition of complete success, that a free way should be open for
the transmission of the vibrations from the flame, backwards, through the
gas-pipe which feeds it. The orifices of the stopcocks near the flame ought
to be as wide as possible."
During the preparation of some lectures on Sound in the spring of last
year, it occurred to me that light would probably be thrown upon these
interesting effects by introducing a manometer on a lateral branch near the
flame. In the path of the gas there were inserted two stopcocks, one only
a little way behind the manometer-junction, the other separated from it by a
long length of india-rubber tubing. When the first cock was fully open and
the flame was brought near the flaring-point by adjustment of the distant
cock, the sensitiveness to external sounds was great, and the manometer
indicated a pressure of ten inches of water. But when the distant cock stood
fully open and the adjustment was effected at the other, high sensitiveness
could not be attained ; and the reason was obvious, because the flame flared
without external excitation while the pressure was still an inch short of that
which had been borne without flinching in the former arrangement. On
opening again the neighbouring cock to its full extent, and adjusting the
distant one until the pressure at the manometer measured nine inches, the
flame was found comparatively insensitive.
It appears, therefore, that the cause of the prejudicial action of partially
opened stopcocks in the neighbourhood of the flame is not so much that they
render the flame insensitive as that they induce premature flaring. There
are two ways in which we may suppose this to happen. It may be that, as
Prof. Barrett suggests (Phil. Mag. April 1867), the mischief is due to the
irregular flow and consequent ricochetting of the current of gas from side to
side of the pipe ; or, again, the cause may lie in the actual production of
sonorous disturbance of the kind to which the flame is sensitive, afterwards
propagated forwards to the burner along the supply-pipe acting as a speaking-
tube. The latter explanation was the one that suggested itself to my mind
at the time, in consequence of the observation that a hissing sound was
easily audible by the ear placed close to the half-open stopcock through
which gas was passing; and it was confirmed when I found that a screw
pinch-cock could be used for adjustment near the flame with impunity, in
which case no sound was perceptible.
102 ACOUSTICAL OBSERVATIONS. [84
Subsequent!}' further experiments were tried with various nozzles inserted
in the supply-tube. These included holes in thin metal plates and drawn-
out glass tubes. Even though the rubber tubes were so bent that the
streams issuing from the nozzles were directed against the sides, no sound
was heard, and no loss of sensitiveness was apparent. It would seem that
mere irregularity of flow produced no marked effect, and that, provided no
sound attended it, the full pressure could be borne without flaring.
These observations in no way impair the value of the practical rule laid
down by Tyndall. In some cases I have found a flame flare without external
excitation when a neighbouring stopcock was partially closed, and in spite of
the increase of pressure recover itself when the stopcock was completely
opened. When the object is to investigate the conditions of flaring, the use
of a manometer near the flame is decidedly to be recommended.
85.
FURTHER OBSERVATIONS UPON LIQUID JETS, IN CON-
TIXUATION OF THOSE RECORDED IN THE ROYAL
SOCIETY'S 'PROCEEDINGS' FOR MARCH AND MAY, 1879.
[Proceedings of the Royal Society, xxxiv. pp. 130145, 1882.]
THE experiments herein described were made in the spring and summer
of 1880, with the assistance of Mrs Sidgwick. Section 2 was indeed written
out as it now stands in August of that year. There were some other points
which I had hoped to submit to examination, but hitherto opportunity has
not been found.
On some of the Circumstances which influence the Scattering of a nearly
Vertical Jet of Liquid.
1. It has been already shown [Art 59, voL I. p. 372] that the normal
scattering of a nearly vertical jet is due to the rebound of the drops when
they come into collision. If, by any means, the drops can be caused to
amalgamate at collision, the appearance of the jet is completely transformed.
This result occurs if a feebly electrified body be held near the place of
resolution into drops, and it was also observed to follow the addition of a
small quantity of soap to the water of which the jet was composed. In
trying to repeat the latter experiment in May, 1880, at Cambridge, I was
astonished to find that even large additions of soap failed to prevent the
scattering. Thinking that the difference might be connected with the
hardness of the Cambridge water at home I had used rain water I
repeated the observations with distilled water, but without finding any
explanation. The jet of distilled water scattered freely, both with and
without soap, and could only be prevented from doing so by electricity.
Eventually the anomalies were traced to differences in the character of the
104 ON LIQUID JETS. [85
soap. That used at Cambridge up to this point was a clarified specimen
prepared for toilet us,e. On substitution for it of common yellow soap, the
old effects were fully reproduced.
Further experiment seemed to prove that the real agent was not soluble
soap at all. If water impregnated with the yellow soap was allowed to stand,
a white deposit separated, after which the supernatant liquid was found to
be inactive. But after shaking up the same effects were produced as at
first. The addition of caustic potash to the unclarified soapy mixture
destroyed its power. On the other hand, sulphuric acid rendered the
clarified soap solution active.
The natural conclusion from these facts would be that the real agent is
unsaponified greasy matter distributed through the liquid ; and this view is
confirmed by the striking results which follow the addition of small quantities
of milk. The experiment may be made conveniently by connecting a Woulf's
bottle with the water tap by a rubber tube fitted to one tubulure, while the
vertical nozzle is in connexion with another tubulure. If a little milk be
placed in the bottle, the jet of opalescent liquid apparently coheres, and
passes the summit in one unbroken stream. After a time the milk is
gradually washed out, and the scattering is re-established. About one drop
of skimmed milk per ounce of water [say one part in 600] is sufficient to
produce the effect.
I must not omit to mention that on several occasions distinct evidence
was obtained that it is possible for soap to be in excess. With a large
quantity the coherence of the jet was imperfect, and was improved by
dilution. The complete elucidation of the subject probably requires more
chemical knowledge and experience than is at my command.
Of the various other substances which have been tried, such as glycerine,
sugar, gum arabic, alcohol, sulphuric acid, none have been found active.
Vertical fountains of mercury were found not to scatter. The head was
about 15 inches [one inch = 2'54 cm.], and various glass nozzles were used
from ^ inch to -^ inch in diameter. Also a nozzle terminating in an amalga-
mated brass plate, through which a hole of ^ inch was pierced. In all these
cases the drops of mercury coalesced at collision, behaving in the same way
as drops of milky water issuing from the same nozzles. Fountains of clean
water issuing from these nozzles under the same pressure scattered freely.
When the diameter of the nozzle from which a water jet issues is reduced
to below T ^ inch, the scattering cannot be completely prevented by the
presentation of an electrified body. One possible reason for'this is evident.
The mutual repulsion of the similarly electrified drops increases rapidly
relatively to the masses as the size is reduced, and thus it may happen
that before the differential electrification sufficient to rupture the separating
85] OH LIQUID JKIS. 105
envelope at contact is arrived at, the repulsion may be powerful enough to
prevent most of the drops from coming into contact at alL In connexion
with this it may be remarked that two perfectly equal and equally electrified
spheres would repel one another at all distances ; but that if there be the
slightest difference in the size or electrification, the repulsion will be
exchanged for attraction before actual contact is attained. This attraction
will be local, and thus the opposed part* of the surfaces may come into
contact with considerable violence, even when the relative motion of the
centres of the masses is small. It is easily shown experimentally (see 4)
that violence of contact tends to promote coalescence, so that we have here a
possible explanation of the action of electricity.
With respect to the persistent scattering of very 6ne jets, however, it
would appear that the principal cause is simply that many of the fine drops
fail to come into contact in any case. The capillary forces act with exagge-
rated power, and doubtless impress upon the minute drops irregular lateral
velocities, which may easily reach a magnitude sufficient to cause them to
clear one another as they pass. At any rate little difference is observable
in this respect between a fine jet- of clean water under feeble electrical
influence, and one to which a little milk has been added, but without
electrification.
With a suitable jet, say from a nozzle about ^ inch diameter, and
rising about 2 feet, the sensitiveness to electricity is wonderful, more
especially when we remember that the effect is differential. I have often
caused a jet to appear coherent, by holding near the place of resolution
a brass ball about 1 inch in diameter, supported by a silk thread, and
charged so feebly that a delicate gold-leaf electroscope would show nothing.
Indeed, some care is necessary to avoid being misled by accidental electrifica-
tions. On one occasion the approach of a person, who had not purposely
been doing anything electrical, invariably caused a transformation in the
appearance of the jet.
The jets hitherto under discussion are such as resolve themselves
naturally into drops soon after leaving the nozzle, or at any rate before
approaching the summit of their path. If the diameter be increased, we
may arrive at a condition of things in which the undisturbed jet passes
the summit unbroken. In such a case the addition of milk, or the presenta-
tion of an electrified body, produces no special effect. One interesting
observation, however, may be made. By the action of a vibrator of suitable
pitch, e.g. a tuning-fork, resolution on the upward path may be effected.
As the vibration gradually dies down, the place of resolution moves upwards,
but it cannot pass a certain point. When the point is reached, resolution
into actual drops ceases, the upper part of the jet exhibiting simple undula-
tions, when viewed intermittently. The phenomenon is in perfect harmony
106 ON LIQUID JETS. [85
with theory. As it leaves the nozzle, the jet is unstable for the kind of
disturbance imposed upon it by the vibrator. The subsequent loss of velocity,
however, shortens the wave-lengths of disturbance, until at length they are
less than the circumference of the jet, after which the disturbance changes
its character from unstable to stable. The vibrator must evidently produce
its effect quickly, or not at all.
Influence of Regular Vibrations of Low Pitch.
2. Towards the close of my former paper on the capillary phenomena
of jets [Art. 60, vol. I. p. 395], I hazarded the suggestion that the double
stream obtained when an obliquely ascending jet is subjected to the influence
of a vibration an octave graver than the natural note, is due to the compound
character of the vibration. At the time of Plateau's researches the fact that
most musical notes are physically composite was much less appreciated than
at present, and it is not surprising that this point escaped attention. I have
lately repeated Plateau's experiments under improved conditions, with results
confirmatory of the view that no adequate explanation of the phenomena can
be given which does not have regard to the possible presence of overtones ;
and I have added some observations on the effects of the simultaneous action
of two notes forming a consonant chord.
In order to make a satisfactory examination of it, it is necessary to
employ some apparatus capable of affording an intermittent view of the jet
in its various stages of transformation. In the experiments formerly described
I used sparks from an induction coil, governed by the same tuning-fork which
determined the resolution of the jet. This has latterly been replaced by a
perforated disk of black cardboard, driven at a uniform speed by a small
water-motor. The diameter of the holes is one-fifth of an inch about that
of the pupil of the eye, and the interval between the holes is about four
inches. Examined under these conditions the jet and resultant drops are
sufficiently well defined, and there is abundant illumination if the apparatus
is so arranged that the jet is seen projected against the sky. The speed of
the motor is regulated so that there is one view through the holes in about
one complete period of the phenomenon to be observed. If the power is a
little in excess, the application of a slight friction to the axle carrying the
disk renders the image steady, or, what is better, allows it to go forwards
through its phases with moderate slowness.
Although the multiple streams are better separated when the jet is
originally directed upwards at an angle of about 45, I preferred to use a
horizontal direction as giving simpler conditions. The velocity and diameter
are then practically constant throughout the transformation, and may be
readily calculated from observations of the head and of the total quantity of
fluid discharged in a given time. The reservoir consisted of a large glass
85] ON LIQUID JETS. 107
bottle, provided with a tubulure near the bottom. Into this was fitted a
1-inch brass tube, closed at the end by a flat plate, in which a circular
aperture was pierced of about -^ of an inch [say, 2 mm.] in diameter.
If h = head, d= diameter of jet, v = velocity of issue, V= volume dis-
charged in unit time, then
=V, v
Again, if N' be the frequency of the most rapid vibration which can
influence the jet, we have by Plateau's theory
jrd
If N be the frequency of the principal note of the jet, then, as explained
in my former paper,
In the present experiment it was found that 1050 cub. centims. were
discharged in four minutes, and the head was 7 inches, so that in c.G.s.
measure
whence
#' = 372, #=259.
As sources of sound tuning-forks, provided with adjustable sliding pieces,
were employed. Except when it was important to eliminate the octave as
far as possible, the vibration was communicated to the reservoir through the
table on which it stood. The forks were either screwed to the table and
vibrated with a bow, or mounted on stands (resting on the table) and
maintained electrically. The former method was quite adequate when on'y
one fork was wanted at a time.
With pitches ranging from 370 to about 180, the observed phenomena
agreed perfectly with the unambiguous predictions of theory. From the point
decidedly below 370 at which a regular effect was first obtained, there
was always one drop for each complete vibration of the fork, and a single
stream, every drop breaking away under the same conditions as its prede-
cessor. After passing 180 it becomes a question whether the octave of the
fork's note may not produce an effect as well as the prime. If this effect be
sufficient, the number of drops is doubled ; and unless the prime be very
subordinate indeed, there is a double stream, alternate drops taking sensibly
different courses. In these experiments the influence of the prime was
usually sufficient to determine the number of drops, even in the neighbour-
hood of pitch 128. Sometimes, however, the octave became predominant,
108 ON LIQUID JETS. [85
and doubled the number of drops. It must be remembered that the relative
intensities with which the two vibrations reach the jet depend upon many
accidental circumstances. The table has natural notes of its own, and even
the moving of a weight upon it may change the conditions very materially.
When the octave is not strong enough actually to double the drops, it often
produces an effect which is very apparent to an observer examining the
transformation through the revolving holes. On one occasion a vigorous
bowing of the fork, which favours the octave, gave at first a double stream,
but this after a few seconds passed into a single one. Near the point of
resolution those consecutive drops which ultimately coalesce as the fork dies
down, are connected by a ligament. If the octave is strong enough, this
ligament breaks and the drops are separated ; otherwise the ligament draws
the half-formed drops together, and the stream becomes single. The transi-
tion from the one state of things to the other could be watched with facility.
In order to get rid entirely of the influence of the octave a different
arrangement is necessary. It was found that the desired result could be
arrived at by holding a 128 fork in the hand over a resonator of the same
pitch resting on the table. The transformation was now quite similar in
character to that effected by a fork of frequency 256, the only differences
being that the drops were bigger and twice as widely spaced, and that the
sph&rule, which results from the gathering together of the ligament, was
much larger. We may conclude that the cause of the doubling of a jet by
the sub-octave of the note natural to it is to be found in the presence of the
second component, from which scarcely any musical notes are free.
When two forks of pitches 128 and 256 were sounded together, the single
or double stream could be obtained at pleasure by varying the relative
intensities. Any imperfection in the tuning is rendered very evident by
the behaviour of the jet, which performs evolutions synchronous with the
audible beats. This observation, which does not require the aid of the
revolving disk, suggests that the effect depends in some degree upon the
relative phases of the two tones, as might be expected a priori. In some
cases the influence of the sub-octave is shown more in making the alternate
drops unequal in magnitude, than in projecting them into very different
paths.
Returning now to the case of a single fork screwed to the table, it was
found that as the pitch was lowered below 128, the double stream was
regularly established. The action of the Twelfth below the principal note
(85) demands special attention. At this pitch we might in general expect
the first three components of a compound note to influence the result. If
the third component were pretty strong it would determine the number of
drops, and the result would be a threefold stream. In the case of a fork
screwed to the table the third component of the note must be extremely
85] ON LIQUID JETS. 109
weak, if not altogether missing ; but the second (octave) component is (airly
strong, and in fact determines the number of drops (190f). At the same
time the influence of the prime (85) is sufficient to cause the alternate drops
to pursue different paths, so that a double stream is observed.
By the addition of a 256 fork there was no difficulty in obtaining the
triple stream, but it was of more interest to examine whether it were
possible to reduce the double stream to a single one with only 85 drops per
second. In order to secure as strong and as pure a fundamental tone as
possible and to cause it to act in the most favourable manner upon the jet,
the air space over the water in the reservoir was tuned to the note of the
fork by sliding a piece of glass over the neck so as partially to cover it.
When the fork was held over the resonator thus formed, the pressure which
expels the jet was rendered variable with a frequency of 85, and overtones
were excluded as far as possible. To the unaided eye, however, the jet still
appeared double, though on more attentive examination one set of drops was
seen to be decidedly smaller than the other. With the revolving disk,
giving about eighty-five views per second, the real state of the case was
made clear. The smaller drops were the spherules, and the stream was
single in the same sense as the streams given by pure tones of frequencies
128 and 256. The increased size of the spherule is of course to be
attributed to the greater length of the ligament, the principal drops being
now three times as widely spaced as when the jet is under the influence of
the 256 fork.
With still graver forks screwed to the table the number of drops con-
tinued to correspond to the second component of the note. The double
octave of the principal note (64) gave 128 drops per second, and the influence
of the prime was so feeble that the duplicity of the stream was only just
recognisable. Below 64 the observations were not carried. Attempts to get
a single stream of 64 drops per second were unsuccessful, but it is probably
quite possible to do so with vibrations of greater power than I could
command.
In the case of a compound note of pitch 64 a considerable variety of
effects might ensue, according to the relative strengths of the various
components. Thus, the stream might be single (though this is unlikely),
double, triple, four-fold, or even five-fold, with a corresponding number of
drops.
Observations were next made on the effects of chords. For the chord of
the Fifth the pitches taken were 256 and f x 256. The two forks could be
screwed to the table and bowed, or, as is preferable (especially in the case of
the chords of the Fourth and Third to be spoken of presently), maintained in
vibration electromagnetically by a periodic current from a break-fork of pitch
85 , standing on another table. The revolving disk was driven at such a
110 ON LIQUID JETS. [85
speed as to give about eighty-five views per second. As was to be expected,
the number of drops was either 256 in a triple stream, or f x 256 in a double
stream, according to the relative intensities of the two vibrations. With the
maintained forks the phenomenon is perfectly under control, and there is
no difficulty in observing the transition from the one state of things to the
other.
In like manner with forks 256 and f x 256, driven by fork 64, and with
sixty-four views per second, the stream is either triple or quadruple ; and
with forks 256 and f x 256, we get at pleasure a four-fold or five-fold stream.
To obtain a good result the intervals must be pretty accurately tuned. In
the case of electrically maintained forks, the relative phase remains un-
changed for any length of time, and the spectacle seen through the revolving
holes is one of great beauty.
The actual results obtained experimentally by Plateau differ in some
respects from mine, doubtless in virtue of the more composite character of
the notes of the violoncello employed by him, but they are quite consistent
with the views above expressed. The only point as to which I feel any
difficulty relates to the single stream, which occasionally resulted from the
action of the Twelfth below the principal note. It seems improbable that
this could have been a single stream of the kind that I obtained with some
difficulty from a pure tone ; indeed the latter would have been pronounced
to be a double stream by an observer unprovided with an apparatus for
intermittent views. I should rather suppose that the number of drops
really corresponded to an overtone, and that from some accidental cause
the divergence of what would generally be separate streams failed to be
sensible.
The Length of the Continuous Part.
3. When a jet falls vertically downwards, the circumstances upon which
its stability or instability depend are continually changing, more especially
when the initial velocity is very small. The kind of disturbance to which
the jet is most sensitive as it leaves the nozzle is one which impresses upon
it undulations of length equal to about four and a half times the initial
diameter. But as the jet falls its velocity increases (and consequently the
undulations are lengthened), and its diameter diminishes, so that the degree
of instability soon becomes small. On the other hand, the kind of disturb-
ance which will be effective in a later stage is altogether ineffective in the
earlier stages. The change of conditions during fall has thus a protective
influence, and the continuous part tends to become longer than would be the
case were the velocity constant, the initial disturbances being unaltered.
I have made many attempts to determine the origin of the disturbances
which remain in operation when the jet is protected from ordinary tremors,
85]
ON LIQUID JETS.
Ill
but with little result. By suspending the reservoir with india-rubber straps,
&c., from the top of a wooden tripod, itself resting upon the stone floor of one
of the lower rooms of the Cavendish Laboratory, a considerable degree of
isolation was attained. A stamp of the foot upon the floor, or the sounding
of a note of suitable pitch of moderate intensity in the air, had no great
effect. Without feeling much confidence I rather incline to the opinion that
the residual disturbances are of internal origin. As the fluid flows up to the
aperture along the inner surface of the plate which forms the bottom of the
reservoir, eddying motions are almost certainly impressed upon it, and these
may very possibly be the origin of the ultimate disintegration. With the
view of testing this point, I arranged an experiment in which the velocity of
the fluid over the solid walls should be as small as possible.
AB (fig. 1) represents a large brass tube, to which a smaller one is soldered
at , suitable for india-rubber connexion. The bottom of the large tube
Fig. l.
consists of a carefully worked plate in which is a circular hole of inch
diameter. When the rubber tube is placed in connexion with the water
supply, a jet drops from A, and may be made exceedingly fine by regulation
of the pinch-cock C. By turning off the supply at C altogether, the jet at A
may be stopped, without emptying the vessel. The stability, due to the
capillary tension of the surface at A, preponderates over the instability due
to gravity. By this device it is possible to obtain a jet whose velocity is
acquired almost wholly after leaving the vessel from which it issues. In this
form of the experiment, however, the jet is liable to disturbance depending
upon the original velocity of the fluid as it passes through the comparatively
narrow rubber tube, and when I attempted a remedy by suspending a closed
reservoir (fig. 2), in which the water might be allowed first to come to rest,
other difficulties presented themselves. The air confined over the surface of
112
ON LIQUID JETS.
[85
the water acts as a spring, and the flow of water below tends to become
intermittent, when rendered sufficiently slow by limiting the admission of
air. A definite cycle is often established, air flowing in and water flowing
Fig. 2.
out alternatively at the lower aperture. The difficulty may be overcome by
careful manipulation, but there is no easy means of making an adequate
comparison with other jets, so that the question remains undecided whether
the residual disturbances are principally of internal or of external origin.
Collision of Two Resolved Streams.
4. In the case of a simple vertical fountain, when the scattering is
prevented by electricity, there is every reason to believe that the action is
differential, depending on a difference of potentials of colliding drops. The
principal electrification, however, of the successive drops must be the same ;
and thus, sensitive as it is, this form of the phenomenon is not by any means
the best calculated to render evident the smallest electrical forces. As was
shown in my former paper [Art. 59, vol. I. p. 374], it is far surpassed by
colliding jets, between which a difference of potential may be established, a
subject to which we shall return in 5. It is possible, however, to experiment
upon the collision of two distinct streams of drops, which are differently
if we please, oppositely electrified from the first. Apart from electrical
influence, the collision of such streams presents points of interest which have
been made the subject of examination.
Two similar brass nozzles, terminating in apertures about ^ inch in
diameter, were supplied from the same reservoir of water, and were held so
that the jets rising obliquely from them were in the same plane and crossed
each other at a moderate angle. The jets were resolved into regular series
of drops by the action of a 256 fork screwed to the table and set in action by
bowing. The periodic phenomenon thus established could be examined with
facility by intermittent vision through a revolving perforated disk ( 2), so
arranged that about 256 holes passed the eye per second.
85] OH LIQUID JETS. 113
When the angle of collision is small, the disposition of the files of drops
may be made such that they rebound without crossing, fig. 3 ; more often,
however, the drops shoulder their waj through after one or more collisions,
Kg. 3.
e o g 8 S o o
somewhat as in fig. 4. In both cases the presentation of an electrified body
to one place of resolution will determine the amalgamation of colliding drops,
with of course complete alteration of the subsequent behaviour. By judicious
Fig. 4.
-
management a feebly electrified body may be held in an intermediate position
between the two points of resolution so as not to produce the effect, con-
firming the view that the action is differentiaL
At a somewhat higher angle of collision amalgamation will usually occur
without the aid of electricity, but the feet may easily escape recognition
when intermittent vision is not employed. The streams do not usually join
into one, as we might perhaps expect, but appear to pass through one
another, much as if no union of drops had occurred. With the aid of the
revolving disk the course of things is rendered evident. The separating
layer is indeed ruptured at contact, and for a short time the drops move as
one mass. There is, however, in general, considerable outstanding relative
Fig. 5.
nil
, i
/
velocity, which is sufficient to bring about an ultimate separation, preceded
by the formation of a ligament (tig. 5). In certain cases, although after
contact a ligament is formed, the relative velocity is insufficient to overcome
5
114 ON LIQUID JETS. [85
its tension, and the drops draw again together and ultimately cohere. If the
impact is very direct, so that the relative velocity is almost entirely in the
line of centres, the drops may flatten against one another and become united
without the formation of a ligament.
In order to determine how small a difference of potential would be
effective in causing the coalescence of streams of drops meeting at a small
angle, the two places of resolution were enclosed in inductor-tubes, between
which with the aid of a battery a difference of potential could be established.
The arrangement is shown in fig. 6. One of the inductors is placed in
Fig. 6.
connexion with the earth, with the reservoir from which the water comes,
and with one pole of the battery. By operating a key, the other inductor
may be placed at pleasure in communication with the first inductor, or with
the other pole of the battery. In the first case the battery is out of use, and
in the second the difference of potential due to the battery is established
between the two inductors.
Experiment showed that the effect depends a good deal upon the exact
manner of collision. In almost all cases twenty cells of a De la Rue battery
sufficed to produce amalgamation, with subsequent replacement of the
original streams by a single one in a direction bisecting the angle between
the original directions. With a less battery power the result may be
irregular, some of the drops coalescing and others rebounding. When the
collisions are very direct, even four cells will sometimes cause a marked
transformation.
The complete solution of the problem of the direct collision of equal
spheres of liquid, though probably within the powers of existing mathe-
matical analysis, is not necessary for our purpose ; but it may give precision
to our ideas to consider for a moment the case of a row of equal spheres, or
cylinders, with centres disposed upon a straight line, and so squeezed
together that the distances between the centres must be less than the
original diameters. By the symmetry, the common surfaces are planes,
and the force between contiguous masses is found by multiplying the
85] ON LIQUID JETS. H5
area of the common surface by the internal capillary pressure. When
the amount of squeezing is small, the internal capillary pressure is ap-
proximately unaltered, and the force developed is simply proportional to
the area of contact. In the case of the cylinder the problem admits of
very simple solution, even when the squeezing is not small ; for, as is easily
seen, the free surfaces are necessarily semicircular, and thus the condition of
unaltered volume is readily expressed. It will of course be noticed that as
regards lateral displacements the equilibrium is unstable.
Collision of Streams before Resolution.
5. The collision of unresolved streams was considered in my former
paper. It appeared that the electromotive force of a single Grove cell, acting
across the common surface, was sufficient to determine coalescence, and that
the addition of a small quantity of soap made rebound impossible. More-
over, the "coalescence of the jets would sometimes occur in a capricious
manner, without the action of electricity or other apparent cause."
As in many respects this form of the phenomenon is the most instructive,
I was desirous of finding out the explanation of the apparent caprice, and
many experiments have been made with this object in view. The observa-
tions on fountains recorded in 1 having suggested the idea that the
accidental presence of greasy matter, removable by caustic potash, might
operate, this point was examined.
"JulyS, 1880.* Colliding Jets. Two large glass bottles, with holes in
the sides, close to the bottom, were fitted by means of corks with glass tubes,
drawn out to nozzles of about ^ of an inch in diameter. The bottles were
well rinsed with caustic potash, to remove any possible traces of grease, and
filled with tap water. The colliding jets coalesced in a manner apparently
entirely capricious, the only principle observable being that they coalesced
even more readily with high pressures (12 inches) than with low, and with
lower pressures would stand collision at greater angles. The addition of
caustic potash sufficient to give a very decided taste to the water produced
no apparent effect." Subsequently the water used was boiled with caustic
potash, but without success.
" July 27, 28, 29, 30. On the theory that when the jets collide without
uniting there is between them a thin film of air, which would be very liable
to be sucked up by water not saturated with air, we tried jets of water
through which a stream of atmospheric air had been passed for several hours.
We tried it three times. The first time the jets seemed very decidedly less
liable to unite capriciously. The second time they behaved even worse than
ordinary tap water usually does. The third time we thought it rather better
than tap water usually is, but not materially so."
* Mrs Sidgwick's Note Book.
82
116 ON LIQUID JETS. [85
Jets of hot water, and of mixtures of alcohol and water in various
proportions, were also tried at this time, but without obtaining any clue
as to the origin of the difficulty.
I had begun almost to despair of success, when a determined attempt to
conjecture in what possible ways one part of the stirred liquid could differ
from another part suggested the idea that the anomalies were due to dust.
"Aug. 1880. We tried dropping dust on to the colliding jets just above
the point of collision, and found that union was always produced. The
following powders were tried powdered cork, sand, lycopodium, plaster of
Paris, flowers of sulphur, sugar, dust that had accumulated upon a shelf, and
later emery and putty powder. The lycopodium was a little more uncertain
in its action than the others, but apparently only because, owing to its
lightness, it was difficult to ensure its falling upon the jets. Whenever we
were sure it did so, union followed."
When mixed with the water, powders acted differently. Emery and
putty powders were not effective, but sulphur caused immediate union.
Much probably depends upon the extent to which the extraneous matter
is wetted. A precipitate of chloride of silver, formed in the liquid itself,
seemed to be without influence.
Acting upon this hint, Mrs Sidgwick made an extended series of observa-
tions upon the behaviour of jets composed of water which had been allowed
to settle thoroughly, and which were protected from atmospheric dust. For
this purpose the jets were enclosed in a beaker glass, the end of which was
stopped by a plug of boxwood, fitted airtight. Through the plug passed
horizontally the two inclined glass nozzles, and underneath a bent tube
serving as a drain. The results, observed under these circumstances, were
such as to render it almost certain that dust is the sole cause of the
capricious unions. The protected jets of settled water were observed for
a total period of 246 minutes, during which the unions were at the average
rate of one in ten minutes. The longest intervals without unions were
thirty-four minutes and twenty-nine minutes. Comparative experiments
were made upon the behaviour of jets from the same nozzles under other
conditions. Thus jets of unsettled water, but protected from atmospheric
dust, united on an average twenty-four times in ten minutes. With un-
settled water the protection from atmospheric dust is not of much use, as
unprotected jets of the same water did not unite more than twenty-six times
in ten minutes. On the other hand, jets of settled water, not protected from
the atmosphere, united only twelve times in ten minutes. Although, no
doubt, somewhat different numbers might be obtained on repetition of these
experiments, they show clearly that the dust in the water is the more
frequent cause of union under ordinary circumstances, but that when this
is removed the atmospheric dust still exerts a powerful influence. The
ON LIQUID JETS. 117
difficulty of getting water free from dust is well known from Tyndall's
experiments, so that the residual tendency to unite under the most favour-
able conditions will not occasion surprise.
Although there is no reason to suppose that any other cause than dust
was operative in the above experiments, it remains true that very little
impurity of a greasy character will cause immediate union of colliding jets.
For this purpose the addition of milk at the rate of one drop of milk to
a pint of water [say one part in 10,000] is sufficient. It may be noticed too
that the effect of milk is not readily neutralised by caustic potash.
With respect to the action of electricity, further experiments have been
made to determine the minimum electromotive force competent to cause
union. The current from a Daniell cell was led through a straight length of
fine wire. One end of the wire was connected by platinum foil with the
liquid in aa insulated glass bottle, from which one of the jets was fed. The
glass bottle supplying the second nozzle was similarly connected with a
moveable point on the stretched wire. The electromotive force necessary to
cause union, as measured by the distance between the two fine wire contacts,
though definite at any one moment, was found to vary on different occasions,
possibly in consequence of forces having their seat at the surfaces of the
platinum oil From one-half to three-quarters of the whole force of the
Daniell was usually required.
With a view to further speculation upon this subject, an important
question suggests itself as to whether or not there is electrical contact
between colliding and rebounding jets. To solve this question it was only
necessary to introduce a fine wire reflecting galvanometer into the arrange-
ment just described, taking care that the electromotive forces employed fell
short of what would be required to cause the union of the jets. Suitable
keys were introduced for more convenient manipulation, and sulphuric acid
was added to the water, in order to make sure that absence of strong
galvanometer deflection could not be due merely to the high resistance
of the thin columns of water composing the jets. Repeated trials under
these conditions proved that so long as the jets rebounded their electrical
insulation from one another was practically perfect.
As to the explanation of the action of electricity in promoting union, it
would be possible to ascribe it to the additional pressure called into play by
electrical attraction of the opposed water-surfaces, acting as plates of a
condenser. But it appears much more natural to regard it as due rather to
actual disruptive discharge, by which the separating skin is perforated and
the equilibrium of the capillary forces is upset. A small electromotive force,
incapable of overcoming the insulation of the thin separating layer, is without
effect. [1900. See however Phil. Mag. XLVIU. p. 328, 1899.]
86.
ADDRESS TO THE MATHEMATICAL AND PHYSICAL SCIENCE
SECTION OF THE BRITISH ASSOCIATION.
[British Association Report, pp. 437441, 1882.]
IN common with some of my predecessors in this chair, I recognise
that probably the most useful form which a presidential address could
take, would be a summary of the progress of physics, or of some important
branch of physics, during recent years. But the difficulties of such a task
are considerable, and I do not feel myself equal to grappling with them.
The few remarks which I have to offer are of a general, I fear it may
be thought of a commonplace character. All I can hope is that they may
have the effect of leading us into a frame of mind suitable for the work
that lies before us.
The diversity of the subjects which come under our notice in this
section, as well as of the methods by which alone they can be adequately
dealt with, although a sign of the importance of our work, is a source of
considerable difficulty in the conduct of it. From the almost inevitable
specialisation of modern science, it has come about that much that is
familiar to one member of our section is unintelligible to another, and
that details whose importance is obvious to the one fail altogether to
rouse any interest in the mind of the other. I must appeal to the authors
of papers to bear this difficulty in mind, and to confine within moderate
limits their discussion of points of less general interest.
Even within the limits of those departments whose foundation is evi-
dently experimental, there is room, and indeed necessity, for great variety
of treatment. One class of investigators relies mainly upon reiterated
appeals to experiment to resolve the questions which appear still to be
open, while another prefers, with Thomas Young, to base its decisions as
far as possible upon deductions from experiments already made by others.
It is scarcely necessary to say that in the present state of science both
86] ADDRESS TO MATHEMATICAL SECTION OF BRITISH ASSOCIATION. 119
methods are indispensable. Even where we may fairly suppose that the
fundamental principles are well established, careful and often troublesome
work is necessary to determine with accuracy the constants which enter
into the expression of natural laws. In many cases the accuracy desir-
able, even from a practical point of view, is hard to attain. In manv
others, where the interest is mainly theoretical, we cannot afford to
neglect the confirmations which our views may derive from the com-
parison of measurements made in different fields and in face of different
experimental difficulties. Examples of the inter-dependence of measure-
ments apparently distinct will occur to every physicist. I may mention
the absolute determinations of electrical resistance, and of the amounts
of heat developed from electrical and mechanical work, any two of which
involve also the third, and the relation of the velocity of sound to the
mechanical and thermal properties of air.
Where a measurement is isolated, and not likely to lead to the solution
of any open question, it is doubtless possible to spend upon it time and
attention that might with advantage be otherwise bestowed. In such a
case we may properly be satisfied for a time with work of a less severe
and accurate character, knowing that with the progress of knowledge the
way is sure to be smoothed both by a better appreciation of the difficulties
involved and by the invention of improved experimental appliances. I
hope I shall not be misunderstood as underrating the importance of great
accuracy in its proper place if I express the opinion that the desire for
it has sometimes had a prejudicial effect. In cases where a rough result
would have sufficed for all immediate purposes, no measurement at all has
been attempted, because the circumstances rendered it unlikely that a high
standard of precision could be attained. Whether our aim be more or less
ambitious, it is important to recognise the limitations to which our methods
are necessarily subject, and as far as possible to estimate the extent to
which our results are uncertain. The comparison of estimates of uncer-
tainty made before and after the execution of a set of measurements may
sometimes be humiliating, but it is always instructive.
Even when our results show no greater discrepancies than we were
originally prepared for, it is well to err on the side of modesty in esti-
mating their trustworthiness. The history of science teaches only too
plainly the lesson that no single method is absolutely to be relied upon,
that sources of error lurk where they are least expected, and that they
may escape the notice of the most experienced and conscientious worker.
It is only by the concurrence of evidence of various kinds and from
various sources that practical certainty may at last be attained, and com-
plete confidence justified. Perhaps I may be allowed to illustrate my
meaning by reference to a subject which has engaged a good deal of my
120 ADDRESS TO THE MATHEMATICAL AND [86
attention for the last two years the absolute measurement of electrical
resistance. The unit commonly employed in this country is founded upon
experiments made about twenty years ago by a distinguished committee
of this Association, and was intended to represent an absolute resistance
of 10 9 . C.G.S., i.e. one ohm. The method employed by the committee at
the recommendation of Sir W. Thomson (it had been originally proposed
by Weber) consists in observing the deflection from the magnetic meri-
dian of a needle suspended at the centre of a coil of insulated wire. This
forms a closed circuit and is made to revolve with uniform and known
speed about a vertical axis. From the speed and deflection, in combina-
tion with the mean radius of the coil and the number of its turns, the
absolute resistance of the coil, and thence of any other standard, can be
determined.
About ten years later Kohlrausch attacked the problem by another
method, which it would take too long to explain, and arrived at the
result that the B.A. unit was equal to T02 ohms about two per cent,
too large. Rowland, in America, by a comparison between the steady
battery current flowing in a primary coil with the transient current de-
veloped in a secondary coil when the primary current is reversed, found
that the B.A. unit was '991 ohms. Lorenz, using a different method again,
found '980, while H. Weber, from distinct experiments, arrived at the
conclusion that the B.A. unit was correct. It will be seen that the
results obtained by these highly competent observers range over about
four per cent. Two new determinations have lately been made in the
Cavendish laboratory at Cambridge, one by myself with the method of the
revolving coil, and another by Mr Glazebrook, who used a modification of
the method followed by Rowland, with the result that the B.A. unit is
'986 ohms. I am now engaged upon a third determination, using a method
which is a modification of that of Lorenz.
In another important part of the field of experimental science, where
the subject-matter is ill understood, and the work is qualitative rather
than quantitative, success depends more directly upon sagacity and genius.
It must be admitted that much labour spent in this kind of work is ill-
directed. Bulky records of crude and uninterpreted observations are not
science, nor even in many cases the raw material out of which science
will be constructed. The door of experiment stands always open; and
when the question is ripe, and the man is found, he will nine times out
of ten find it necessary to go through the work again. Observations
made by the way, and under unfavourable conditions, may often give rise
to valuable suggestions, but these must be tested by experiment, in which
the conditions are simplified to the utmost, before they can lay claim to
acceptance.
86] PHYSICAL SCIENCE SECTION OF THE BRITISH ASSOCIATION. 121
When an unexpected effect is observed, the question will arise whether
or not an explanation can be found upon admitted principles. Sometimes
the answer can be quickly given; but more often it will happen that an
assertion of what ought to have been expected can only be made as the
result of an elaborate discussion of the circumstances of the case, and this
discussion must generally be mathematical in its spirit, if not in its form.
In repeating, at the beginning of the century, the well-known experiment
of the inaudibility of a bell rung tn vacua, Leslie made the interesting
observation that the presence of hydrogen was inimical to the production
of sound, so that not merely was the sound less in hydrogen than in air
of equal pressure, but that the actual addition of hydrogen to rarefied air
caused a diminution in the intensity of sound. How is this remarkable
fact to be explained ? Does it prove, as Herschel was inclined to think,
that a mixture of gases of widely different densities differs in its acous-
tical properties from a single gas ? These questions could scarcely be
answered satisfactorily but by a mathematical investigation of the process
by which vibrations are communicated from a vibrating solid body to the
surrounding gas. Such an investigation, founded exclusively upon prin-
ciples well established before the date of Leslie's observation, was under-
taken years afterwards by Stokes, who proved that what Leslie observed
was exactly what ought to have been expected. The addition of hydrogen
to attenuated air increases the wave-length of vibrations of given pitch,
and consequently the facility with which the gas can pass round the edge
of the bell from the advancing to the retreating face, and thus escape
those rarefactions and condensations which are essential to the formation
of a complete sound wave. There remains no reason for supposing that
the phenomenon depends upon any other elements than the density and
pressure of the gaseous atmosphere, and a direct trial, e.g. a comparison
between air and a mixture of carbonic anhydride and hydrogen of like
density, is almost superfluous.
Examples such as this, which might be multiplied ad libitum, show
how difficult it often is for an experimenter rightly to interpret his
results without the aid of mathematics. It is eminently desirable that
the experimenter himself should be in a position to make the calcula-
tions, to which his work gives occasion, and from which in return he
would often receive valuable hints for further experiment. I should like
to see a course of mathematical instruction arranged with especial refer-
ence to physics, within which those whose bent was plainly towards experi-
ment might, more or less completely, confine themselves. Probably a year
spent judiciously on such a course would do more to qualify the student
for actual work than two or three years of the usual mathematical cur-
riculum. On the other side, it must be remembered that the human
mind is limited, and that few can carry the weight of a complete mathe-
122 ADDRESS TO THE MATHEMATICAL AND [86
matical armament without some repression of their energies in other direc-
tions. With many of us difficulty of remembering, if not want of time for
acquiring, would impose an early limit. Here, as elsewhere, the natural
advantages of a division of labour will assert themselves. Innate dexterity
and facility in contrivance, backed by unflinching perseverance, may often
conduct to successful discovery or invention a man who has little taste
for speculation ; and on the other hand the mathematician, endowed
with genius and insight, may find a sufficient field for his energies in
interpreting and systematising the work of others.
The different habits of mind of the two schools of physicists sometimes
lead them to the adoption of antagonistic views on doubtful and difficult
questions. The tendency of the purely experimental school is to rely
almost exclusively upon direct evidence, even when it is obviously im-
perfect, and to disregard arguments which they stigmatise as theoretical.
The tendency of the mathematician is to overrate the solidity of his
theoretical structures, and to forget the narrowness of the experimental
foundation upon which many of them rest.
By direct observation, one of the most experienced and successful ex-
perimenters of the last generation convinced himself that light of definite
refrangibility was capable of further analysis by absorption. It has hap-
pened to myself, in the course of measurements of the absorbing power
of various media for the different rays of the spectrum, to come across
appearances at first sight strongly confirmatory of Brewster's views, and
I can therefore understand the persistency with which he retained his
opinion. But the possibility of further analysis of light of definite refran-
gibility (except by polarisation) is almost irreconcilable with the wave
theory, which on the strongest grounds had been already accepted by
most of Brewster's contemporaries ; and in consequence his results, though
urgently pressed, failed to convince the scientific world. Further experi-
ment has fully justified this scepticism, and in the hands of Airy, Helmholtz,
and others, has shown that the phenomena by which Brewster was misled
can be explained by the unrecognised intrusion of diffused light. The
anomalies disappear when sufficient precaution is taken that the refrangi-
bility of the light observed shall really be definite.
On similar grounds undulationists early arrived at the conviction that
physically light and invisible radiant heat are both vibrations of the same
kind, differing merely in wave-length; but this view appears to have been
accepted slowly, and almost reluctantly, by the experimental school *.
* [1900. The reader may refer to a paper on "The History of the Doctrine of Eadiant
Energy," Phil. Mag, xxvn. p. 265, 1889.]
86] PHYSICAL SCIENCE SECTION OF THE BRITISH ASSOCIATION. 123
When the facts which appear to conflict with theory are well defined
and lend themselves easily to experiment and repetition, there ought to
be no great delay in arriving at a judgment. Either the theory is upset,
or the observations, if not altogether faulty, are found susceptible of
another interpretation. The difficulty is greatest when the necessary con-
ditions are uncertain, and their fulfilment rare and uncontrollable. In
many such cases an attitude of reserve, in expectation of further evidence,
is the only wise one. Premature judgments err perhaps as much on one
side as on the other. Certainly in the past many extraordinary observa-
tions have met with an excessive incredulity. I may instance the fire-
balls which sometimes occur during violent thunderstorms. When the
telephone was first invented, the early reports of its performances were
discredited by many on quite insufficient grounds.
It would be an interesting, but too difficult and delicate a task, to
enumerate and examine the various important questions which remain
still undecided from the opposition of direct and indirect evidence. Merely
as illustrations I will mention one or two in which I happen to have
been interested. It has been sought to remedy the inconvenience caused
by excessive reverberation of sound in cathedrals and other large unfur-
nished buildings by stretching wires overhead from one wall to another.
In some cases no difference has been perceived, but in others it is thought
that advantage has been gained. From a theoretical point of view it is
difficult to believe that the wires could be of service. It is known that
the vibrations of a wire do not communicate themselves in any ap-
preciable degree directly to the air, but require the intervention of a
sounding-board, from which we may infer that vibrations in the air
would not readily communicate themselves to stretched wires. It seems
more likely that the advantage supposed to have been gained in a few
cases is imaginary than that the wires should really have played the part
attributed to them.
The other subject on which, though with diffidence, I should like to
make a remark or two, is that of Front's law, according to which the
atomic weights of the elements, or at any rate of many of them, stand
in simple relation to that of hydrogen. Some chemists have reprobated
strongly the importation of a priori views into the consideration of the
question, and maintain that the only numbers worthy of recognition are
the immediate results of experiment. Others, more impressed by the
argument that the close approximations to simple numbers cannot be
merely fortuitous, and more alive to the inevitable imperfections of our
measurements, consider that the experimental evidence against the simple
numbers is of a very slender character, balanced, if not outweighed, by
the a priori argument in favour of simplicity. The subject is eminently
124 ADDRESS TO MATHEMATICAL SECTION OF BRITISH ASSOCIATION. [86
one for further experiment; and as it is now engaging the attention of
chemists, we may look forward to the settlement of the question by the
present generation. The time has perhaps come when a redetermination
of the densities of the principal gases may be desirable an undertaking
for which I have made some preparations*.
If there is any truth in the views that I have been endeavouring to
impress, our meetings in this section are amply justified. If the progress
of science demands the comparison of evidence drawn from different sources,
and fully appreciated only by minds of different order, what may we not
gain from the opportunities here given for public discussion, and, perhaps
more valuable still, private interchange of opinion ? Let us endeavour, one
and all, to turn them to the best account.
* [1899. See Proc. Roy. Soc. XLHI. p. 356, 1888 ; L. p. 449, 1892 ; LIII. p. 134, 1893.1
87.
ON THE TENSION OF MERCURY VAPOUR AT COMMON
TEMPERATURES.
[British Association Report, p. 441, 1882.]
THE anther called attention to the difficulty of reconciling the values of
Regnault and Hagen with the phenomena observed by Crookes relating to
the viscosity of gases at high exhaustions. The total gaseous pressure in
the working chamber cannot be less than that of the mercury at the pump.
If the penetration of mercury vapour be prevented by chemical means, some
other gas must be present in equivalent quantity. If the value of Regnault
and Hagen is substantially correct, it does not appear how the phenomena [of
viscosity] could vary so much as they are observed to do at the highest
degrees of exhaustion as measured by the M c Leod gauge. The question
then arises whether the value of mercury tension hitherto received may not
be much in excess of the truth. In Hagen's researches it is assumed without
reason that the pressure in a chamber of variable temperature is governed by
the temperature of the coldest part, but this consideration tells in the wrong
direction. It was suggested that possibly a change in the capillary constant,
or currents in the fluid mercury at the chilled surface of the meniscus, might
have had something to do with the minute changes of level which have been
attributed to differences of pressure in the mercury vapour.
ON THE ABSOLUTE MEASUREMENT OF ELECTRIC CURRENTS.
[British Association Report, pp. 445, 446, 1882.]
THE accurate absolute measurement of currents seems to be more difficult
than that of resistance. The methods hitherto employed require either
accurate measurements of the earth's horizontal intensity, or accurate
measurements of coils of small radius and of many turns. If in the latter
measurement we could trust to the inextensibility of the wire, as some
experimenters have thought themselves able to do, the mean radius could be
accurately deduced from the total length of wire and the number of turns ;
but actual trial has convinced me that fine wire stretches very appreciably
under the tension necessary for winding a coil satisfactorily. Kohlrausch's
method, in which the same current is passed through an absolute galvano-
meter and through a coil suspended bifilarly in the plane of the meridian,
is free from the above difficulty ; but it is not easy so to arrange the propor-
tions that the suspended coil shall be sufficiently sensitive, and the galvano-
meter sufficiently insensitive. In this method, as in that of the dynamo-
meter, the calculation of the forces requires a knowledge of the moment of
inertia of the suspended parts.
When the electromagnetic action is a simple attraction or repulsion, it
can be determined directly by balancing it against known weights. In
Mascart's recent determination a long solenoid is suspended vertically in the
balance, and is acted upon by a flat coaxial coil of much larger radius, whose
plane includes the lower extremity of the solenoid. This arrangement,
though simple to think about, does not appear to be the one best adapted to
secure precise results. It is evident that a large part of the solenoid is
really ineffective, those turns which lie nearly in the plane of the flat coil
being but little attracted, as well as those which lie towards the further
extremity. The result calculated from the total length of wire (even if this
88] Olf THE ABSOLUTE MEASUREMENT OF ELECTRIC CURRENTS. 127
could be trusted), the length of the solenoid, and the number of tarns, has
an appearance of accuracy which is illusory unless it can be assumed that the
distribution of the wire over the length is strictly uniform. It would appear
that all the turns of the suspended coil should operate as much as possible,
that is, that the suspended coil should be compact, and should be placed in
the position of maximum effect.
There is a farther incidental advantage in this arrangement which it is
the principal object of the present note to point out. The expression for the
attraction involves as factors the product of the numbers of turns, the square
of the current, and a function of the mean radii of the two coils and of the
distance between their mean planes. Now, as may be seen from the feet
that the square of a current is already of the dimensions of a force, this
function of three linear quantities is itself of no dimensions. In determining
its actual value we should in general be subject to three errors; but when
the position is such that the function (for two given coils) is a maximum, the
result is practically dependent only upon the two mean radii, and being of no
dimensions can involve them only in the form of a ratio. In order then to
calculate the result, all that it is necessary to know with precision is the ratio
of the mean radii of the two coils. This ratio can be obtained electrically,
with full precision, and without any linear measurements. For, if the two
coils considered as galvanometer coils are brought coaxially into the same
plane, the ratio of their constants can be found by the known method of
dividing a current between them in such a way that no effect is produced
upon a small magnet suspended at their common centre. The ratio of the
resistances in multiple arc gives the ratio of the currents, and this again
(subject to small corrections for the finite size of the sections), gives the ratio
of the mean radii.
It appears that in this way all that is necessary for the absolute determi-
nation of currents can be obtained without measurements of length, or of
moments of inertia, or even of absolute angles of deflection. In practice it
will be desirable to duplicate the fixed coil, placing the suspended coil
midway between two similar fixed ones, through which the current passes in
opposite directions. A rough approximation to the condition of things
above described will be quite sufficient.
89.
ON THE DURATION OF FREE ELECTRIC CURRENTS
IN AN INFINITE CONDUCTING CYLINDER.
[British Association Report, pp. 446, 447, 1882.]
TAKING the axis of the cylinder as that of z, we suppose that the currents
are functions of V(&' 2 + 2/ 2 )> ur r > on ly> an d flow in the circles r = constant.
From the equations given in Maxwell's Electricity, vol. n. 591, 598, 607,
610, 611, we may deduce for a .conductor of constant ^
( fc & d?\ n dc
3-^ + jo + j^ c = 47m(7 -j- ,
\da? dy* dz 2 J dt
with similar equations for b and a.
In the present case the magnetic forces b and a vanish, and c is a function
of r only. Thus
/ d 2 I d
(d^ + -rd-
or, if c varies as e~ nt ,
the solution of which, subject to the condition of finiteness at the centre, is
c = AJ {x/(47r/LmCV } = A J (kr).
To determine the admissible values of n, we have only to form the
condition which must be satisfied at the boundary of the cylinder r = R. It
is evident that the magnetic force must here be zero, so that the condition is
} =0.
The roots of the function are,
2-404, 5-520, 8'654, 11-792, &c.
89] ON THE DURATION OF FREE ELECTRIC CURRENTS. 129
For the principal mode of longest duration
c = AJ (2-404 r/R),
and
2-404 2
If T be the time in which the amplitude sinks in ratio e : 1,
1
T =
n~ (2-404)* '
For copper in C.G.S. measure C= , ft , 9 , p.
and thus
T= 8oo nearly -
In order that T should be one second, the diameter of the cylinder would
have to be about two feet.
[1900. In the case of iron subjected to small magnetic forces we may
take (see Phil, Mag. voL XXIIL p. 235, 1887)
so that
P?
r = - nearly.]
90.
ON THE EQUILIBRIUM OF LIQUID CONDUCTING MASSES
CHARGED WITH ELECTRICITY.
[Philosophical Magazine, xiv. pp. 184186, 1882.]
IN consequence of electrical repulsion, a charged spherical mass of
liquid, unacted upon by other forces, is in a condition of unstable equi-
librium. If a be the radius of the sphere, Q the charge of electricity, the
original potential is given by
V=Q/a .
If, however, the mass be slightly deformed, so that the polar equation
of its surface, expressed by Laplace's series, becomes
r = a(l + F l + F,+ ... + F n + ...),
then
and the potential energy of the system reckoned from the equilibrium
position is
In actual liquids this instability, indicated by the negative value of P',
is opposed by stability due to the capillary force. If T be the cohesive
tension, the potential energy of cohesion is given by
If F n oc cos (pt + e), we have for the motion under the operation of both
sets of forces,
P a s
* See Proc. Roy. Soc. May 15, 1879 [vol. i. p. 400].
90] ON THE EQUILIBRIUM OF LIQUID CONDUCTING MASSES. &C. 131
If r> Q*16a* J , the spherical form is stable for all displacements.
When Q is great, the spherical form is unstable for all values of n below
a certain limit, the maximum instability corresponding to a great, bnt
still finite, value of n. Under these circumstances the liquid is thrown
out in fine jets, whose fineness, however, has a limit.
The case of a cylinder, subject to displacement in two dimensions only,
may be treated in like manner.
The equation of the contour being in Fourier's series
r=a(l + F t + ...+F u +...\
we find as the expression for the potential energy of unit length
Q being the quantity of electricity resident on length /..
The potential energy due to capillarity is
and for the vibration of tvpe n under the operation of both sets of forces.
The influence of electrical charge in diminishing the stability of a
cylinder for transverse disturbances may be readily illustrated by causing
a jet of water from an elliptical aperture to pass along the axis of an
insulated inductor-tube, which is placed in connexion with an electrical
machine. The jet is marked with a recurrent pattern, fixed in space,
whose wave-length represents the distance travelled by the water in the
time of one vibration of type n = 2*. When the machine is worked, the
pattern is thrust outwards along the jet, indicating a prolongation of the
time of transverse vibration. The inductor should be placed no further
from the nozzle than is necessary to prevent the passage of sparks, and
must be short enough to allow the issue of the jet before its resolution
into drops.
The value of T being known (81 C.GJ&). we may calculate what elec-
trification is necessary to render a small rain-drop of, say, 1 millimetre
diameter unstable. The potential, expressed in electrostatic measure, is
given by
V = Q /a, = V (16a,T) = 20.
The electromotive force of a Daniell cell is about "004: so that an
electrification of about 5000 cells would cause the division of the drop
in question.
[Ait. GO, voL L p. 377.]
91.
ON AN INSTRUMENT CAPABLE OF MEASURING THE
INTENSITY OF AERIAL VIBRATIONS.
[Philosophical Magazine, xiv. pp. 186, 187, 1882.]
THIS instrument arose out of an experiment described in the Proceedings
of the Cambridge Philosophical Society*, Nov. 1880, from which it appeared
that a light disk, capable of rotation about a vertical diameter, tends with
some decision to set itself at right angles to the direction of alternating
aerial currents. In Fig. I, A is a brass tube closed at one end with a glass
Fig. l.
plate B, behind which is a slit C backed by a lamp. D is a light mirror
with attached magnets, such as are used for reflecting-galvanometers, and
is suspended by a silk fibre. The light from the slit is incident upon
the mirror at an angle of 45, and, after reflection, escapes from the tube
through a glass window at E. It then falls upon a lens F, and throws
an image of the slit upon a scale G. At a distance DH, equal to DO,
the tube is closed by a diaphragm of tissue paper, beyond which it is
acoustically prolonged by a sliding tube 7.
When the instrument is exposed to sounds whose half wave-length is
equal to OH, H becomes a node of the stationary vibrations, and the
* See also Proc. Eoy. Soc. May 5, 1881, p. 110. [Art. 79, vol. n. p. 7.]
91] ON AN INSTRUMENT FOR MEASURING, &C. 133
paper diaphragm offers but little impediment. Its office is to screen the
suspended parts from accidental currents of air. At D there is a loop ; and
the mirror tends to set itself at right angles to the tube under the
influence of the vibratory motion. This tendency is opposed by the magnetic
forces ; but the image upon the scale shifts its position through a distance
proportional to the intensity of the action.
As in galvanometers, increased sensitiveness may be obtained by com-
pensating the earth's magnetic force with au external magnet. Inasmuch,
however, as the effect to be measured is not magnetic, it is better to
obtain a small force of restitution by diminishing the moment of the
suspended magnet rather than by diminishing the intensity of the field in
which it works. In this way the zero will be less liable to be affected by
accidental magnetic disturbances.
So far as I have tested it hitherto, the performance of the instrument is
satisfactory. What strikes one most in its use is the enormous disproportion
that it reveals between sounds which, when heard consecutively, appear to be
of the same order of magnitude.
92.
COMPARISON OF METHODS FOR THE DETERMINATION OF
RESISTANCES IN ABSOLUTE MEASURE.
[Philosophical Magazine, xiv. pp. 329346, 1882.]
AT the present time, and in view of the projected conference at Paris,
the subject of the present paper is engaging a large share of attention ;
and Prof. G. Wiedemarm* has published an interesting discussion of some
of the methods that have been employed. I have thought it might be
of service if I also were to place upon record the views that I have been
led to entertain, and which are the result of a good deal of experience.
Resistance being of the dimensions of velocity, its absolute measurement
involves the absolute measurement of a length and of a time. The latter is
usually the time of a vibration of a suspended magnet, and it can be
determined without much difficulty. In the B.A. method it is the time of
rotation of the revolving coil, and it can be obtained with all desirable
accuracy. In this respect there is not much to choose between one method
and another ; but when we come to consider the manner in which the linear
measurement enters, important differences reveal themselves. These will be
discussed in detail presently ; but for the moment it will be sufficient to say
that the presumption is in favour of any method which requires only a
single linear measurement. It is true that this question cannot be decided
without regard to the subject of the measurement; but, with scarcely an
exception, it is necessary to know the mean radius of a coil of several layers
of insulated wire. This is apparently the measurement which fixes the limit
of final accuracy ; and, in comparison with it, determinations of the distances
of mirrors and scales &c. are of secondary difficulty.
* "Ueber die bisherigen Methoden zur Feststellung des Ohm." Separatabdruck aus der
Electrotechnischen Zeitschrift, July 1882. Phil. Mag. for October, p. 258.
92] DETERMINATION OF RESISTANCES IN ABSOLUTE MEASURE. 135
It will be convenient now to enumerate the principal methods which have
been proposed for determining absolute resistances. Minor details, which
are not likely to influence the final value of the results, will in general be
passed over.
I. Kirchhoff's Method, Maxwell's Electricity and Magnetism, 759.
The magnitude of a continuous battery-current in a primary coil is
compared with that of the transient current induced in a secondary coil
when the primary circuit is removed. Rowland* effected an important
improvement by simply reversing the battery-current without motion of the
primary coil. The time of vibration of the ballistic galvanometer employed
for the transient current is the principal time-measurement. In Rowland's
investigation a second galvanometer was employed for the battery-current,
and the ratio of constants had to be found by auxiliary experiments. In
Glazebrook'sf recent determination by this method only one galvanometer
was used, the battery-current being reduced in a known manner by shunting.
It is shown that the evaluation of the resistance -ratios presents no serious
difficulty.
Let h denote the ratio in which the primary current is reduced when it
produces a deflection a upon the galvanometer, Q the throw from rest due
to the induction-current when the battery is reversed, T the time of vibration
of the needle measured from rest to rest, M the coefficient of induction ;
then the resistance of the secondary circuit in absolute measure is given by
D TT M tan a ,
jB = TTsinp^ A -
Whenever, as in this method, the conductor whose resistance in absolute
measure is first determined is composed of copper, frequent comparisons
are necessary with standards of German silver or platinum-silver. Other-
wise a variation of temperature of about \ of a degree Cent., which can
hardly be detected with certainty by thermometers, would influence the
result by as much as one part in a thousand.
If it be granted that the comparison of currents and the reference to the
standard of resistance can be effected satisfactorily, we have only to consider
the amount of error involved in the determination of M, the coefficient of
mutual induction between the two circuits, which is the fundamental linear
measurement. If the two coils are of very nearly the same size, it appears
from symmetry that the result is practically a function of the mean of
the mean radii only, and not of the two mean radii separately. It is also
of course a function of the distance between the mean planes b. Leaving
out of consideration the small corrections necessary for the finite size of the
* American Journal, vol. xv. 1878.
t Proc. Boy. Soc. June 1882.
136
COMPARISON OF METHODS FOR THE
sections, we consider M as equal to 4fir^(Aa) multiplied by the function
of 7 given in tables appended to the second edition of Maxwell's Electricity,
where
or, if we identify A and a with their mean (A ),
tan 7 = 2,4 /6.
The error in M will depend upon the errors committed in the estimates
of A and 6. If we write
dM
dA db
then, since M is linear,
Thus, if b were great relatively to A , \ = 4, /* = 3,
a very unfavourable arrangement, even if it did not involve a great loss
of sensitiveness. The object must be so to arrange matters that the errors
in A and b do not multiply themselves unnecessarily in M. But since //, is
always negative, X must inevitably be greater than unity.
The other extreme case, in which 6 is very small relatively to A , may
also be considered independently of the general tables; for we may then
take approximately (Maxwell's Electricity, 705)
whence
\og(8A /b)-2'
showing that as b diminishes JJL approaches zero, and accordingly X approaches
unity, as is indeed otherwise evident. But when 6 is small, it is the absolute
error db which we must regard as given rather than the relative error db/b ;
and thus we are directed to stop at a moderate value of b, even if the
increased correction necessary for the size of the sections were not an
argument in the same direction.
The following intermediate cases, calculated by the tables, will give an
idea of the actual conditions suitable for a determination by this method :
7
b!2A
X
M
M
60
577
2-61
-1-61
316
70
364
2-18
-1-18 -597
75
268
1-98
-0-98 '829
80
176
1-76
-0-76
1-186
92] DETERMINATION OF RESISTANCES IN ABSOLUTE MEASURE. 137
We may say that the error in the distance of mean planes will reproduce
itself something like proportionally in the final result, and that the error
of mean radius will be doubled.
Any uncertainty in the actual position of the mean planes relatively
to the rings on which the wire is wound may be eliminated, as Glazebrook
has shown, by reversing the rings relatively to the distance-pieces.
This method is subject to whatever uncertainty attaches to the use of a
ballistic galvanometer*. In its favour it may be said that the apparatus and
adjustments are simple, and that no measurement of distances between
mirrors and scales is necessary for the principal elements. It should be
noticed also that the error due to faulty determination of the distance of
mean planes can be eliminated in great measure by varying this quantity,
which can be done over a considerable range without much difficulty or
expense.
With reference to the capabilities of the method for giving results of the
highest accuracy when cairied out in the most ambitious manner, it is
important to consider the effect of increasing the size of the coils. The coils
used by Glazebrook have a mean radius of about 26 centim.; the axial and
radial breadths of the section are each about 2 centim. If we suppose
the mean radius and the sides of the section to be doubled, the number
of turns (about 800) remaining unaltered, the sensitiveness would be
increased both by the doubling of M and by the diminished resistances
of the coils, while at the same time the subjects of the linear measurements
would be of more favourable magnitudes. To enhance the latter advantage,
it would probably be an improvement to diminish the radial breadth of
the section, on which much of the uncertainty of mean radius depends. In
either case it is clear that the limit of accuracy obtainable by this method
has not yet been reached.
II. Weber's Method by Transient Currents, Maxwell 760.
" A coil of considerable size is mounted on an axle so as to be capable of
revolving about a vertical diameter. The wire of this coil is connected
with that of a tangent-galvanometer so as to form a single circuit. Let
the resistance of this circuit be jR. Let the large coil be placed with its
positive face perpendicular to the magnetic meridian, and let it be quickly
turned round half a revolution. There will be an induced current due to the
earth's magnetic force ; and the total quantity of electricity in this current
in electro-magnetic measure will be
'-
* See Phil. Tran*. 1882, p. 669. [Art. 80, vol. n. p. 48.]
138 COMPARISON OF METHODS FOR THE [92
where g^ is the magnetic moment of the coil for unit current, which in the
case of a large coil may be determined directly by measuring the dimensions
of the coil and calculating the sum of the areas of its windings ; H is the
horizontal component of terrestrial magnetism ; and R is the resistance
of the circuit formed by the coil and galvanometer together. This current
sets the magnet of the galvanometer in motion."
"If the magnet is originally at rest, and if the motion of the coil
occupies but a small fraction of the time of a vibration of the magnet,
then, if we neglect the resistance to the motion of the magnet, we have,
by 748,
where G is the constant of the galvanometer, T is the time of vibration
of the magnet, and 6 is the observed elongation. From these equations
we obtain
R -
~
The value of H does not appear in this result, provided it is the same at the
position of the coil and at that of the galvanometer. This should not be
assumed to be the case, but should be tested by comparing the time of
vibration of the same magnet, first at one of these places, and then at
the other."
If a be the mean radius of the coil of the inductor and A that of the
galvanometer, we may write, neglecting the corrections for the finite sizes
of the sections*,
g = Tra 2 , G = 2-rr/A ;
so that
gG = 2-n*a*/A.
This is the linear quantity of the method. With respect to the chances
of error in determining it, we see that the error of the mean radius of the
inductor enters doubly, and that of the mean radius of the galvanometer
enters singly. Probably in this respect there is not much to choose between
this method and the use in method I. of the same coils placed at a moderate
distance apart.
A colossal apparatus for the use of the present method has been con-
structed and tested by MM. W. Weber and F. Zollnerf, the coils of which
are as much as 1 metre in diameter. The principal difficulty arises in
connexion with the galvanometer-magnet. Two magnets were used whose
* [1899. The factors expressive of the number of convolutions in the two coils are here
omitted.]
t Per. d. Kon. Sachs. Ges. zu Leipzig, 1880, vol. n. p. 77.
92] DETERMINATION OF RESISTANCES IN ABSOLUTE MEASURE. 139
lengths were respectively 200 millim. and 100 millim.; and the results
obtained in the two cases differed by as much as 2 per cent. The dis-
crepancy is doubtless due to the influence of the finite length of the magnets
causing the magnetic poles to be sensibly distant from the centre of the coil,
for which point the effects are calculated : and the disturbance will be
proportional to the square of the distance between the poles, or more
properly to the " radius of gyration " of the ideal magnetic matter about
the axis of rotation. But to assume that the disturbance from this source
was exactly four times as great in the one case as in the other, and thence to
deduce the result corresponding to an infinitely short magnet, appears to me
to be a procedure scarcely consistent with the degree of accuracy aimed at.
If this method is to give results capable of competing with those obtainable
in other ways, it will be necessary to use a much shorter magnet ; or, if that
is not practicable, to devise some method by which the distance of the poles
can be determined and a suitable correction calculated.
In cam-ing out the observations in the usual manner, it is necessary
to measure the distance between a mirror and a scale. By using a double
mirror with two scales and telescopes, MM. Weber and Zollner avoid the
principal cause of difficulty, i.e. the unsteadiness of the suspended mirror,
all that is then necessary to know with accuracy being the distance between
the two scales.
In using this and the three following methods great pains must be taken
with the levelling of the earth inductor, since the deviation of the axis of
rotation from the vertical (at least in the plane of the meridian) gives rise to
an error of the first order with (in these latitudes) a high coefficient. In this
respect it would be a decided advantage to carry out the experiments in
a locality nearer to the magnetic equator (see " Account of Experiments
to determine the Value of the B.A. Unit in Absolute Measure," Phil. Trans.
1882) [vol. n. p. 63]. It is to be hoped that the measurements commenced
by Weber and Zollner will be carried to a successful issue, as it is only
by the coincidence of results obtained by various methods that the question
can be satisfactorily settled. At present no value in absolute measure
of the B.A. unit or of the Siemens unit has been published as the result
of their work.
III. Method of Revolving Coil.
This method, first, it would appear, suggested by Weber, was carried into
execution by the celebrated Electrical Committee of the British Association*,
and more recently by myself with the assistance of Dr Schuster and others f.
The greater part of what I have to say upon this subject has been put
* Brit. Aaoc. Reports, 1862-1867. Reprint, Spon, 1873.
t Proc. Roy. Soc. May 1881, Feb. 1882; Phil. Tram. 1882. [Arts. 79, 80.]
140 COMPARISON OF METHODS FOR THE [92
forward already in the papers referred to, from which alone the reader
can form a complete opinion on the merits or demerits of the method as
hitherto practised. On the present occasion I must take many of the
conclusions there arrived at for granted, or at most give a mere indication
of the nature of the arguments by which they may be supported.
Method III. differs from II. mainly in the fact that in III. the earth-
inductor is, so to speak, its own galvanometer, the needle whose deflections
measure the currents being suspended at the centre of the revolving coil
itself instead of at the centre of another galvanometer-coil forming part
of the same circuit. If, as in II., the inductor-coil were simply twisted
through 180 when the needle passes its position of equilibrium, the dis-
advantages of the simplification would probably preponderate over the
advantages. The diminution of effect due to the oblique position of the
coil relatively to the needle (except at the moment of passing the magnetic
meridian) would indeed be compensated by the diminished resistance of
the complete circuit, and, as will presently appear, considerable advantage
would arise in respect of errors in the measurement of the coil ; but
an almost fatal uncertainty would be introduced from the influence of
self-induction.
The important advantage of III., obtained, as I believe, without any
really important sacrifice, arises only when the inductor is set into uniform
rotation. In II., if the connexions were maintained without a commutator,
the current in the galvanometer-coil would be alternating, and therefore
unsuitable for measurement with a magnetic needle ; but in III., although
the current in the coil itself alternates, the reversal of the coil relatively to
the needle causes all the impulses to operate finally in the same direction.
When, therefore, the coil is caused to revolve in a periodic time small
relatively to that of the free vibration of the needle, a steady deflection
is obtained which varies inversely with the absolute resistance of the coil.
If we omit for the moment all secondary considerations, although some
of them may not be without importance, the formula by which the resistance
(R) of the revolving circuit is given in terms of the mean radius (a), the
number of turns (ri), the angular velocity of rotation (&>), and the angle of
deflection (<), runs
R = 7r 2 w 2 a&) cot 0;
from which it appears that, in respect of errors arising from the measure-
ments of the coil, this method is much superior to those hitherto discussed.
There is only one linear quantity concerned ; and the error committed in its
determination enters but singly into the final result. Indeed we may say
that in this respect no improvement is possible, unless it be in the direction
of substituting for the mean radius of a coil of several layers some other kind
of linear quantity more easy to deal with.
92] DETERMINATION OF RESISTANCES IN ABSOLUTE MEASURE. 141
In requiring the absolute measurement of angle, II. and HI. stand
precisely upon a level.
The time of vibration in the experiments of MM. Weber and Zollner was
17 seconds or 30 seconds none too long relatively to the time (2 seconds)
occupied in turning the inductor. If we suppose the coil to be uniformly
rotated at the rate of, say, 2 revolutions per second, there would be 68 or 120
impulses upon the needle in the time of 1 vibration. It would no doubt be
a great exaggeration to represent the increase of sensitiveness as being in
anything like this proportion, since by the method of recoil it is possible to
make several observations of impulses during the time required for one
observation of steady deflection. Nevertheless it cannot be doubted that the
advantage of IIL in respect of sensitiveness is very considerable.
Experience has shown that there is no difficulty in controlling and
measuring the rotation of the coil ; but of course some auxiliary apparatus
is required for the purpose. Against this may be set the escape from
observations of the time of vibration : and from any uncertainty which may
attach to the ballistic use of a galvanometer-needle. The suspended magnet
may easily be made of such dimensions that no appreciable error can arise
from supposing it to be infinitely small.
On the other hand, some new complications enter in method III. which
I desire to state in full. In the first place we have to take account of the
fact that the inductor moves in a field of force due not only to the earth, but
also to the suspended magnet itself. I do not think that the correction thus
rendered necessary (about 4 parts per thousand in my experiments) adds in
any appreciable degree to the uncertainty of the final result ; but we may
take note of the fact that an auxiliary determination must be made of
the ratio of the magnetic moment of the suspended magnet to the earth's
horizontal force.
If the metal ring on which the wire is wound be on a large scale and
sufficiently massive for strength, currents may be developed in it, even
although it is divided into two parts by ebonite insulation. In my
experiments the effect of these currents was very sensible, and had to be
allowed for by careful observations of the deflection produced when the ring
was rotated with wire circuit open. In any future repetition it will be
worthy of consideration whether the ring should not be formed of less
conducting material. It does not appear, however, that the final result can
be prejudicially influenced ; and the effect produced by secondary closed
circuits allows us to verify the insulation of contiguous layers or turns of the
wire by comparing the deflections obtained before the wire is wound with
those obtained after winding, but with main circuit open, any difference
being due to leakage.
142 COMPARISON OF METHODS FOR THE [92
But the most serious complication in method III., and one which in the
eyes of some good judges weighs strongly against it, is the disturbing
influence of self-induction. With respect to this, the first point to be
noticed is that the action is perfectly regular, and that the only question
which arises is whether its magnitude can be determined with such accuracy
that the final result does not suffer. Now the operation of self-induction
is readily submitted to calculation if a certain coefficient (L) be known.
We find
R = 7r 2 w 2 aw cot <f> [I - U tan 2 <j> - U 2 tan 4 <},
where U is a numerical quantity dependent upon L, so that the influence
of self-induction is approximately proportional to the square of the speed
of rotation. The same law applies also to any disturbances depending
upon mutual induction between the wire circuit and subordinate circuits
in the ring.
It will be seen that, if the law of squares may be depended upon, the
influence of self-induction (and mutual induction) can be satisfactorily
eliminated by combining observations taken at different speeds. In my
experiments four speeds were used, of which the greatest and the least were
in the ratio of 2 : 1. The effect of self-induction was therefore four times as
great at the high speed as at the low speed. In other words, the quantity
(about 1 per cent.) by which the low-speed result is to be corrected in order
to eliminate the influence of self-induction is only one-third of the dis-
crepancy between the uncorrected results of the extreme speeds. If, there-
fore, the observations are good for anything at all, they are good enough
to determine this correction with all desirable precision. If a check be
considered necessary, it is supplied by the results of the intermediate
speeds.
The above reasoning proceeds upon the supposition that we have no
independent knowledge of the magnitude of the coefficient U. In point
of fact, this coefficient can be calculated with considerable accuracy from the
data of construction, so that the empirical correction is applied only to a
small outstanding residue.
In considering the disadvantageous influence of self-induction as an
argument in favour of II. as against III., we must remember that the
magnitude of the influence can be greatly attenuated by simply diminishing
the speed of rotation. At half the lowest speed above spoken of, for which
the correction for self-induction would be reduced to | per cent., the
deflection (over 100 millim. at a distance of 2670 millim.) would probably
correspond to a much greater sensitiveness than it is possible to obtain
under II. If we prefer the higher speed, it is because we estimate the
advantage of doubled sensitiveness as outweighing the disadvantage of a
fourfold correction for self-induction.
92] DETER3HXATIOX OF RESISTANCES IX ABSOLUTE MEASURE. 143
The fourth objection which may be taken to this method, and H is one
from which IL is free, lies in the necessary creation of mechanical dis-
turbance in the neighbourhood of the suspended magnet.
How far these complications may be supposed to prejudice the result of
carefully conducted experiments must be left to the estimation of the reader
of my paper, in which very full data for a judgment are given. Mv own
opinion is, that while in the aggregate they most be allowed to have some
weight, they are far from preponderating over the advantages which the
method pussesses in comparison with IL
If we take the view that the method itself is trustworthy, the principal
error will arise in connexion with the mean radius of the coil ; and it
becomes an interesting question to consider whether advantage may be
expected from a further increase in the dimensions of the apparatus. For
this purpose we may regard tan <f> as given. The total resistance R will be
proportional to ri*a f S, where S denotes the aggregate section of the copper,
from which it follows that S may be regarded as given, while a is left
undetermined by the consideration of sensitiveness. Thus, if we retain &
and .S unaltered in a magnified apparatus, we shall have the same sensitive-
ness as before, while the increased diameter of the coil and the relatively
decreased dimensions of the section will conduce to a more accurate de-
termination of the mean radius.
The angular deflection being given, the correction for self-induction is
nearly constant whatever may be the proportions of the coil
If we are of opinion that there is danger in the operation of self-
induction, the case becomes strong for the introduction of a second coil
in a plane perpendicular to that of the first*. By this means the relative
correction for self-induction would be reduced to one quarter, while the
deflection remained unaltered. It scarcely needs to be remarked that
this use of a second coil would not, as in IL, increase the uncertainty
depending upon the linear measurements, the two mean radii entering into
the result as parts, and not as factors.
This combination would lend itself especially well to low speeds of
rotation; for the deflecting force, being uniform in respect to time, would
not give rise to forced vibrations of the needle. The latter would have
nothing further to do than to indicate the direction of a constant field
of force.
IV.
This method, which was proposed by Fostert, and more recently by
Lippmann, and to a certain extent executed by the former, is a modification
* Prof. Boy. Soe. May 1881, p. 133 [rot n. p. 19].
t Brit. Aaoe. Report, 188L
144 COMPARISON OF METHODS FOR THE [92
of III., in which the electromotive force generated during the rotation of the
inductor is balanced by an external electromotive force, and thus not allowed
to produce a current. The external electromotive force is due to the
passage of a battery-current through certain resistance-coils; and the
current is compared with the earth's horizontal intensity (H) by an absolute
tangent-galvanometer. The difference of potential at the two points of
derivation is thus known in terms of the included absolute resistance (R)
and H. The circuit is continued through a sensitive galvanometer and the
coil of the inductor, and is closed only when the latter coil is nearly in the
plane of the meridian. When balance is obtained, the electromotive force of
induction n . TTO? . H . a> is equal to (RH/G) tan a, where is the constant
of the tangent-galvanometer and a the angle of deflection. The result,
from which H disappears, if it may be assumed to be the same in the two
places, is thus
R = nira?G . w . cot a,
or, if A be the mean radius of the galvanometer-coil,
R = 2tt7r 2 < cot a . a? I A,
from which the value of the resistance-coils is obtained in absolute measure.
One advantage of this method, which it shares with VI. below, is that the
resistance immediately expressed may be that of well-constructed coils of
German silver or of platinum-silver at a known temperature.
This method is nearly free from the secondary objections to III. discussed
above. The self-induction of the revolving wire-circuit does not enter, as no
appreciable current is allowed to form itself; but there would appear to be a
possibility of disturbance from mutual induction between the wire-circuit
and secondary circuits in the ring. It would certainly be necessary to
prevent the flow of currents round the ring by the insertion of an insulating
layer; and even with this precaution some control in the way of a variation
of speed would almost be necessary. Again, it is a question whether
disturbance from thermo-electricity for instance, may not arise at the place
where the contacts are made and broken.
It is to be hoped that a complete series of observations may be made
by this method, which certainly possesses considerable merits ; but at best
it remains open to the objection mentioned under II., with which in this
respect it stands upon a level, i.e. that errors may enter from the measure-
ments of both coils, the error of A entering singly into the result, and that of
a entering doubly.
In respect of requiring absolute measurements of angle, there is nothing
to choose between II., III., IV., and V.
92] DETERMINATION OF RESISTANCES IN ABSOLUTE MEASURE. 145
V. Weber's Method by Damping.
This is the method followed by Kohlrausch* in his investigations upon
this subject. It is founded upon II. ; but in order to avoid the difficulty
arising from the necessity of using a magnet small relatively to the coil
in which it is suspended, no attempt is made to determine the constant from
the data of construction. The inductor is connected with a sensitive gal-
vanometer, and the constant of the latter is deduced from observations of the
logarithmic decrement of the vibrations of the magnet when the circuit is
closed (A), and when it is open (\). The result, however, involves H the
horizontal intensity, K the moment of inertia of the needle, as well as
the time of vibration T. Expressed roughly, in the notation previously
employed, it is
AB
where R is the resistance of the circuit composed of the inductor and
galvanometer, A and B are the arcs of vibration in the method of recoil.
Interesting as this method is in some respects, I cannot but agree with
Rowland in thinking that the final formula is enough to show that it cannot
compete with others on equal terms, if the object be to obtain a result of
high accuracy. The horizontal intensity itself is perhaps nearly as difficult
to determine as absolute resistance: and the error thence arising doubles
itself in the result. There is in addition the error of K. But even if H and
K were not subject to error at all, I believe that the occurrence of the fourth
power of the radius of the inductor is a fatal defect, and tends to explain
the discrepant result obtained by Kohlrausch-h It is also worthy of note
that the error of levelling enters twice as much as in IL, III., and IT.
VI. Lorenz's Method.
This method, which, with the introduction of certain modifications not
affecting its essential character, I am disposed to consider the best of all, was
proposed and executed by Lorenz, of Copenhagen, in 1873*. A circular
disk of metal, maintained in rotation about an axis passing through its
* Pogg. Am. Erginzungsband TI; Phil. Hag. 1874, April and May.
t Oct. 1882. It i* very satiafrctory to note that Kohlrauseh (G5tt. G. Sept. 1882) has
recently detected an error in the value of the area of the windings of the inductor assumed
in his previous calculations. Introducing the new value, obtained by an electrical process
analogous to that described in Maxwell's Electricity, 754, he finds
1 B. A. unit= -990 x HP.
+ Pogg. Amu. voL czuz. p. 251.
B. IL 10
146 COMPARISON OF METHODS FOR THE [92
centre at a uniform and known rate, is placed in the magnetic field due
to a battery-current which circulates through a coaxal coil of many turns.
The revolving disk is touched near its centre and circumference by two wires.
If the circuit were simply closed through a galvanometer, the instrument
would indicate the current due to the electromotive force of induction acting
against the resistance of the circuit. The electromotive force corresponding
to each revolution is the same as would be generated in a single turn of wire
coincident with the circumference of the disk by the formation or cessation
of the battery-current. If this be called 7, and M be the coefficient of
induction between the coil and the circumference, m the number of revo-
lutions per second, the electromotive force is mMy. For the present
purpose, however, the circuit is not simply closed, but its terminals are
connected with the extremities of a resistance R through which the battery-
current flows, and the variable quantities are so adjusted that the electro-
motive force .fry exactly balances that of induction. When the galvanometer
indicates no current, the following relation, independent, it will be observed,
of the magnitude of the battery-current, must be satisfied,
and from this, M being known from the data of construction, the absolute
resistance R of the conductor is determined.
It will be seen that this method has pretty close affinity to I. The
secondary circuit is here, in a sense, reduced to a single turn, or rather
to as many turns as the disk makes revolutions in a time comparable with
the time of swing of the ballistic galvanometer; but the disadvantage of
a reduced number of turns is probably more than compensated for by the
continuous character of the induced current, which allows of its being
brought into direct opposition to that of the battery. During the months
from April to August of the present year I have been occupied in carrying
out a determination by this method. Space will not permit of a detailed
consideration of the various questions which presented themselves; and I
must content myself with a brief statement of the procedure, and with such
a discussion of the sources of error as will allow a comparison of this method
with others. I hope shortly to communicate a detailed paper upon the
subject to the Royal Society*.
One of the principal difficulties to be overcome arises from the exceeding
smallness of the resistance R, less than ^ B.A. in my experiments.
Lorenz employed an actual column of mercury of known dimensions, so
that the result is given at once in terms of mercury. I had intended to
follow the same course, but, after some trials, came to the conclusion that
there would be difficulties in the way of thus obtaining the degree of
* [See Phil. Trans. 1883; Art. 94 below.]
92] DETERMINATION OF RESISTANCES IN ABSOLUTE MEASURE. 147
accuracy aimed at, and ultimately adopted a method of shunting. The
main current from the battery was divided into two parts, the larger of
which passed through a resistance of half a unit, formed by combining two
singles in multiple arc. The resistance traversed by the other part of the
main current was much larger (from 10 to 20) ; and it was to two points on
this branch distant -^ that the wires of the derived circuit were connected.
With proper precautions this arrangement was found satisfactory, and the
equivalent resistance R could be accurately expressed in terms of the
standard B.A. units. The adjustment for obtaining the balance was effected
by varying a large resistance placed in multiple arc with one of the others ;
or rather two effective resistances were used, one on either side of that
required for balance, the latter being finally calculated by interpolation
from the indications of the galvanometer.
By observing only the effect of reversing the battery-current the results
are freed from the influence of terrestrial magnetism, and from the very
sensible thermoelectric force having its seat at the sliding contact. These
contacts were made by means of brushes of copper wire. One brush pressed
against the cylindrical edge of the disk, which was about inch broad ; and
the other pressed against the shaft on which the whole turned. The
area included by the secondary circuit was therefore not exactly that
of the disk, but required a small correction, as to which, however, there
is no difficulty.
The arrangements for driving the disk and for observing the speed were
the same as for the revolving coil of method III. The results, which in the
same arrangement have not differed by so much as y^- on different days,
show that the sensitiveness was sufficient.
After these explanations I come to the main subject of the present
remarks, viz. the degree of accuracy likely to be attained in the fundamental
linear measurement. In the present case the quantity to be determined
is M ; and so far there is no difference between this method and I. But the
fact that the secondary circuit is here represented by a disk whose diameter
can be measured much more accurately than that of a coil introduces a
certain modification. It is necessary also that the arrangements be sym-
metrical with respect to the middle plane of the disk, as, on account of
the width of the brush, the place of contact cannot be considered as well
defined. The necessary condition can be satisfied with a single coil by
placing it so that its mean plane coincides with that of the disk. In this
position slight errors of adjustment produce effects of the second order only,
and everything depends upon the radii.
Preparatory to the design of the apparatus for my experiments, I made
some calculations of the values of the induction-coefficient and of its rates
102
L48
COMPARISON OF METHODS FOR THE
[92
of variation for various ratios of the radius of the coil (A) to that of the
disk (a). The angle y (see method I.) is here (6 = 0) determined by
' " 1 a/A. If we write
M
A
Ba
a
the sum of \ and v will be unity. The following are the values found.
Those under M are proportional only, and relate to the case in which A is
constant.
a/A
X
V
M
5
-1-2
+ 2-2
4-37
6
-1-36
+ 2-36
6-65
7
-1-5
+ 2-5
9-80
8
-2-0
+ 3-0
14-4
In Lorenz's apparatus the value of a/A was even larger than the last
in the table, and the radial dimension of the coil was no small fraction
of (A a). On this account, as has already been pointed out by Rowland,
no very accurate result could be expected.
In my experiments two similar coils were used [in series] whose radius
(A) = about 26 cm., and in two distinct arrangements. In the first arrange-
ment the two cells were placed close together ; so that the case corresponded
pretty closely with that just spoken of. The radius of the disk is about
16 cm. ; and thus the proportions are nearly those of the second example in
the table. It will be seen that the circumstances are not unfavourable to
accuracy, the error of mean radius of the coil entering into the result to
a less extent than in any of the methods hitherto described, except III.
and IV. The disk is so much more easily measured, that the larger
coefficient 2 '36, applicable to it, should not lead to much error in the
result.
This arrangement was worked at two speeds of rotation in the proportion
of 10 : 16, and gave with close accordance
1 B.A. unit = -9867 x 10 9 c.G.s.
In the other arrangement the two coils were separated to a considerable
distance, and the induction-coefficient depended not only upon the mean
radii of the coils (and of the disk), but also upon the distance of their mean
planes. The peculiarity of this arrangement, to which I wish to draw
special attention, is that it is possible so to proportion the quantities that
the error of mean radius of the coil does not affect the result, which accordingly
'2 DETERMDfATIOX OF RESISTAXCES IX ABSOLUTE MEASURE. 149
depends only upon the diameter of the disk and the distance of the coil's
mean planes. How this may come about will be readily understood bj
considering the dependence of M upon A when a and 6 are given. It is
clear that M vanishes, both when A is very small and when it is very
large; from which it follows that there must be some value of A for
which the effect is a maximum and therefore independent of small varia-
tions of A.
In earning out this idea it is not necessary to approach the above-
defined state of things very closely: for of course we have in realitv
a good approximate knowledge of the value of A. In my apparatus the
distance of mean planes was about 30 cm., so that 6 = about 15 cm. With
the actual proportions a calculation of the effects of the various errors
shows that
S If ~ J XA
ojn , _ o-id. _ _ oo oa
so that the error of A enters in quite a subordinate degree. The positive
coefficient of SA shows that with the given coils and disk the separation
was somewhat too great to secure the greatest independeuce of BA.
The success of this arrangement depends principally upon the degree
of accuracy with which 6 can be determined. The two rings on which the
wire is coiled are separated by distance-pieces ; and, as in I., by reversing the
rings relatively to the distance-pieces the result may be made to depend
upon the mean length of these pieces and the mean thicknesses of the rings
at the places of contact. The three distance-pieces were held together in
one length and measured under microscopes; and the thicknesses of the
rings were taken with verified callipers. There can hardly be a doubt but
that this determination is much more accurate than that of the mean radius
of a coil ; and, what is also of some importance, it admits of repetition at
pleasure with comparatively little trouble.
The value of the B.A. unit resulting from the measurement with this
arrangement was "9869 x 10' C.G.S.*
There seems no reason why a further increase of accuracy should not be
obtainable by enlarging the scale of the apparatus. If we suppose the scale
doubled, the number of turns in the coil and the angular speed of the
disk being unaltered, the value of M would be doubled; and thus with
the same batterv-current the sensitiveness would be improved. Or, if
we suppose the circumferential linear speed of the disk rather than its
angular speed to be constant, the sensitiveness would be unchanged. If
the larger coil were made of the same kind of wire as the smaller, its
* The redactions not being yet finally completed, these numbers are liable to a change of
one or two units in the fourth place of decimal*.
150 COMPARISON OF METHODS FOR THE DETERMINATION, &C. [92
resistance would be augmented; but if the dimensions of the section were
also doubled, so as to keep the proportions throughout, the advantage in this
respect would lie with the larger apparatus.
On the whole, I am of opinion that if it is desirable at the present time
to construct apparatus on the most favourable scale, so as to reach the
highest attainable accuracy, the modification of Lorenz's method last
described is the one which offers the best prospect of success. Before this
is done, however, it appears to me important that the value now three
times obtained in the Cavendish Laboratory by distinct methods should be
approximately verified (or disproved) by other physicists. To distinguish
between this value and those obtained, for instance, by Kohlrausch, by
Lorenz, or by the first B.A. Committee, should not require the construction
of unusually costly apparatus. Until the larger question is disposed of, it
appears premature to discuss the details of arrangements from which the
highest degree of precision is to be expected.
93.
OX THE DARK PLANE WHICH IS FORMED OVER A
HEATED WIRE IN DUSTY AIR
[Proceedings of the Royal Society, xxxiv. pp. 414 118, 1882.]
Ix the course of his examination of atmospheric dust as rendered evident
by a convergent beam from the electric arc. Professor Tyudail noticed the
formation of streams of dust-free air rising from the summits of moderatelv
heated solid bodies*. f 'To study this effect a platinum wire was stretched
across the beam, the two ends of the wire being connected with the two
poles of a galvanic battery. To regulate the strength of the current a
rheostat was placed in the circuit. Beginning with a feeble current, the
temperature of the wire was gradually augmented : but before it reached the
heat of ignition, a flat streain of air rose from it, which, when looked at edge-
ways, appeared darker and sharper than one of the blackest lines of Fraun-
hofer in the solar spectrum. Right and left of this dark vertical band the
floating matter rose upwards, bounding definitely the non-luminous stream
of air."
" When the wire is white hot, it sends up a band of intense darkness.
This, I sav, is due to the destruction of the floating matter. But even when
its temperature does not exceed that of boiling water, the wire produces a
dark ascending current. This, I say, is due to the distribution of the floating
matter. Imagine the wire clasped by the mote-filled air. My idea is that it
heats the air and lightens it, without in the same degree lightening the
floating matter. The tendency, therefore, is to start a current of clean air
through the mote-filled air. Figure the motion of the air all round the wire.
Looking at its transverse section, we should see the air at the bottom of the
wire bending round it right and left in two branch currents, ascending its
sides, and turning to fill the partial vacuum created above the wire. Now
* Pmt. Bay. Im*t. voL . p. 3, 1870.
152 ON THE DARK PLANE WHICH IS FORMED [93
as each new supply of air, filled with its motes, comes in contact with the hot
wire, the clean air, as just stated, is first started through the inert motes.
They are dragged after it, but there is a fringe of cleansed air in advance of
the motes. The two purified fringes of the two branch currents unite above
the wire, and, keeping the motes that once belonged to them right and left,
they form by their union the dark band observed in the experiment. This
process is incessant. Always the moment the mote-filled air touches the
wire, the distribution is effected, a permanent dark band being thus produced.
Could the air and the particles under the wire pass through its mass, we
should have a vertical current of particles, but no dark band. For here,
though the motes would be left behind at starting, they would hotly follow
the ascending current, and thus abolish the darkness."
Professor Frankland*, on the other hand, considers that what is proved
by the above described observations is that " a very large proportion of the
suspended particles in the London atmosphere consists of water and other
volatile liquid or solid matter."
Last summer (1881) I repeated and extended Tyndall's beautiful experi-
ment, not feeling satisfied with the explanation of the dark plane given by
the discoverer. Too much stress, it appeared to me, is placed upon the
relative lightening of the air by heat. The original density is probably not
more than about T ^ part of that of the particles, and it is difficult to see
how a" slight further lightening could produce so much effect. In other
respects, too, the explanation was not clear to me. At the same time I was
not prepared to accept Professor Frankland's view that the foreign matter is
volatilised.
The atmosphere of smoke was confined within a box (of about the size of
a cigar-box), three of the vertical sides of which were composed of plates of
glass. A beam of sunlight reflected into the darkened room from a heliostat
was rendered convergent by a large lens of somewhat long focus, and made
to pass in its concentrated condition through the box. The third glass side
allowed the observer to see what was going on inside. It could be removed
when desired so as to facilitate the introduction of smoke. The advantages
of the box are twofold. With its aid much thicker smoke may be used than
would be convenient in an open room, and it is more easy to avoid draughts
which interfere greatly with the regularity of the phenomena to be observed.
Smouldering brown paper was generally used to produce the smoke, but
other substances, such as sulphur and phosphorus, have been tried. The
experiment was not commenced until the smoke was completely formed, and
had come nearly to rest." In some respects the most striking results were
obtained from a copper blade, about -inch broad, formed by hammering flat
one end of a stout copper rod. The plane of the blade was horizontal, and
* Proc. Roy. Soc. vol. xxv. p. 542.
93] OVER A HEATED WIRE IN DUSTY AIR. 153
its length was in the line of sight. The unhammered end of the rod
projected from the box, and could be warmed with a spirit-lamp. The dark
plane was well developed. At a moderate distance above the blade it is
narrow, sometimes so narrow as almost to render necessary
a magnifying-glass : but below, where it attaches itself to the
blade, it widens out to the full width, as shown in the figure.
Whether the heated body be a thin blade or a cylindrical
rod, the fluid passes round the obstacle according to the
electrical law of flow, the stream-lines in the rear of the
obstacle being of the same form as in front of it. This
peculiarity of behaviour is due to the origin of the motion
being at the obstacle itself, especially at its hinder surface. If a stream
be formed by other means and impinge upon the same obstacle without
a difference of temperature, the motion is of a different character altogether,
and eddies are formed in the shadow.
The difference of temperature necessary to initiate these motions with
this dark plane accompaniment is insignificant. On July 20, 1881, a glass
rod, about ^-inch [6 mm.] in diameter, was employed. It was heated in a
spirit-lamp, and then inserted in the smoke-box. The dark plane gradually
became thinner as the rod cooled, but could be followed with a magnifier for
a long time. While it was still quite distinct the experiment was stopped.
and on opening the box the glass rod was found to be scarce ly warmer than
the fingers. It was almost impossible to believe that the smoky matter had
been evaporated.
In order to test the matter more closely, smoke was slowly forced through
a glass tube heated near the end pretty strongly by a spirit-lamp, and then
allowed to emerge into the concentrated sunshine. Xo distinct attenuation
of the smoke could be detected even under this treatment.
It is not necessary to dwell further upon these considerations, as the
question may be regarded as settled by a decisive experiment tried a few
days later. The glass rod before used was cooled in a mixture of salt and ice,
and after wiping was placed in the box. In a short time a dark plane,
extending downwards from the rod, clearly developed itself and persisted for
a long while. This result not merely shows that the dark plane is not due to
evaporation, but also excludes any explanation depending upon an augmenta-
tion in the difference of densities of fluid and foreign matter.
The experiment was varied by using a U-tube, through which cooled
water could be made to flow. When the water was not very cold, the
appearances were much the same as with the solid rod ; but when, by means
of salt and ice, the tube was cooled still further, a curious complication
presented itself. Along the borders of the dark plane the smoke appeared
considerably brighter than elsewhere. Sometimes when the flow was not
154 ON THE DARK PLANE WHICH IS FORMED OVER A HEATED WIRE, &C. [93
very regular it looked at first as if the dark plane had been replaced by a
bright one, but on closer examination the dark plane could be detected
inside. There seems no doubt but that the effect is caused by condensation
of moisture upon the smoke, due to the chilling which the damp air under-
goes in passing close to the cold obstacle. Where the fog forms, more light is
scattered; hence the increased brightness. That the fog should not form
within the smoke-free plane itself is what we might expect from the interest-
ing observations of Aitken.
With respect to the cause of the formation of the dark plane, the most
natural view would seem to be that the relatively dense particles are thrown
outwards by centrifugal force as the mixture flows in curved lines round the
obstacle. Even when the fluid is at rest, a gradual subsidence must take
place under the action of gravity ; but this effect could at first only manifest
itself at the top where the upper boundary of the gas prevents the entrance
of more dust from above. It is known that air in a closed space will
gradually free itself from dust, but the observation of a thin dust- free stratum
at the top of the vessel is difficult. If we conceive a vessel full of dusty air
to be set into rapid rotation, the dust might be expected to pass outwards in
all directions from the axis, along which a dust-free line would form itself.
I have tried this experiment, but looking along the axis through the glass
top of the vessel I could see no sign of a dark line, so long as the rotation
was uniform. When, however, the vessel was stopped, a column of compara-
tively smoke-free air developed itself along the axis. This I attributed to
the formation of an inward flow along the top of the vessel, combined with a
downward flow along the axis after the manner described and explained by
Professor James Thomson, so that the purified air had been in intimate
proximity with the solid cover. It would almost seem as if this kind of contact
was sufficient to purify the air without the aid of centrifugal force.
The experiments made hitherto in order to elucidate this question have
given no decisive result. If the thin convex blade already spoken of be held
in the smoke-box in a vertical instead of in a horizontal plane, the lines of
motion are much less curved, and we might expect to eliminate the influence
of centrifugal force. I have not succeeded in this way in getting rid of the
dark plane ; but since under the magnifier the curvature of the motion was
still quite apparent, no absolute conclusion can be drawn.
[1900. The reader is referred to interesting papers by Aitken (Edin.
Trans, xxxn. p. 239, 1884) and by Lodge and Clark (Phil Mag. xvn. p. 214,
1884) in which this question is further discussed. It seems clear that
gravitation and a movement from hot to cold, somewhat as in Crookes'
radiometer, are both concerned.]
94.
EXPERIMENTS, BY THE METHOD OF LOREXZ, FOR THE
FURTHER DETERMINATION OF THE ABSOLUTE VALUE
OF THE BRITISH ASSOCIATION UNIT OF RESISTANCE,
WITH AN APPENDIX ON THE DETERMINATION OF THE
PITCH OF A STANDARD TUNING-FORK.
[Phil Trans. CLXXIV. pp. 295322, 1883.]
By Lord RAYLEIGH and Mrs H. SIDGWICK.
1. IN this method, which was employed by Lorenz in 1873*, a
circular disc of metal is maintained in rotation at a uniform and known rate
about an axis passing through its centre, and is placed in the magnetic field
due to a battery current which circulates through a coaxal coil of many turns.
The revolving disc is touched at its centre and circumference by two wires.
If the circuit were simply closed through a galvanometer, the instrument
would indicate the current due to the electromotive force of induction acting
against the resistance of the circuit. The electromotive force corresponding
to each revolution is the same as would be generated iii a single turn of wire
coincident with the circumference of the disc by the formation or cessation of
the battery current. If this be called 7, and M be the coefficient of induc-
tion between the coil and the circumference, m the number of revolutions
per second, the electromotive force is mMy. In the actual arrangement,
however, the circuit is not simply closed, but its terminals are connected with
the extremities of a resistance R, traversed by the battery current, and the
variable quantities are so adjusted that the electromotive force Ry exactly
balances that of induction. When the galvanometer indicates no current, the
* Pogg. Ann. vol. cxtrs. p. 251.
156 ON THE ABSOLUTE VALUE OF THE [94
following relation, independent, it will be observed, of the magnitude of the
battery current, must be satisfied
and from this, M being known from the data of construction, the absolute
resistance R of the conductor is determined.
One of the principal difficulties to be overcome arises from the smallness
of the resistance R, necessary for a balance, even when m and M are both
increased as far as possible. Lorenz employed three resistances, ranging
from '0008 to '002 of a mercury unit, and he evaded the necessity of com-
paring these small resistances with ordinary standards by constructing them
of actual columns of mercury. His result was accordingly obtained directly
in terms of mercury, and was to the effect that
1 mercury unit = '9337 x 10 9 C.G.S.
differing nearly 1 per cent, from the value ('941) obtained by ourselves.
2. Under the conviction that this method offers in some respects im-
portant advantages, and influenced also by the fact that the arrangements for
producing and measuring the uniform rotation necessary were ready to our
hands, we determined to give it a trial, in the hope of obtaining confirmation
of the results already arrived at by ourselves and by Glazebrook with other
methods. At first the intention was to follow Lorenz in using for the re-
sistance a glass tube full of mercury, with two points of which contact would
be made by platinum wires passing through the glass. It appeared, however,
that there would be difficulty in making the measurements with the degree
of accuracy aimed at. If the wires were sealed into the glass, the section
would probably be rendered irregular. An attempt was made to avoid this
difficulty by using a tube from which the ends had been cut with the aid
of heat. After small nicks had been filed sufficiently deep to receive the
platinum wires, the ends were replaced in their original positions and secured
with shellac. In this way a satisfactory uniformity of section near the points
of derivation could be attained, but the measurement of the distance between
these points, which is required to be known with full accuracy, was rendered
difficult by the presence of the cement. It is possible that these difficulties
might have been overcome, but at this point a method of shunting occurred
to us, allowing the use of mercury to be dispensed with. Merely for the
purpose of connecting the mercury unit with the B.A. unit or other standard
of resistance, it would not be desirable to use tubes of such large bore*.
This problem may more conveniently be taken by itself, and has already been
treated by us in a former communication to the Society f.
* If the distance between the points of derivation were 1 metre, Jv = -002 mercury unit would
require a section equal to 500 square rnillims.
t Phil. Trans. 1883, p. 173 [vol. n. p. 78],
94] BRITISH ASSOCIATION UNIT OF RESISTANCE.
157
3. In the shunt method the greater part of the main current 7 passes
on one side through a relatively small resistance a (see fig. 1), and the
difference of potentials at the points of derivation B, C, is due to the
rir. 1.
of a small fraction only of the total current, the resistance (6 + c) being
great compared with a. If at the same time 6 be small relatively to c, the
difference of potentials is doubly attenuated- Its value for a given main
current 7 is found at once from the consideration that the current divide?
itself between the two branches in the inverse ratio of the resistances. The
current through 6 is thus ^ 7, and the difference of potentials at the
points of derivation is j 7. The quantitv thus takes the
a+b+c ' a 4-6 - c
place of jR in the simple formula, and is called the effective resistance. By
taking for instance a = $, 6 = 1, c = 100, we get an effective resistance of
about ^f : and the resistances employed may be those of ordinary resistance
coils, capable of accurate comparison with the standards.
4. In designing the apparatus we were influenced by the fact that we
had at our disposal two very suitable coils of large radius, wound some year>
ago by Professor Chrystal, the same in fact as were used by Mr Glazebrook
in his investigation by another method. By bringing the two coils close to
one another and to the plane of the disc, the inductive effect is rendered a
maximum. This arrangement accordingly was the one first experimented
with, as being the most likely to prove successful.
The diameter of the disc is limited by two considerations. If it be too
small, the whole inductive effect, and with it the sensitiveness of the arrange-
ment, suffers. On the other hand if it be too large, the circumference enters
the more intense region of magnetic force which lies near the wire, and the
coefficient of induction changes its value rapidly when any alteration occurs
in the mean radius of the coils, or in the diameter of the disc, and thus the
final result becomes too sensitive to errors in the magnitude of these elements.
In the PkiL Mag. for Kov., 1882, [Art 92] the reader will find a calculation
of the values of M for various cases, and a general comparison of the principal
methods for determining absolute resistance, especially in respect of errors
arising in connexion with the fundamental linear measurements. For the
]58 ON THE ABSOLUTE VALUE OF THE [94
experiments now to be described, the diameter of the disc was chosen so as to
be somewhat more than half that of the coils ( 22, 23).
5. The disc was of brass and turned upon a solid brass rod as axle.
This axle was mounted vertically in the same frame that carried the re-
volving coil in the experiments described in a former communication to the
Society* [see Vol. II. p. 39], an arrangement both economical and convenient,
as it allowed the apparatus then employed for driving the disc and for ob-
serving the speed to remain almost undisturbed. The coils were supported
horizontally upon wooden pieces screwed on the inner side of the three up-
rights of the frame.
During the earlier trials, extending over the month of May, 1882, the
edge of the disc was bevelled, and contact was made with it by means of a
brush of fine copper wires held in a nearly vertical position. No sufficiently
regular results could be obtained until the sliding surfaces were amalgamated,
and even then there were discrepancies between the work of one day and that
of another, whose cause was not discovered until a later period. It soon be-
came manifest, however, that the bevelled edge would not answer the purpose,
for it cut its way by degrees into the wires of the brush in such a manner as
to render the effective radius uncertain. The substitution of a cylindrical for
a bevelled edge promised better results. The width of the edge (equal to
the thickness of the disc) was 4^ millims. and allowed sufficient room for
the contact of the brush though placed tangentially. In this way broader
bearing surfaces were available, and the small extension of the contact in the
direction of the axis is unobjectionable, provided everything be arranged
symmetrically with respect to the middle plane of the disc.
As will presently appear, the success of the method is independent of any
constant thermo-electric force at the sliding contact, but it is evident that
good readings cannot be taken if the thermo-electric force changes its magni-
tude often and suddenly. It was found advisable to renew the amalgamation
of the edge at the commencement of each day's work. The excess of
mercury, if any, attaches itself to the brush, and does not appear to render
the diameter of the disc uncertain.
The inner contact was made in a similar manner by a brush pressing
against the shaft itself at a place a little below that at which the disc was
attached. The coefficient of induction to be employed in the calculation is
the difference between the coefficients for the coil and the outer and inner
circles of sliding contact respectively, but the latter is quite subordinate
( 25)-
6. The disc was driven by the same water-engine that was employed
for the revolving coil of former deter minationsf, the connexion being made
* PhU. Trans. Part II. 1882 [Art. 80].
t Proc. Roy. Soc. May 5, 1881 [Art. 79]; Phil. Trans. Part II. 1882 [Art. 80].
BRITISH ASSOCIATION UXTT OF RESISTANCE.
159
by a long cord pas-ing round a wooden puller attached to the lower part of
the shaft. To the upper face of the disc was cemented a circle of paper on
which were marked a series of circles of alternately black and white teeth.
One observer looking through the prongs of an electro-magnetically main-
tained fork regulated the speed of the disc by application of the necessary
friction to the driving-coni which passed through his fingers. When one of
the series of circles is seen to be stationary, a simple and easily expressed
relation is established between the frequency of the fork and that of revolu-
tion. At intervals the number of beats per minute is counted between the
notes of a standard fork, and (the octave of) the electric fork. There is no
difficulty in thus determining the speed of rotation to within one part in
10,000. With respect to the absolute pitch of the standard fork iis<elf_ see
the Appendix to this Memoir.
When the disc is caused to rotate, and the galvanometer circuit is
closed, a deflexion is observed, although the battery which generates the
main current is not in action. This deflexion is due to two causes thermo-
electric force at the sliding contact, and induction dependent upon the vertical
component of the earth's magnetism. Although not a direct sourw of error,
this deflexion is better avoided, both for convenience in reading the galvano-
meter and because it implies the actual passage of a not insensible current
through the sliding contacts and thus brings into consideration the regigtanf?
of these contacts. The compensation was effected by the introduction of an
opposing electromotive force : for which purpose two terminals of the gaivano-
meter circuit J, K, fig. 2, instead of being connected directly, were attached
by binding screws to two points on a stout copper wire forming part of a
circuit which included a sawdust Daniell (L) and a resistance coil of
100 ohms (M). By shifting one of the binding screws, the galvanometer
reading, in the absence of the main battery current, and after attainment of
160 ON THE ABSOLUTE VALUE OF THE [94
the proper speed, was made to be nearly the same as when the galvanometer
contact was broken.
8. The general plan of the connexions and the modus operandi will
now be intelligible from fig. 2. The poles of the battery A, consisting of
20 Daniell cells, were connected with a mercury reversing key B, the two
positions of which were distinguished by the letters E and W (east and
west). From thence the current passed through the induction coils C
and the equivalent resistance R, of which the details are reserved for
the moment. The reflecting galvanometer, G, is placed at a considerable
distance in order to avoid the direct influence of the coils, and is con-
nected with the inner sliding contact, F. Its resistance is about ^ ohm ;
and by the aid of the compensating magnet the vibrations of the needle
were made slow enough to be readily observed. The terminals of the
galvanometer branch, which includes also a commutator, /, are connected
to the extremities of the resistance, R.
If, while the disc is maintained in uniform rotation, the reading of
the galvanometer is the same whichever way the battery key may stand
(correction being made, if necessary, for a direct effect upon the needle),
it is a proof the contemplated balance is actually attained. In this way
all disturbance from the earth's magnetism, and from thermo-electric forces
whether situated at the sliding contacts, or within the resistance coils of
which R is composed, or at any other part of the galvanometer circuit, is
eliminated from the result. The adjustment is effected by varying a com-
paratively large resistance, taken from a box, and placed in multiple arc
with one of the components of R.
9. In actual work, however, it is not necessary, or even desirable, to
hit off the balance with great accuracy. An unmistakeable difference of
readings when the battery key is put over, is rather an advantage than
otherwise, as giving an indication that the circuits are properly closed.
The plan adopted was to take a series of readings of the effect (E W)
of reversing the battery current with an effective resistance R l} not very
different from R. Single readings were liable to considerable irregularity
in consequence of change in the friction at the sliding contacts, and
of momentary variations in the speed. These errors cannot possibly be
systematic, and are in great measure eliminated in the mean of a series.
Having thus obtained the difference of galvanometer readings (E W)
corresponding to R 1} we altered the resistance in multiple arc so as to
change R l into R 2 , the difference being some such fraction as T ^- of the
whole, and in such a direction that the sign of E W is changed. The
two series give by simple interpolation (after correction for the direct
effect) the true value of R, that is the effective resistance corresponding
to the balance. In order to get the best result relatively to the time
94] BRITISH ASSOCIATION UNIT OF RESISTANCE. 161
occupied, the number of observations of E W in each set was taken
roughly in inverse proportion to the values. To diminish the influence of
a progressive change in the strength of the battery current, the obser-
vations with RZ were interspersed between those with /, as effective
resistance. The readings were usually taken continuously, with no more
delay than was necessary to allow the vibrations of the needle to become
of moderate extent after each change. When they were completed, the
driving cord was reversed, as well as the commutator, /, and a similar set
of observations was taken with rotation in the opposite direction.
10. In the earlier experiments the resistance coils composing the
effective resistance were arranged as in fig. 1, in which A, B, C may be
supposed to represent mercury cups, the bottoms of which were formed
of amalgamated copper discs. On these discs rested the amalgamated
terminals of the various resistance coils and connecting wires. The shunt
a consisted of two unit coils in multiple arc, between which the greater
part of the main current was equally divided. The magnitude of the
main current was less than ^ ampere. The resistance b between the
points of derivation was a unit, while the third resistance c was alternately
105 and 106.
In reckoning the resistance of the galvanometer circuit we have to
include b. The remainder scarcely exceeds the ^ ohui due to the gal-
vanometer itself. It appears therefore that the deflections obtained with
the arrangement described are only one-third part as great as they would
be if a quite small resistance were substituted for the unit in 6. As the
sensitiveness appeared likely to be inadequate, we afterwards replaced the
unit by ^, using for c a coil of ten units. As in this case the addition
or subtraction of a whole ohm in c would make too great a difference, the
adjustment was obtained by varying a comparatively large resistance placed
in multiple arc with a.
In the light of subsequent experience it is doubtful whether this
change was an improvement. The increase of galvanometer deflection was
not really of much advantage, since the difficulty of getting sharp results
arose from electromotive disturbances, and these were magnified in the
same proportion. It would probably have been better to have retained
the unit in 6, and to have replaced the galvanometer by one of higher
11. Preliminary trials having given apparently satisfactory results,
we proceeded to make regular series of observations in the manner already
described. We had not gone far before anomalies revealed themselves of
such a character as to prove that we were not yet masters of the method.
It usually happened that each days observations agreed well together,
showing that the sensitiveness was sufficient; but when we came to com-
11
162 ON THE ABSOLUTE VALUE OF THE [94
pare the results obtained on different days unaccountable discrepancies
became apparent. The first result of the more severe criticism to which
the arrangements were then subjected was to show that sufficient thought
had not been given to the question of insulation. The wire composing
the induction coils, or rather one extremity of it, is necessarily at a high
potential, and a very moderate leakage from the coils to the frame, and
thence to the disc, might cause great disturbance. Some such leakage was
in fact detected on application of appropriate tests. Ebonite insulation
was accordingly introduced into the supports of the coils. The battery
was carefully insulated from the ground, as was also the frame carrying
the revolving disc, and other precautions were taken which it is unnecessary
here to detail. For the sake of definiteness one point of the galvanometer
commutator was connected to earth. With these improvements tests were
satisfied more severe than that of actual use, and these tests were renewed
at intervals during the spinnings.
The results however still showed that some defect existed which we had
not yet succeeded in detecting. It made no appreciable difference which
way the disc rotated, but the means of different days' work failed to exhibit
the desired accordance. Two months' work had already been spent upon
the experiments, and we had begun to despair of a satisfactory issue, when
it occurred to us that the connexion of the coils for compounding the
effective resistance was faulty.
12. By reference to fig. 1 it will be seen that the main current
traverses part of the cup G, and that part of the same cup is also included
in 6. Now, although for all ordinary purposes the resistance of the parts
of the cup might be neglected, in the present case it is the small effective
resistance R with which it comes into comparison. If we aim at an accu-
racy of TjfijB^, we cannot afford to overlook a resistance entering in this
manner, even though it may not exceed ^ffinh^ onm - The discrepancies
were doubtless due to small differences in the position of the wires and
coils in cup G, moved as they were from day to day in order to verify the
soundness of the contacts.
In order to avoid the difficulty we have only to take care that no part
of 6 can possibly be traversed by the main current, and this is easily done
by the introduction of another mercury cup. Fig. 3 shows the arrange-
ment adopted. The main current enters at the cups A and D, and the
greater part is taken by the two unit coils in multiple arc whose ter-
minals rest in these cups. The galvanometer terminals are led into two
other cups J5 and G. The ends of these are beaten fiat and the legs of
the ^ rest upon them. The connexion between C and D was through a
stout copper rod, which may be regarded as part of c. For the first series
the connexion between A and B was through a single coil of 10 units'
94] BRITISH ASSOCIATION UNIT OF RESISTANCE. 163
resistance, replaced in subsequent series by other coils giving altogether
16 and 20 units' resistance respectively.
CALK
To make the necessary adjustment and variation of resistance, a box, E,
was placed in multiple arc with the two unit coils. The resistances taken
from the box were afterwards carefully determined, but they enter into
the final results in quite a subordinate manner.
13. Further trials now led to the satisfactory conclusion that the
defect was remedied, for the means obtained on different days agreed
well together, even although the resistance coils were taken down and
remounted in the interval. As we had now every reason to suppose
that our experiments would have a successful issue, we proceeded to
make the final adjustments preparatory to a complete series of obser-
vations.
In the first and second series the two [induction-] coils were near one
another, separated only by three slips of glass, and held firmly together
by wooden clamps. The adjustments presented no particular difficulty. By
means of an iron finger clamped to the disc and carried gradually round,
it could be verified that the coils and disc were concentric and in parallel
planes. The coils were gradually wedged into their places, and secured
when their mean planes occupied the desired symmetrical positions relatively
to the disc. It is evident that errors of maladjustment influence the result
only in the second order.
14. Experience in this series having shown that the arrangement
was satisfactory, and that the sensitiveness was fully sufficient, we pro-
ceeded to make a second series of observations without displacement of
the induction coils, but at a speed of rotation lower than before in about
the ratio of 16-10 This, of course, entailed a corresponding change
112
164 ON THE ABSOLUTE VALUE OF THE [94
in R, which was effected by increasing the component c. An agreement
between the final results of the two series would give an important con-
firmation, inasmuch as leakage of electricity from the main circuit into
the galvanometer branch would exert a different influence in the two
cases. The observations were not reduced until some time afterwards, and
it then appeared that the agreement was even better than it would have
been reasonable to expect.
15. The final number, '9867 x 10 9 , expressing the value of the B.A.
unit in absolute measure as determined by these two series of observa-
tions, is almost identical with that previously obtained by ourselves, and
by Glazebrook using other methods. With respect to the independence
of these determinations, the only thing calling for notice is the fact
that the same induction coils were employed both by Glazebrook and in
the present investigation. In other respects there has been, we believe,
scarcely any point of contact. But it is evident that an error in the
measurements of mean radius of these coils must propagate itself into
both results. The point to which we now wish to direct attention, is that
the error of mean radius will influence the final number in opposite direc-
tions. In the method employed by Glazebrook, an under-estimate of the
mean radius would lead to an under-estimate of the induction coefficient,
whereas with us it would lead to an over-estimate of that quantity. So
far, therefore, as the error of mean radius is concerned, it would ap-
pear that the use of the same coils is far from impairing the value of
the results. Even with respect to the number of turns, an error, if that
be supposed possible, would affect the results in a different manner, for
Glazebrook was concerned with the product of the numbers for the two
coils, while we evidently are concerned with the sum.
16. In researches of this kind it is proper to calculate the influence
upon the result of errors in the fundamental measurements. The value of
M depends upon three linear quantities: the radius of the disc (a), the
mean radius of the two coils (A), and the distance between their mean
planes (26). In the present case, however, the latter element enters in a
very subordinate degree. From 25 it appears that
dM . dA . . da
^ = -1-4 -^- + 2-4 .
M A a
It has been shown* that these conditions compare favourably with those
of most of the other methods that have been employed. From its nature
a is much more easily measured than the diameter of a coil.
17. The results deduced from the several days' observations, when
corrected for slight variations of temperature of the resistance coils, &c.,
* Phil. Mag. Nov. 1882 [Art. 92].
94] BRITISH ASSOCIATION TJXTT OF RESISTANCE. 165
exhibit a remarkable accordance. By reference to the tables < 27) the
reader will see that the maximum divergence from the mean in Seeks I.
is only about one part in 4000, while in Series LL it is even leas. We
were thus encouraged to carry out a modification of the method which we
had had in view all along, and the results of which would be in great
measure independent of those of Series I. and IL
18. The modification referred to relates to the position of the in-
duction coils relatively to the disc. In the arrangement with which we
have been dealing hitherto, the mean planes of the coils are nearlj
coincident with that of the disc, and the accuracy of the final number
depends upon an exact knowledge of the mean radius of the coils. It has.
on the other hand, the advantages of being practically independent of
measurements parallel to the axis, and of giving the maximniu coeffi-
cient of induction. In the new arrangement the coils are separated to
such a distance that the result w nearly independent of a knvHrltflpe >f tine
mean radius. How this may come about will be readily understood by
considering the dependence of the coefficient of induction J/ upon A T
when a and 6 are given. It is clear that M vanishes, both when A is
very small, and also when it is very large; from which it follows skai
there must be some value of .1 for which the effect is a maximum, and
therefore independent of small variations of A.
In carrying out this idea, it is not necessary to approach she ar>-ve
defined state of things very closely; for of course we have in iw:y a
good approximate knowledge of the value of A. In our appara:u the "dis-
tance of mean planes was about 30 centime, so that b = abat 15 ceutims,
(A =26, a = 16). From the calculations in | 25 it appears that with the
actual proportions
dM . dA db .,.8*1.
T =+12 J"" ^T* *^ r
so that the error rf A enters in quite a subordinate degree. The positive
coefficient of dA shows that with the given coils and the given disc the
separation was somewhat too great to secure the utmost independence
of dA.
19. The success of this arrangement depends principally upon the
degree of accuracy with which 6 can be determined. The two rings upon
which the coils are wound were held apart by three equal distance-pieces,
against which they were firmly pressed by wooden clamps. The distance-
pieces were hollow, of massive* brass, and the terminal faces were carefully
turned. Central marks upon them facilitated the adjustment of the coils
into the symmetrical positions. The distance of mean planes does not
however depend solely upon the distance-pieces. Even if we could assume
166 ON THE ABSOLUTE VALUE OF THE [94
that the mean planes are symmetrically situated relatively to the grooves
in which the wire is wound, we should still have to take account of the
thicknesses of the flanges. All uncertainty in this matter is eliminated by
following the plan adopted by Glazebrook of reversing the rings (without
interchange), and then repeating the measurements. Whatever may be
the situation of the mean planes and the thicknesses of the flanges, the
mean result thus obtained corresponds to a distance equal to the length
of the pieces plus half the total outside thicknesses of the rings. These
quantities can all be measured with great precision, and as easily after
the coils are wound as before. Full particulars are given in 24. There
can hardly be a doubt but that the determination is much more accurate
than that of the mean radius of a coil ; and, what is also -of some importance,
it admits of repetition at pleasure with comparatively little trouble.
20. The sensitiveness of this arrangement was about the same as in
Series II., and the table shows a good agreement among the results obtained
on different days. The final number from this series is '9868, almost the
same as from Series I. and II.
The small difference of effective resistances required for balance in the
two positions of the induction coils, amounting to about one part per
thousand, is almost exactly accounted for by the small difference of distances
of mean planes in the two cases, as deduced from Professor Chrystal's
measurements of the thicknesses of the flanges. In the first position (see
24) the coils are nearer together by almost exactly one part per thousand,
a difference which, according to the formula given above ( 18), should be
reproduced almost without change in M and therefore in R, the greater
values of M and R corresponding to the smaller distance.
21. If we combine all the results of the present investigation, giving
equal weights to the two arrangements of the induction coils, we have
1 B.A. unit = -98677 x 10 c.G.s.
With use of the ratio between the mercury unit and the B.A. unit found
by us (Proc. Roy. Soc., May, 1882 [Art. 81]), this gives
1 mercury unit = '94150 x 10 9 c.G.s. ;
or, which is the same thing, the ohm is the resistance of a column of mercury
at centigrade whose section is 1 square millim., and whose length is
1062-14 millims.
We now pass on to the details of the measurements.
94] BRITISH ASSociATiox ranr OF RESISTANCE. 167
DETAHJS OF MEASUREMENTS.
Diameter of disc.
22. Preliminary measurements of the disc while still mounted were
made on August 11, 1882, with callipers by Mesas Elliott. Bead by the
vernier of the instrument itself the mean diameter was
2a = 310-T6 millims.
The opening of the callipers was also determined independently by
reference with the aid of microscopes to a verified scale of millimetres In
this way
2o = 310-77 millim*
The circumference was also measured by a steel tape, afterwards com-
pared with the millimetre scale. Correction being made for the thickness of
the tape, the resort was
2a = 310-84 millims.
After the disc had been dismounted, the diameter could be determined
more advantageously by direct observation through microscopes focussei
upon its edge with subsequent reference to the standard scale. It was foaod
(August 19, 1882) that a very appreciable difference existed between the
diameter of the upper and lower faces, showing that the edge was somewhat
conical. At the upper edge the diameter was 310-80, and at the lower e*ige
310*58. These were the extremes. At the middle of the thickness trie
diameter was 310^5. This departure from the truly cylindrical form was
undoubtedly a defect in the apparatus, which could easily have been avoided
if detected in time. When the apparatus was first set np r the success of the
experiment was problematical, and a minute examination of the disc seemed
premature. The diameter to be adopted is an average "taken with reference
to the conductivity of brush contact. The whole width of the brush being
decidedly less than the thickness of the disc, and the pressure being greatest
at the central parts, we decided (of course without knowing to what precise
final result the estimate would lead) to take the mean of 310-75 and
| (310-58 + 31 0-80> Thus
2a = 310-72 millims.
The error due to the conicality of the edge cannot exceed one part in
5000 at the worst, and thus it appeared scarcely worth while to correct the
defect and repeat the spinnings.
The diameter of the shaft at the place where the other brush contact was
made, was found to be -825 inch, or 20-96 milHms.
168
ON THE ABSOLUTE VALUE OF THE
[94
The induction coils.
| 23. These are the same as were used in Mr Glazebrook's measurement,
and were wound by Professor Chrystal in 1878. The following are the
dimensions; for further particulars reference may be made to Mr Glaze-
brook's Memoir*.
A
B
Mean
Mean radius in centims. (A)
25-753
25-766
25-760
Eadial width of section (27t)
1-92
1-90
1-91
Axial width of section (2k)
1-896
1-899
1-897
797
791
\ x 1588
Resistance (approximate) in B.A. units .
84
83
|x!67
Since the coils are so nearly similar and were used symmetrically, it is
sufficient to use the numbers in the last column. The section of the ring is
shown in fig. 4 full size.
To find the distance of mean planes the following measurements of the
thicknesses of the rims are required. They are given in centimetres.
A
B
Eim (marked side) ....
Channel
478
1-896
446
1-899
Eim (unmarked side) . . .
Total thickness of ring . .
488
2-862
465
2-810
Now that the rings are wound it is difficult to verify these numbers.
However, the total thickness of the rings at the places touched by the
distance-pieces in the arrangement used for Series III. was taken, with the
result
A
B
Mean of three places . . .
2-8625
2-8067
These latter values of the thicknesses will be used in the calculation of
Series III.
* Phil. Trans. 1883, p. 223.
94]
BRITISH ASSOCIATION UXTT OF
In Series I. and IL the rings were not reversed, and we must
numbers above given for the thicknesses of rims which
were contiguous to the slips of glass ; bat in this case
the result is not at all sensitive to changes in the
distance of mean planes. The rims contiguous to the
glass were for both coils the marked rims, of which the
aggregate thickness is "924. If we add to this the
thickness of the glass strips "454, we obtain 1-378 as
the distance between the wire sections. Again, adding
the mean axial width of section 1'897, we find as the
distance of the mean planes
2 = 3-275 centime
169
the
The distance-pieces.
24. The measurement of the distance-pieces used for the third series
was made with great care. As only the mean is required, the three piece*
were held under the microscopes in one length by a nut and a long boh
running through. Readings were taken in several positions, as the pieciss
were turned round, and reference was finally made to the standard scale.
Two independent measurements gave 83'580 and 83^579, mean S3"57f-> con-
tains., as the aggregate length. This was further verified by measuring; each
piece separately with callipers, the sum of the lengths thus found l-riug
83*582. For the mean length of these distance-pieces we take
27-8598 centims.
As has been already explained, the rings were used in two positions
relatively to the distance-pieces, with the view of eliminating any uncertainty
as to the situation of the mean planes, and of rendering the final result
independent of all measurements of thickness except that of the total thick-
nesses of the rings. Thus the mean distance of mean planes in the two
positions is
27-8598 + ^ (2-8625 + 2-8067) = 3<r6944 centims.
To compare the partial results for the two positions separately, we must
use the thicknesses of the rims which were in contact with the distance-
pieces. In the first position these were the marked rims, and thus the
distance of mean planes
= 27*60+ 478+ 446 + 1-897 = 30-681 centims.
In like manner for the second position we find
- 97-860 + -488 + -465 + 1*97 = 30710 cenUros.
170 ON THE ABSOLUTE VALUE OF THE [94
The induction-coefficients.
25. Series I. and II. The distance (b) of the mean planes of the coils
from the middle plane of the disc is
6 = 1-637 centim.
The extreme distances, required to be known for the quadrature, are
b + k = 2-585 centims., b - k = '689 centim.
The extreme and mean radii are
A - h = 24-805 centims., A = 25760 centims., A + h = 26715 centims.
while
a 15'536 centims.
The coefficient of induction between the disc and the middle turn of the
coil, denoted by M (A, a, b), is equal to 4?r V (A a) ./(y), where f(^) is a
function of 7 given by tables*. The angle 7 itself is defined by
2 *Aa
n ^
It is not necessary to give the details of the calculations, which have been
carefully checked. The tabular interval being 6', it was found desirable in
many cases to proceed beyond the simple interpolation by first differences.
The results are
M(A,a, b) = 215-4674
M(A+h, a, 6) = 205-1917
M(A - h, a, b) = 226'9835
M(A, a, 6 + &) = 2117246
M(A,a, 6 -A;) = 217-5972.
The mean coefficient for the area of the section is found by doubling the
first of these values, adding in the others, and then dividing by 6.
Thus
M = 215-405 f.
The separate values allow us to form an estimate of the effect of errors in
the fundamental data. If we write
dM ^ dA db da
~JTF = * ~T~ + P ~1~ + v >
M A ^ b a
* Maxwell's Electricity and Magnetism, 2nd edition, 706.
t The factor expressing the number of windings is omitted.
94] BRITISH ASSOCIATION UlflT OF RESISTANCE. 171
we may take approximately
M
In like manner, /* = O2, whence, since X+ /* + * = 1, = + 2*38.
Series IIL In this case the data remain precisely as before, except that
we now have b = 15'3472.
We find
+A,a, 6) =111-2573
-A, a, 6) =110-2442
, a,6 + t) = 104 5571
If (4, a, 6-*) = 117-6519,
whence
Jf= 110-926.
Determining X, /&, r, as in the former case, we find
From these values, calculated for the circumference of the disc, we have
to subtract the value (Jf t ) applicable to the small circuit touched by the
inner brush. The area of this is %T (2-096 f. For the first and second series
we have
For the third series in like manner
JT.= 534.
Thus finally for the first and second series
JT- If. = 214-569,
and for the third series
J/- I/, = 11039*
7%e resistance-coils.
26. In all three series the resistance 6, fig. 3 r was a German-silver coil
of about ^j, referred to for brevity as the [^] : and the resistance a was
composed of three resistances in multiple arc, the first two being standard
lingipB, and the third a resistance such as 7 BJL units taken from a box.
To make the necessary change, according to the plan already explained in
9, the 7 would be replaced by & The value of o is of course determined
principally by the unit resistance-coils, and only secondarily by the resistance
taken from the box.
172 ON THE ABSOLUTE VALUE OF THE [94
The third element of the system of resistances was varied in the different
series. In the first series c was a [10], in the second series it was
[10] + [5] + [1], and in the third series [10] + [5] + [5']. Besides the
standard singles, whose values at various temperatures was already known
in terms of the mean B.A. unit, we had to determine accurately the values
of the [-$], the [10], the [5], and the [5'J, as well as the small resistances of
the various connecting pieces employed.
The [10] has been determined in various ways, but principally by means
of the device referred to in the former paper*. Three German-silver wires
of about 3 units each are wound on the same tube, and their terminals are so
arranged that by means of a base board containing mercury cups they can be
combined either in multiple arc or in series. In the former combination they
are compared with a standard single, and the resistance is found to be (say)
1 + a, where a is small. The coils are now without loss of time combined in
series, a change which can be effected in a moment. The resistance in series
is very approximately 9 + 9a ; by the addition of the standard single it
becomes 10 + 9a, and can now be compared with the [10]. If the difference
observed be /3 we have [10] = 10 + 9a+ 8. By this method it is easy to
obtain an accuracy of at least y^^y.
The [5]'s were determined in two ways. Five singles were combined in
series and compared with one of the [5]'s ; afterwards the two [o]'s were
compared with one another. In the second method, which is probably
preferable, the sum of the two [5]'s was found by comparison with' the [10].
From the sum and difference the separate values can of course be deduced.
The measurement of the [^] demanded some precaution on account of
its smallness. Two standard singles, the [10], and the [^], were combined
with four insulated mercury cups, and without the use of connecting pieces,
Fig. 5.
GALV.
BATTERY
GALV
so as to form a Wheatstone's balance (fig. 5), care being taken to bring the
associated battery and galvanometer terminals into immediate contact with
the legs of the [^] (see 12). To get the means of adjustment, a box,
* Phil. Trans. Part II. 1882, p. 697 [vol. n. p. 75.].
94] BRITISH ASSOCIATION UNIT OF RESISTANCE. 173
giving resistances up to 10,000, was placed in multiple arc with one of the
singles. If, as was the case, the four coils be so nearly in proportion that a
resistance of several hundreds from the box is needed for balance, the
delicacy of the arrangement is all that can be desired. Readings are
taken also with battery reversed, to eliminate thermo-electric disturbances.
Especial pains were taken with the measurement of the [jL], and of the [10],
errors of which would be propagated into the results of all three series.
27. The various temperatures of the coils at the time of use, and
the fluctuations from day to day, complicate the calculation of the effective
resistances ^ and /L, which in principle is simple enough. The results
are given in column II. of the Tables. Thus in the first series on July 14,
when the effective resistance was 0044076 B.A., as calculated from the
values of a, b, c, for the observed temperatures of the coils, the effect (E W)
of reversing the battery key (corrected for direct effect) was 30 divisions of
the galvanometer scale, the direction of rotation being positive. When the
effective resistance was altered to '0044430, the difference E W became
+ 10 divisions. From these results we infer that E W would vanish for
the effective resistance 0044341, as given in column V. The corresponding
result with negative rotation is given in column VI. These resistances
relate to the actual speed of rotation determined by the frequency of
vibration of the electric fork ( 6). To render the results of different days
fairly comparable, two small corrections have to be introduced, the first
relating to small alterations in the relative frequencies of the two forks,
as shown by the number of beats per minute (column VII.), the second to
variations in the frequency of the standard fork itself, dependent upon
change of temperature. The temperatures were read by a thermometer
which stood between the prongs of the standard, and are given in column IX.
The corrections necessary for reduction to a standard number of beats
(16 per minute) and to a standard temperature (16') are tabulated in
columns VI1L and X., and the corrected results themselves in XI. and XII.
In all cases the electric fork vibrated more quickly than the standard.
The degree of accordance in the numbers entered in these columns shows
the success of the observations, so far as relates to errors of a casual character.
In column XIII. the results of the positive and negative rotations are
combined, so as to exhibit the total result of the day's work.
The Table, showing the results of the third series, is divided into two
parts, corresponding to the two positions of the induction coils, before
and after reversal ( 19). In each position, it will be seen that two sets
of observations were taken upon one of the days. Both sets, however,
were complete, and in the interval between them the resistance-coils
were all dismounted. A similar precaution was taken at least once in each
of Series I. and II.
174
ON THE ABSOLUTE VALUE OF THE
[94
W
-S
Ji -
Sit
T3 CD Pi
O PH P-
O co <!
vJfU
iilli
o a s
l"s
'
rrec
to!
11
111
CO IO
+ + + + +
CO p CO C5
U5 O O
JO
p cp a
S'
+ 1 +1 +1 + I
50 O.CO -*5 ^O
Ol~ (Mi-l IMi-H IMr-l IMi-H
OO P'P T* 1 "? 1 T*' 1 ;' 9 s ?* I* 1 ? 5
60 4t<6 oco t-co coco ^tjco
COiH CQH CfliH OTrH OJf t OJi-H
1+ 1+ 1+ 1+ 1+ 1 +
g s:
94]
BRTTISH ASSOCIATIOS UXIT OF RESIST A VCK.
175
Il|
J ^l
|
5
i s
111
J
|
|
j
C X
? : ?
!~3
_s
"S -s-
yuiir.oo.
OO'JTHOH
I i
~
I
j!
t 'OOOOOOfi
1 '0000004
i 'oooooort
MnniiB , ,
i
1
1
o
1
Temperature
of
MUiulni'il fork
^
S
a
ruvotuiiut
1
S
1
B
=
***""
1
=
1
S
ao
J I
J
_-
iji
s
e
s
- =
f 1
r ~H
8
j
lip.
j
f
?
9
|
_5 f_
2
j
I1 S
I-
ii i;.'.,.iio J
I
llii
~
s
3
3
-* j=
r
?r
lif
""^ **
j-
: 'i +
T5
ii +
2
i 1
II
(iPtHROO.
tr.Hir.uo>
If
II
II f-f.f.
176
ON THE ABSOLUTE VALUE OF THE
[94
THIRD SERIES.
Coils separated.
Speed of disc about 12'8 revolutions per second.
Approximate resistances a = , b = -^j, c=20.
* SH
ilSPi
r-l <M <M
00 ^ D
?1 01 01
Means of all the
observations with
direction of rotation
1'
1 II
| II
'jt +
I IS
i Si
liifj-j
s ,3 2 a, ^ a "5
Ullill
& *"8 J=
W3 iO
CD < 1 t>- i ( rH O
d <M C^l Ol O1 <M 9
Effective resistance
(in B.A. units)
as finally corrected
1'
ill ill
a
I*
M
o
*
2
'o
1
PI
*o
1
1
k
2
S3
o
o
g
P
'%
+ + + ^ + + +
ck "^
111 -a
III- 8
H ^
tl-H
-tf "* T 1 O co -ft -H
00 GO 00 rt t- t- l>
J
Correction
to
72 beats
o ^
o ^ooo
o C3
8
1,1
1
cc
S S S g ?2 g
Effective resistance
(in B.A. units)
corresponding to
zero difference
in galvanometer
1,
1
lO >O O 0s GO O
| +
C<J 1C *O ^o CD CD
Difference
of reading of
galvanometer on
reversal of
current
1,
I
CO -^ Oi O D -H 00 t- r-l O "*
" O5 i-H O ' -! Ot?-'c<)i^liN
1+ 1+ 1+ ++++ +
a
o
i-l O T 1 ^- T 1 ? 5 03O5 OOOJ 00(M
rH pH nHOS ,H ' rH ' O
+ 1 1 1 + 1 II II II
gS .
.5 c-u-^.j
H M -a s
g^
11 If 11 (I 11 11
^
Q
II
-^
A- ^ '4 .. . 5 - -5 . -5 -
Tjl~lO~iO. CQ^C'-^t*^
94] BRITISH ASSOCIATION C3FTT OF RESISTANCE. 177
28. The results given in these tables are the effective resistances
required to obtain a balance, expressed in terms of the BuL unit To reduce
them to absolute measure we must multiply bj 10*, and bj a factor, which
we may call JT, expressing the absolute value of the BJL unit in terms
of 10*. and which it is our object to determine
The actual value of the same quantities in absolute measure is found
by multiplying the coefficients of induction (J/ Jf t ) already given (| 25)
by the number of turns in the coils 1588. and by the number of revolutions
per second.
In the first series the frequency of vibration (f) of the electric tuning-
fork was in the standard case (see Appendix)
/= (128-140 + f) = } x 128-407
and the number of revolutions per second is equal to 2/"-=- 10. In the
second and third series 2/= 129-340, a number which in the second series
is to be divided by 16, and in the third series by 10, in order to obtain the
number of revolutions per second.
The equation to determine x is thus for the first series of observations
214-569 x 1588 x 12-8407 = x x -00443407 x 10* r
whence
* = -98674.
From the second series
1588 x 129 340
16 x 10* x -00279157
From the third series
Ilfr392 x 1588 x 129 340
10 x 10* x D0229762
These are the final results already considered in | 21.
APPEXDEX.
Frequency of Vibration of Standard Fork.
All our measurements, both by this method and by that of the revolving
coil, being dependent upon the pitch of a standard tuning-fork, we have
considered it advisable to determine this element afresh. As in the fin*
determination*, a fork vibrating about 32 times per second rendered inter-
mittent an electric current, which, passing through the coils of small
- Pne. Soy. SBC. May 1881, p. 137 [roL n. p. 33}.
n.
178 ON THE ABSOLUTE VALUE OF THE [94
electromagnets, maintained in vibration not only the interrupter fork itself,
but also a second fork of pitch about 128. After the apparatus has been
a short time in operation, the vibrations of the second fork are exactly
four times as quick as those of the first, independently of any precise
tuning ; and they give rise to audible beats when the standard fork is
simultaneously excited. In the presence of extraneous noises the obser-
vation of the beats is much facilitated by the use of resonators, with one
of which the ear may be connected by an indiarubber tube. The object
to be aimed at is to make the intensities of the two sounds (as they
reach the ear) very nearly equal. The moment of antagonism is then
marked by a well-defined silence, whose occurrence can be timed to within
a second, although the whole duration of the beat may be 20 seconds or
more. Without fresh bowing of the standard, the silences can be observed
satisfactorily for at least a minute.
In the first determination the comparison between the fork of frequency
32 and the pendulum of the clock was made directly. The observer,
looking over a plate carried by the upper prong of the fork, obtained 32
views per second, i.e., 64 views of the pendulum in one complete vibra-
tion. The immediate subject of observation is a silvered bead attached
to the bottom of the pendulum, upon which as it passes the position of
equilibrium the light of a paraffin lamp is concentrated. Close in front
of the pendulum is placed a screen perforated by a somewhat narrow
vertical slit. If the period of the pendulum were a precise multiple of
that of the fork, the flash of light which to ordinary observation would
be visible at each passage, would either be visible, or be obscured, in a
permanent manner. If, as in practice, the coincidence be not perfect, the
flashes appear and disappear in a regular cycle, whose period is the time
in which the fork gains (or loses) one complete vibration. This period can
be determined with any degree of precision by a sufficient prolongation of
the observations.
On account of the large number of views per second, the interval be-
tween successive visible positions of the bead, even when it is moving
with maximum velocity, is rather small ; and thus the adjustment of the
apparatus is somewhat delicate*. In order to meet this objection, a modi-
fication has been introduced, which must now be explained f.
* In the earliest use of this method (Nature, vol. xvii. p. 12, 1877) [vol. i. p. 333] the
break-fork had a frequency of about 13, and no difficulty of this kind was experienced.
t July, 1883. It should be stated, however, that the wheel may easily be dispensed with,
if proper care be taken in the illumination of the bead and in the management of the fork.
The vibration should be vigorous, and the screens so arranged that the view past the fork
at the moment of greatest elongation should be of short duration. Determinations by this
method (without the wheel) have often been made successfully by students in the Cavendish
Laboratory.
94] BRITISH ASSOCIATION UNIT OF RESISTANCE. 179
A few years ago it was shown almost simultaneously by La Cour and
by Lord Rayleigh [Art. 56, vol. I. p. 355], that an electromagnetic engine
could be accurately governed by an interrupter-fork. The construction
(fig. 6) which has been found most suitable is similar to that of Froment's
engine. A horizontal shaft revolving upon steel points carries a number
of parallel soft iron armatures, disposed symmetrically round the circum-
ference. In the course of the revolution these armatures pass in succession
between the poles of a vertical horse-shoe electromagnet, so as almost to
complete the magnetic circuit. It is much better that the armatures
should pass between the poles than over them, as in the most usual arrange-
ment, for in the latter case the bearings are subjected to an unnecessary
and prejudicial strain. The wheel may be used either with or without
an independent driving power. In the former case the power should be
very steady, and adjusted so as to give by itself nearly the speed in-
tended. The currents from the interrupter-fork are passed also through
the electromagnet of the engine, and give the force required to accelerate
or retard the motion so that it may exactly synchronise with the fork,
one armature passing for each complete vibration. If the independent
power is in excess, the phase of the motion is such that the electromagnet
is excited principally after the armatures have passed through the electro-
magnet; if the independent power is in defect, the electromagnet is ex-
cited principally while the armatures are approaching it. Within certain
limits any necessary acceleration or retardation is obtained by suitable
self-acting adjustment of phase.
Fig. 6.
If when the wheel is moving steadily under the influence of the inter-
mittent currents, a slight disturbance is communicated to it, oscillations
will set in, the wheel being alternately in advance and in the rear of its
proper position. In some cases these oscillations are very persistent, and
interfere seriously with the utility of the instrument. To check them, a
hollow ring filled with water is attached to the shaft and revolves with
it When the rotation is perfectly regular, the water behaves as if it
12 2
180 ON THE ABSOLUTE VALUE OF THE [94
were a rigid body and offers no impediment to the motion, but it tends
to check variations of speed of moderate period. The oscillations, when
they exist, are usually audible ; and in any case the behaviour of the wheel
in this and other respects may be examined by looking at the interrupter-
fork through a paper disc carried by the wheel and perforated symmetrically
along a circle with holes equally numerous with the armatures. When all is
regular, the prongs of the fork are seen in one phase only, so long as the eye
retains a position fixed in space.
When the wheel runs lightly, independent driving power may be dis-
pensed with, a sufficient amount of work being obtainable from the inter-
mittent governing current. In the present case the whole apparatus,
consisting of the two forks and the wheel, was driven by one current
supplied from three Grove cells. The only difficulty experienced is in
starting the wheel. By means of string passed once round the shaft,
alternately tightened for the advance and slackened for the return, it is
easy to cause the wheel to achieve a speed in excess of the necessary
eight revolutions per second. But it will not usually happen, every time
the speed falls through the proper value, that the wheel will engage with
the fork. For this purpose it is necessary that at the moment in question
the phase of the wheel should be correct, within limits, which may be narrow
when there is no great margin of power ; and this can only happen by
chance. Several attempts may be necessary before success is reached. With
a little practice, however, there is no great loss of time, the ear learning to
recognise, by the gradual slowing and subsequent quickening of a sort of
beat, when the wheel has passed through the right speed without engage-
ment. A fresh impulse is then given without waiting further. After a start
is once effected, the wheel will usually run, keeping perfect time with the
fork, until the battery is exhausted.
The wheel employed in the experiments we are now concerned with,
has four soft iron armatures, and is governed by the interrupter-fork of
frequency 32. The speed of the wheel is thus eight revolutions per second ;
and a single hole in a paper disc carried round with it allows eight views of
the pendulum per second, the smallest number of views obtainable by direct
use of the fork being 32. Altogether we may regard the frequency of the
interrupter-fork as being multiplied four times precisely in the frequency
of the auxiliary fork, and as divided four times precisely in the frequency of
the wheel. The former is directly comparable with the standard fork, and
the latter with the clock. The standard fork was screwed to the table
precisely as during the electrical measurements. A thermometer placed
between the prongs gave the temperature with fair accuracy.
The calculation of the results is very simple. Supposing in the first
instance that the clock is correct, let a be the number of cycles per second
94] BRITISH ASSOCIATION UNIT OF RESISTANCE. 181
(perhaps ^) between the wheel and the clock. Since the period of a cycle
is the time required for the wheel to gain, or to lose, one revolution upon
the clock, the frequency of revolution is 8 a. The frequency of the
auxiliary fork is precisely 16 times as great, .., 128 16o. If 6 be the
number of beats per second between the two forks, the frequency of the
standard is
128 16a 6.
To give an idea of the magnitudes of the numbers concerned, it will be
advisable to quote in detail the results of one day's observations. On
October 19, with a certain loading of the interrupter-fork, the cycle of the
pendulum occupied about 78 seconds, and the beats were at the rate of about
six per minute. The interrupter was then sharpened, after which several
observations were taken of the duration of five cycles of the pendulum,
and of 16 beats between the forks. For the former the times found were
210, 210, 212 seconds; for the latter by simultaneous observation 58, 58 t 59,
59, 59, 60, 60 seconds. The temperature, as given by the thermometer,
ranged from !7 c- 2 to 17 0- 4. After the sharpening of the interrupter, the
frequency both of the wheel and of the auxiliary fork was increased, so that
the sign of 16a in the expression written above is determined to be + and
that of 6 to be . Using the mean values we find
16a = -3797, 6 = -2712,
whence
128 + 16a - b = 128-108.
To this we must add O09, making altogether 128-117, to allow for the
gaining rate of the clock, which was 6 seconds per diem. This corresponds
to a mean temperature 17 = -:3.
The procedure adopted was quite good enough for our purpose : but if
it were desired to push the power of the method to its limit, the work
should be undertaken at an astronomical observatory, and extended over the
whole time required to rate the clock by observations of the stars. In this
way the comparison of the period of vibration of the standard fork with the
mean solar second could be effected with the same degree of accuracy as
that to which the former quantity is capable of definition. Without this
precaution we cannot be quite sure that the rate of the clock at the time of
the observations is identical with the mean rate employed in the calculation.
It is scarcely necessary to say that the uncertainty which arises under
this head is common to every method by which absolute pitch could be
determined.
The results obtained, including those recorded previously*, are given in
the accompanying table. They are well represented by the formula
128 140 x (1 - (t - 16) x TO011},
* Pne. Say. Soe. My 1881, p. 138 [roL n. p. 33}.
182
ON THE ABSOLUTE VALUE OF THE
[94
in which the temperature coefficient used ('00011) is that found by M'Leod
and Clarke*. The numbers in the fourth column are calculated from the
formula.
Date
Temperature
Frequency by
observation
Frequency by
calculation
1881
13
128-180
128-182
1881
14*6
128-161
128-160
October, 1882 . .
15-98
128-141
128-140
October, 1882 . .
17-45
128-122
128-120
October, 1882 . .
17'6
128-119
128-118
October, 1882 . .
17-3
128-117
128-122
Of the small discrepancies which the table exhibits it is probable that
the larger part is due to imperfect knowledge of the actual temperatures
of the standard fork. The use of screens to cut off radiation from the
observers would probably have effected an improvement. For the highest
accuracy some sort of jacket, or chamber, would have to be contrived.
SECOND APPENDIX.
(Added July, 1883.)
On the Effect of the Imperfect Insulation of Coils.
In a former paper (Phil. Trans. 1882 [vol. II. p. 51]), it was pointed out
that the method of the revolving coil, employed by the first B.A. Committee,
possesses the important advantage that it is possible to detect the existence
of leakage from turn to turn, or from layer to layer, of the coil of wire.
The general influence of such leakage, if undetected, upon the final number
x expressing the ratio of-|he resistance of the coil when measured (R) in
absolute units to its resistance r x 10 9 as referred to B.A. units, is easily
seen by supposing that one turn of the coil is simply short-circuited. The
formula in c.G.s. measure is
R _ 7r 2 w 2 a at cot </>
rx 10 9 = rx 10 9
(1)
During the revolutions the short-circuited turn produces its full effect in
deflecting the magnet, and error arises only in the comparison with the
standard of resistance. The quantity r will evidently be under-estimated
* Phil. Trans. Part I. 1880.
94] BRITISH ASSOCIATION UNIT OF RESISTANCE. 183
by 1/jj, and this will lead to an over-estimate of a?, also by 1/n. This result,
however, is modified, if as in practice we take only the difference of effects
observed when the wire contact is open and closed. The short-circuited
turn will produce its effects in both cases, and its influence will therefore
disappear from the result. For all purposes it will be virtually non-existent,
and the error produced is the same as if n had simply been miscounted.
The final number x will thus be over-estimated by the fraction 2/n.
In Lorenz's method the effect of a short circuit in the induction coil
is in the same direction. Af, and therefore R and x, will be over-estimated
by 1/n.
If we examine the formulae applicable to determinations by other
methods, we shall see that a similar conclusion holds good, so that in
every case leakage leads to an over-valuation of x, at least whenever the
result is calculated from the number of turns of wire in a coil*. Even
without such an examination, it is pretty evident from consideration of the
magnitudes involved that the large factor 10 9 in the denominator of the
formula corresponding to (1) can only be compensated by one or more
large factors expressive of the number of windings in a coil or coils. An
over-valuation of these factors, due to leakage, will therefore lead to an
over-valuation of x.
In carefully constructed coils serious leakage is, perhaps, not likely to
occur, but its presence in a smaller degree is more probable, and is usually
difficult of detection. So far as this argument applies, we may say that the
smaller values of the number expressive of the B A. unit, or of the mercury
unit, in absolute measure are to be preferred to the larger.
* The case is different when the constants of a coil of many turns are determined by
electrical comparison, as for instance in Kohlraosch's recent correction of the constant of
his earth-inductor.
95.
ON THE MEAN RADIUS OF COILS OF INSULATED WIRE.
[Proceedings of the Cambridge Philosophical Society, IV. pp. 321324, 1883.]
IN electrical work it is often necessary to use coils of such proportions
that their constants cannot well be obtained from the data of construction,
but must be determined by electrical comparison with other coils whose
proportions are more favourable. A method for comparing the galvanometer-
constants of two coils, i.e. of finding the ratio of magnetic forces at their
centres when they are traversed by the same current, is given in Maxwell's
Treatise, vol. n. 753.
I have used a slight modification of Maxwell's arrangement which is
perhaps an improvement, when the coils to be compared are of copper and
therefore liable to change their resistance pretty quickly in sympathy with
variations of temperature. The coils are placed as usual approximately in
the plane of the meridian so that their centres and axes coincide, and a very
short magnet with attached mirror is delicately suspended at the common
centre. If the current from a battery be divided between the coils, connected
in such a manner that the magnetic effects are opposed, it will be possible by
adding resistance to one or other of the branches in multiple arc to annul the
magnetic force at the centre, so that the same reading is obtained whichever
way the battery current may circulate. The ratio of the galvanometer
constants is then simply the ratio of the resistances in multiple arc.
To obtain this ratio in an accurate manner, the two branches already
spoken of are combined with two other resistances of german silver, so as to
form a Wheatstone's balance. Of these resistances both must be accurately
known, and one at least must be adjustable. The electromagnetic balance is
first secured by variation of the resistance associated with one of the given
coils, which resistance does not require to be known. During this operation
the galvanometer of the Wheatstone's bridge is short-circuited. Afterwards
95] ON THE MEAN RADIUS OF COILS OF INSULATED WIRE. 185
the galvanometer is brought into action and the resistance-balance is
adjusted. The ratio of the galvanometer-constants is thus equal to the
ratio of the german silver resistances. The two adjustments may be so
rapidly alternated as to eliminate any error due to changes of temperature in
the copper wires. Indeed, if desired, the final tests of the electromagnetic
and resistance-balances might be made simultaneously.
If the ratio of galvanometer-constants be the final object of the measure-
ment, there is nothing more to be done ; but if we desire to know the ratio of
the mean radii of the coils we must introduce certain small corrections for
the finite dimensions of the sections. In the first place, however, it will be
desirable to consider a little more closely what should be understood by the
mean radius of a coil.
In Maxwell's treatment of the subject ( 700) the mean radius of a coil is
considered to correspond with the geometrical centre of its rectangular
section, that is to say, the windings are assumed to be uniformly distributed
over the section. In practice absolute uniformity is not attainable, and it is
therefore proper to take into account the effect of a small imperfection
in this respect. The density of the windings, i.e. the number of windings per
unit area, may be denoted by p, and is to be regarded as approximately
constant.
The introduction of the factor p makes but little difference in the
investigation of 700. If we take the origin of co-ordinates x and y, no
longer at the geometrical centre, but at what may be called the centre
of density of the section, we shall have (as in the ordinary theory of the
centre of gravity)
ffpxdxdy = 0, ffpydxdy = 0,
the integrations being extended over the area of the section. If P be any
function of x and y, P the mean value of the function (with reference to p),
P, the value at the origin, we have
the terms of the first order disappearing in consequence of the choice of
origin. In the terms of the second order we may neglect the effect of
variable density, and write
ffpxydxdy = 0,
, ri being the breadths in the directions of* and y of the rectangular section.
Thus
186 ON THE MEAN RADIUS OF COILS OF INSULATED WIRE. [95
The form of this expression is the same as when the windings are
supposed to be distributed with absolute uniformity, but the mean radius
and mean plane are to be reckoned with reference to the density of the
windings.
In the application to the galvanometer-constant of a coil, we have, if A be
the mean radius, the radial and 77 the axial dimension of the section,
by means of which, and 77 being approximately known, G l may be inferred
from A, or conversely A may be inferred from GV If the ratio of galvano-
meter-constants of two coils has been determined by the electrical process,
the ratio of mean radii can be accurately deduced by use of the above
formula.
When the mean radius of a coil has been determined in this manner by
comparison with another of proportions more favourable for calculation from
the data of construction, other quantities relating to the coil may be deduced
by mere calculation. For instance, the important constant g lt denoting the
mean area included by the windings, is connected with the mean radius A by
the equation
A more direct process for determining g l electrically is given by Maxwell
754, and has recently received an important application in the hands of
Kohlrausch. In this method the quantity sought is proportional to the cube
of a distance not very easy of precise measurement ; and it is possible
that the less direct method explained above may be the more accurate
in practice.
96.
ON THE INVISIBILITY OF SMALL OBJECTS IN A BAD
LIGHT.
[Proceedings of the Cambridge Philosophical Society, iv. p. 4, 1883.]
IN a former communication to the Society (March 6, 1882) [Art. 82,
voL IL p. 92] I made some remarks upon the extraordinary influence of
apparent magnitude upon the visibility of objects whose 'apparent bright-
ness' was given, and I hazarded the suggestion that in consequence of
aberration (attending the large aperture of the pupil called into operation in
a bad light) the focussing might be defective. Further experiment has
proved that in my own case at any rate much of the effect is attributable to
an even simpler cause. I have found that in a nearly dark room I am
distinctly short-sighted. With concave spectacles of 36 inches negative
focus my vision is rendered much sharper, and is attended with increased
binocular effect. On a dark night small stars are much more evident with
the aid of the spectacles than without them.
In a moderately good light I can detect no signs of short-sightedness. In
trying to read large print at a distance I succeeded rather better without the
glasses than with them*. It seems therefore that the eflvct is not to be
regarded as merely an aggravation of permanent short-sightedness by increase
of aperture.
The use of spectacles does not however put the small and the large
objects on a level of brightness when seen in a bad light, and the outstanding
difference may still be plausibly attributed to aberration.
* [1899. It may be worthy of record that sixteen years later, baring now lost nearly all
power of accommodation, I find lenses of about 36 inches negative focus
to see distant objects perfectly.]
97.
ON MAINTAINED VIBRATIONS.
[Philosophical Magazine, xv. pp. 229235, 1883.]
WHEN a vibrating system is subject to dissipative forces, the vibrations
cannot be permanent, since they are dependent upon an initial store of
energy which suffers gradual exhaustion. In the usual equation
' '
K is positive, and the solution indicates the progressive decay of the
vibrations in accordance with the exponential law. In order that the
vibrations may be maintained, the vibrating body must be in connexion
with a source of energy. This condition being satisfied, two principal
classes of maintained vibrations may be distinguished. In the first class
the magnitude of the force acting upon the body in virtue of its connexion
with the source of energy is proportional to the amplitude, and its phase
depends in an approximately constant manner upon the phase of the
vibration itself; in the second class the body is subject to influences whose
phase is independently determined.
The first class is by far the more extensive, and includes vibrations
maintained by wind (organ-pipes, harmonium-reeds, ffiolian harps, &c.), by
heat (singing flames, Rijke's tubes, &c.), by friction (violin-strings, finger-
glasses, &c.), as well as the slower vibrations of clock-pendulums and of
electromagnetic tuning-forks. When the amplitude is small, the force acting
upon the body may be divided into two parts, one proportional to the
displacement (or to the acceleration), the second proportional to the
velocity ddjdt. The inclusion of these forces does not alter the form of (1).
By the first part (proportional to 6} the pitch is modified, and by the second
the coefficient of decay*. If the altered K be still positive, vibrations
* For more detailed application of this principle to certain cases of maintained vibrations,
see Proceedings of the Royal Institution, March 15, 1878. [Art. 55, vol. i. p. 348.]
97] OUT MAINTAINED VIBRATIONS. 189
gradually die down ; bat if the effect of the included forces be to render
the complete value of K negative, vibrations tend on the contrary to increase.
The only case in which according to (1) a steady vibration is possible, is
when the complete value of is zero. If this condition be satisfied, a
vibration of any amplitude is permanently maintained.
When K is negative, so that small vibrations tend to increase, a point is
of course soon reached after which the approximate equations cease to be
applicable. We may form an idea of the state of things which then arises by
adding to equation (1) a term proportional to a higher power of the velocity.
Let us take
in which K and K are supposed to be small The approximate solution
of (2) is
6 = A sin nt + ^7^- cos 3nf, ........................ (3)
O
in which A is given bv
* + 'n*4*=0 .............................. (4)
From (4) we see that no steady vibration is possible unless * and *' have
different signs. If * and K" be both positive, the vibration in all cases dies
down; while if K and *' be both negative, the vibration (according to (2
increases without limit. If * be negative and *' positive, the vibration
becomes steady and assumes the amplitude determined by (4). A smaller
vibration increases up to this point, and a larger vibration falls down to it.
If, on the other hand, * be positive, while f is negative, the steady vibration
abstractedly possible is unstable, a departure in either direction from the
amplitude given by (4) tending always to increase.
Of the second class the vibrations commonly known as forced have the
first claim upon our attention. The theory of these vibrations has long been
well understood, and depends upon the solution of the differential equation
formed by writing as the right-hand member of (1) Pcaspt in place of zero.
The period of steady vibration is coincident with that of the force, and
independent of the natural period of vibration : but the amplitude of
vibration is greatly increased by a near agreement between the two periods
In all cases the amplitude is definite and is proportional to the magnitude of
the impressed force. When the force, though strictly periodic, is not of the
simple harmonic type, vibrations may be maintained by its operation whose
period is a snbmultiple of the principal period.
There is also another kind of maintained vibration which from one point
of view may be considered to be forced, inasmuch as the period is imposed
from without, but which differs from the kind just referred to in that the
imposed periodic variations do not tend directly to displace the body from its
configuration of equilibrium. Probably the best-known example of this kind
190 ON MAINTAINED VIBRATIONS. [97
of action is that form of Melde's experiment in which a fine string is main-
tained in transverse vibration by connecting one of its extremities with the
vibrating prong of a massive tuning-fork, the direction of motion of the point
of attachment being parallel to the length of the string*. The effect of the
motion is to render the tension of the string periodically variable ; and at
first sight there is nothing to cause the string to depart from its equilibrium
condition of straightness. It is known, however, that under these circum-
stances the equilibrium position may become unstable, and that the string
may settle down into a state of permanent and vigorous vibration, whose
period is the double of that of the point of attachment^.
The theory of vibrations of this kind presents some points of difficulty,
and does not appear to have been treated hitherto. In the present investiga-
tion we shall start from the assumption that a steady vibration is in progress,
and inquire under what circumstances the assumed state of things is
If the force of restitution, or ' spring,' of a body susceptible of vibration
be subject to an imposed periodic variation, the differential equation becomes
0, .................. (5)
in which K and a are supposed to be small. A similar equation would apply
approximately in the case of a periodic variation in the effective mass of the
body. The motion expressed by the solution of (5) can only be regular when
it keeps perfect time with the imposed variations. It will appear that the
necessary conditions cannot be satisfied rigorously by any simple harmonic
vibration ; but we may assume
= A 1 sin pt + Bi cospt + A 3 sin 3pt + B 3 cos 3pt + A 5 sin 5pt 4- . . ., . . .(6)
in which it is not necessary to provide for sines and cosines of even multiples
of pt. If the assumption is justifiable, the series in (6) must be convergent.
Substituting in the differential equation, and equating to zero the coefficients
ofsinpt, cospt, &c., we find
A, (n 2 - p 2 ) - K pB l - aA -I- ctB 3 = 0,
B l O 2 - p 2 ) + xpA l -aA l -aA 3 = 0,
A 3 (n 2 - 9p 2 ) - 3*p5 8 - aB, + aB 5 = 0,
B 3 (n* - 9/) 2 ) + 3*pA 3 + aA l - aA 5 = 0,
A, (n 2 - 25p 2 ) - 5xpB 5 - aB s + aB 7 = 0,
5tcpA s + aA 3 - aA 7 = 0,
* When the direction of motion is transverse, the case falls under the head of ordinary
forced vibrations.
t See Tyndall's Sound, 3rd ed. ch. in. 7, where will also be found a general explanation
of the mode of action.
97] OS MAKTADfED VIBRATIONS. 191
These equations show that relatively to A t , B,, A,, B* are of the older a:
that relatively to A^ B* t A s , B, are of the order a, and so on. If we omit
J,, 1?, in the first pair of equations, we find as a first approximation,
whence
(*>-j?JP=rf-*y. .............................. (8)
Thns T if a be given, the value of p necessary for a regular motion is definite ;
and p having this value, the regular motion is
in which e being equal to tan" 1 (B^A^ is also definite. On the other hand,
as is evident at once from the linearity of the original equation, there is
nothing to limit the amplitude of vibration.
These characteristics are preserved however far it may be necessary t:
pursue the approximation. If A*^^ ^- rl , may be neglected, the first m
pairs of equations determine the ratios of all the coefficients, leaving the
absolute magnitude open ; and they provide further au equation connecting
p and a, by which the pitch is determined.
For the second approximation the second pair of equations gives
A *** JL **
* +=ip' * **^v'
whence
d=PsinOrf+6) + ^^cos(3 / rf + );
and from the first pair
tane=j*-p*- ;|f ^!-( a + * / >), ......... (10)
while p is determined by
Returning to the first approximation, we see from (8) that the solution is
only possible under the condition that a>rp. If a = *p. then p = * : i>. the
imposed variation in the 'spring" must be exactly twice as quick as the
natural vibration of the body would be in the absence of friction. From (7)
it appears that in this case e = 0, which indicates that the spring is a
minimum one-eighth of a period after the body has passed its position of
192 ON MAINTAINED VIBRATIONS. [97
equilibrium, and a maximum one-eighth of a period before such passage.
Under these circumstances the greatest possible amount of energy is
communicated to the system ; and in the case contemplated it is just
sufficient to balance the loss by dissipation, the adjustment being evidently
independent of the amplitude.
If a < Kp, sufficient energy cannot pass to maintain the motion, whatever
may be the phase-relation ; but if a > Kp, the equality between energy
supplied and energy dissipated may be attained by such an alteration of
phase as shall diminish the former quantity to the required amount. The
alteration of phase may for this purpose be indifferently in either direction ;
but if e be positive, we must have
while if e be negative,
p* = n 2 + V( 2 - K-p z ).
If a be very much greater than Kp, e = \ir, which indicates that when the
system passes through its position of equilibrium the spring is at its maximum
or at its minimum.
The inference from the equations that the adjustment of pitch must be
absolutely rigorous for steady vibration will be subject to some modification
in practice ; otherwise the experiment could not succeed. In most cases in? is
to a certain extent a function of amplitude ; so that if n 2 have very nearly
the required value, complete coincidence is attainable, without other altera-
tion in the conditions of the system, by the assumption of an amplitude of
large and determinate amount.
When a particular solution of (5) has been found, it may be generalized
by a known method. Thus, if 6 =A0 1} we have as the complete solution
= A0 1 + B0 l t'e^e-^dt,
Jo
which may be put into the form
= P0 1 -B0J e^e- Kt dt ...................... (12)
J t
When t is great, the second term diminishes rapidly, and the solution tends
to assume the original form 6 = P0 l .
The number of cases falling under the present head which have been
discovered and examined hitherto is not great. The mysterious son rauqiie
of Savart, which sometimes accompanies the longitudinal vibrations of bars,
and which is attributed by Terquem to an associated transverse vibration, is
doubtless of this character. Just as in Melde's experiment already spoken of,
the periodic variations of tension accompanying the longitudinal vibrations
97] ON MAINTAINED VIBRATIONS. 193
will throw the bar into lateral vibration, if there happen to be a mode of
such vibration whose pitch is nearly enough coincident with the suboctave of
the principal note.
For a lecture illustration we may take a pendulum formed of a bar of soft
iron and vibrating on knife-edges. Underneath the pendulum is placed
symmetrically a vertical bar electromagnet, through which is caused to pass
an electric current rendered intermittent by an interrupter whose frequency
is twice that of the pendulum. The magnetic force does not tend to displace
the pendulum from its equilibrium position, but produces the same sort of
effect as if gravity were subject to a periodic variation.
A similar result is obtained by causing the point of support of the
pendulum to vibrate in a vertical path. If we denote this motion by
t] = y8 sin 2pt, the effect is as if gravity were variable by the term 4/? 2 /3 sin 2pt.
Of the same nature are the crispations observed by Faraday and others on
the surface of water which oscillates vertically. Faraday arrived experi-
mentally at the conclusion that there were two complete vibrations of the
support for each complete vibration of the liquid. This view has been
contested by Matthiessen *, who maintains that the vibrations are isoperiodic.
By observations, which I hope to find another opportunity of detail ing f,
I have convinced myself that in this matter Faraday was perfectly correct.
The vibrations of water standing upon a horizontal glass plate, which was
attached to the centre of a vibrating iron bar, were at the rate of 15 per
second when the vibrations of the bar were at the rate of 30 per second. The
only difference of importance between this case and that of the pendulum is
that, whatever may be the rate of vibration of the plate, there is always
possible a free water-vibration of nearly the same frequency, and that conse-
quently no special tuning is called for.
* Pogg. Ann. vol. cm. 1870.
t [1899. See Art. 102 below. It should be remarked that corrections have been introduced
in equations (10), (11) above.]
13
98.
THE SOAKING OF BIRDS.
[Nature, xxvu. pp. 534, 535, 1883.]
THE recent correspondence in Nature upon this subject ought not to
close without some reference to a possible explanation of soaring which does
not appear to have been yet suggested.
I premise that if we know anything about mechanics it is certain that a
bird without working his wings cannot, either in still air or in a uniform
horizontal wind, maintain his level indefinitely. For a short time such
maintenance is possible at the expense of an initial relative velocity, but this
must soon be exhausted. Whenever therefore a bird pursues his course for
some time without working his wings we must conclude either (1) that the
course is not horizontal, (2) that the wind is not horizontal, or (3) that the
wind is not uniform. It is probable that the truth is usually represented by
(1) or (2); but the question I wish to raise is whether the cause suggested
by (3) may not sometimes come into operation*.
In Nature, Vol. xxiu. p. 10, Mr S. E. Peal makes very distinct statements
as to the soaring of pelicans and other large birds in Assam. The course is
in large and nearly circular sweeps, and at each lap some 10 or 20 feet of
elevation is gained. When there is a wind, the birds may in this way
" without once flapping the wings " rise from a height of 200 to a height of
8000 feet.
That birds do not soar when there is no wind is what we might suppose,
but it is not evident how the existence of a wind helps the matter. If the
wind were horizontal and uniform, it certainly could not do so. As it does
not seem probable that at a moderate distance from the ground there could
Under this head reference may be made to Langley's Memoir on the Internal Work
of the Wind, Smithsonian Contributions to Knowledge, 1893.]
98] THE SOARING OF BIRDS. 195
be a sufficient vertical motion of the air to maintain the birds, we are led to
inquire whether anything can be made of the difference of horizontal veloci-
ties which we know to exist at different levels.
In a uniform wind the available energy at the disposal of the bird
depends upon his velocity relatively to the air about him. With only a
moderate waste this energy can at any moment be applied to gain elevation,
the gain of elevation being proportional to the loss of relative velocity
squared. It will be convenient for the moment to ignore the waste referred
to, and to suppose that the whole energy available remains constant, so that
however the bird may ascend or descend, the relative velocity is that due to
a fall from a certain level to the actual position, the certain level being of
course that to which the bird might just rise by the complete sacrifice of
relative velocity.
For distinctness of conception let us now suppose that above and below a
certain plane there is a uniform horizontal wind, but that in ascending
through this plane the velocity increases, and let us consider how a bird
sailing somewhat above the plane of separation, and endowed with an initial
relative velocity, might take advantage of the position in which he finds
himself.
The first step is, if necessary, to turn round until the relative motion is to
leeward, and then to drop gradually down through the plane of separation.
In falling down to the level of the plane there is a gain of relative velocity,
but this is of no significance for the present purpose, as it is purchased by
the loss of elevation ; but in passing through the plane there is a really
effective gain. In entering the lower stratum the actual velocity is indeed
unaltered, but the velocity relatively to the surrounding air is increased.
The bird must now wheel round in the lower stratum until the direction of
motion is to windward, and then return to the upper stratum, in entering
which there is a second increment of relative velocity. This process may
evidently be repeated indefinitely; and if the successive increments of
relative velocity squared are large enough to outweigh the inevitable waste
which is in progress all the while, the bird may maintain his level, and even
increase his available energy, without doing a stroke of work.
In nature there is of course no such abrupt transition as we have just
now supposed, but there is usually a continuous increase of velocity with
height. If this be sufficient, the bird may still take advantage of it to
maintain or improve his position without doing work, ou the principle that
has been explained. For this purpose it is only necessary for him to descend
while moving to leeward, and to ascend while moving to windward, the
simplest mode of doing which is to describe circles on a plane which inclines
downwards to leeward. If in a complete lap the advantage thus obtained
132
196 THE SOARING OF BIRDS. [98
compensates the waste, the mean level will be maintained without expendi-
ture of work ; if there be a margin, there will be an outstanding gain of level
susceptible of indefinite repetition.
A priori, I should not have supposed the variation of velocity with height
to be adequate for the purpose ; but if the facts are correct, some explanation
is badly wanted*. Mr Peal makes no mention of the circular sweeps being
inclined to the horizon, a feature which is essential to the view suggested.
It is just possible, however, that the point might escape attention not
specially directed to it.
However the feat may be accomplished, if it be true that large birds can
maintain and improve their levels without doing work, the prospect for
human flight becomes less discouraging. Experimenters upon this subject
would do well to limit their efforts for the present to the problem of gliding
or sailing through the air. When a man can launch himself from an eleva-
tion and glide long distances before reaching the ground, an important step
will have been gained, and until this can be done, it is very improbable that
any attempt to maintain the level by expenditure of work can be successful.
Large birds cannot maintain their levels in still air without a rapid horizontal
motion, and it is easy to show that the utmost muscular work of a man is
utterly inadequate with any possible wings to allow of his maintenance in a
fixed position relatively to surrounding air. With a rapid horizontal motion,
the thing may perhaps be possible, but for further information bearing upon
this subject, I must refer to a paper on the resistance of fluids published in
the Philosophical Magazine for December, 1876.
[1899. The maintenance of a fixed position in still air, whether by a
bird or by a man or by an engine, can only be secured by the generation of a
downward current of air, e.g. by a screw, whose momentum shall balance the
weights to be supported. If v denote the velocity, S the section of the
stream, p the density of air, the momentum generated in unit time is Spv 2 ;
and the work done in the same time is ^Spv 3 . Thus if gM be the whole
weight sustained,
gM=Spv* (1)
* [1899. A good deal depends upon the velocity of flight. If this reckoned relatively to the
surrounding air be called v, and if it become v', whether owing to a passage of the bird into
another stratum or to a freshening of the wind in the same stratum, the gain (h) of potential
elevation is given by
v'*-v*=2gh,
from which we see that the effect of a given change (v'-v) increases with v.
If we suppose that v = 3Q miles per hour, and that ft = 10 feet, we find /=34'7 miles per hour;
so that at this speed a gain of 10 feet requires a freshening of the wind amounting to 4-7 miles
per hour.
See further a letter on the Sailing Flight of the Albatross, Nature, XL. p. 34, 1889.]
THE SOARING OF BIRDS. 197
Again, if V denote the rate at which the weight must be lifted in order to
represent the work done by the driving engine,
(2)
Thus v=2V, and
So far as these equations are concerned, any weight can be sustained by a
limited expenditure of work, but the smaller the power available the larger
must be the section of the stream of air and consequently of the mechanism
by which the air is set in motion. Again, from (3)
......... .................... <>
so that, if S be given, the whole power required varies as (gMy.
To obtain numbers applicable to the case of a man supporting himself by
his own muscular power, we take in C.G.S. measure,
M = 68000, F=15, p = -sfo, = 981,
thus finding
S = 6-0 x 10 r sq. cm.
This represents the cross-section of the descending column of air. If we
equate S to Jwd 2 , d will be the diameter of the screw required, and we get
d = 90 metres. It is to be observed that the assumed nature of V corre-
sponds to the power which a man may exercise when working for 8 hours
a day. But even if he could do ten times as much for a few minutes, d
would still amount to 9 metres, and in this estimate nothing has been
allowed for the weight of the mechanism, or for frictional losses.
The present subject is further discussed in the Wilde Lecture on the
Mechanical Principles of Flight (Manchester Proceedings 1900).]
99.
DISTRIBUTION OF ENERGY IN THE SPECTRUM.
[Nature, xxvu. pp. 559, 560, 1883.]
IN the reaction against the arbitrariness of prismatic spectra there seems
to be danger that the claim to ascendancy of the so-called diffraction
spectrum may be overrated. On this system the rays are spaced so that
equal intervals correspond to equal differences of wave-length, and the
arrangement possesses indisputably the advantage that it is independent of
the properties of any kind of matter. This advantage, however, would not
be lost, if instead of the simple wave-length we substituted any function
thereof; and the question presents itself whether there is any reason for
preferring one form of the function to another.
On behalf of the simple wave-length, it may be said that this is the
quantity with which measurements by a grating are immediately concerned,
and that a spectrum drawn upon this plan represents the results of experi-
ment in the simplest and most direct manner. But it does not follow that
this arrangement is the most instructive.
Some years ago Mr Stoney proposed that spectra should be drawn so that
equal intervals correspond to equal differences in the frequency of vibration.
On the supposition that the velocity of light in vacuum is the same for all
rays, this is equivalent to taking as abscissa the reciprocal of the wave-length
instead of the wave-length itself. A spectrum drawn upon this plan has as
much (if not more) claim to the title of normal, as the usual diffraction
spectrum.
The choice that we make in this matter has an important influence upon
the curve which represents the distribution of energy in the spectrum. In
all cases the intensity of the radiation belonging to a given range of the
spectrum is represented by the area included between the ordinates which
correspond to the limiting rays, but the form of the curve depends upon what
99] DISTRIBUTION OF ENERGY IK THE SPECTRUM. 199
function of the ray we elect to take as abscissa. Thus in the ordinary pris-
matic spectrum of the sun, the curve culminates in the ultra-red, but in the
diffraction spectrum the maximum is in the yellow, or even in the green,
according to the recent important observations of Prof. Langley. If we wish
to change the function of the ray represented by the abscissa, we can of
course deduce by calculation the transformed curve of energy without fresh
experiments. To pass from the curve with abscissae proportional to wave-
length to one with abscissae proportional to reciprocals of wave-length, we
must magnify the ordinates of the former in the ratio of the square of the
wave-length, and this will give us an energy curve more like that obtained
with a prismatic spectrum.
There is another method of representation intermediate between these
two, which is not without advantage. In the diffraction spectrum the space
devoted to a lower octave (if we may borrow the language of acoustics) is
greater than that devoted to a higher octave. In Mr Stoney's map the
opposite is the case. If we take the logarithm of the wave-length (or of the
frequency) as abscissa, we shall obtain a map in which every octave occupies
the same space, and this perhaps gives a fairer representation than either of
the others. To deduce the curve of energy from that appropriate to the
diffraction spectrum, we should have to magnify the ordinates in the ratio of
the first power of the wave-length.
My object, however, is not so much to advocate any particular method of
representation, as to point out that the curve of energy of the diffraction
spectrum has no special claim to the title of " normal"
100.
INVESTIGATION OF THE CHARACTER OF THE EQUILIBRIUM
OF AN INCOMPRESSIBLE HEAVY FLUID OF VARIABLE
DENSITY*
[Proceedings of the London Mathematical Society, xiv. pp. 170 177, 1883.]
THE well-known condition of equilibrium requires that the fluid be
arranged in horizontal strata, so that its density a is a function of the vertical
coordinate z only. If this state of things be slightly departed from, we may
regard the actual density at any point x, y, z as equal to a + p, where p is a
function of x, y, z, and the time t, which always remains small during the
period contemplated. The component velocities u, v, w are equally to be
regarded as small ; they are connected by the equation of continuity
++
dx dy dz
The equilibrium pressure p is a function of z only. If the actual pressure
be called p + Bp, the dynamical equations become, with omission of the
squares of small quantities,
d&p _ du d8p _ dv dBp _ dw
One further equation is supplied by the condition that the density of
every particle remains unchanged.
Thus
* These calculations were written out in 1880, in order to illustrate the theory of cirrous
clouds propounded by the late Prof. Jevons (Phil. Mag. xiv. p. 22, 1857). Pressure of other work
has prevented me hitherto from pursuing the subject.
100] INVESTIGATION OF THE CHARACTER OF THE EQUnjBRITJM, ETC. 201
By Fourier's theorem and the general theory of disturbed equilibrium,
we know that the complete solution of the present problem may be decom-
posed into partial solutions, for any one of which the variable quantities
considered as functions of x vary as e**, as functions of y vary as *, and as
functions of t vary as e"**. The wave-lengths of the disturbances parallel to x
and y are X, X', where X = 2-jr/t, X' = l-r.k'.
The introduction of these suppositions into (1), (2), and (3) leads to
iku 4- t*'r + d' ( dz = ............................ (4)
k&p = ntru, k'&p = nor, dSp dz = gp intnc, ............ (5)
inp + to cUr-cLz = ............................... (6)
Eliminating u and v between (4) and the two first of equations (5), we
get
......................... (7)
Next eliminating Sp between (7) and the last of equations (5), we find
................ (S)
Finally between (6) and (8) we eliminate p, and thus obtain
or, as it may be also written,
We will first apply this equation to the well-known case of two uniform
fluids of densities tr 1 . <r 2 , separated by a horizontal boundary (z = OX and for
brevity we will omit to write k'. For both regions of fluid, the general
equation (10) reduces to
of which the solution is
w = Ae t! + Be-*. .............................. (12)
By the condition at infinity, we are to take for the upper fluid ^1=0,
and for the lower B=0. Moreover by continuity the value of w must be the
same for both fluids at the separating surface. Thus we may write for the
upper fluid w = Be~**, and for the lower w = B**. The second boundary
condition is obtained by integrating equation (9) across the surface of transi-
tion. Thus
202 INVESTIGATION OF THE CHARACTER OF THE EQUILIBRIUM [100
whence
n . = ^?LZ?, .......................... (13)
' a-i + ff,'
the known solution.
If the upper fluid be the lighter, cr 2 < cr l , and n 2 is positive. This indi-
cates stability with harmonic oscillations, whose frequency increases without
limit with k ; that is, as the wave-length diminishes. If, on the other hand,
cr 2 > <r l9 the equilibrium is unstable, and the instability (measured by the
rate at which a small disturbance is multiplied in a given time) is greater
the smaller the wave-length. If the disturbance be not limited to two
dimensions, we have simply to replace k by \/(& 2 + &' 2 ).
We know from the general theory that only real values of n 2 are admissible
in (9), and that if da/dz be negative throughout, all the values of n 2 are
positive, but if da/dz be positive throughout, all the values of n 2 are negative.
In order to prove this from the equation, suppose that w and w' are two solu-
tions corresponding to different values of n 2 , say n 2 and n' 2 . Then
f , d ( dw\ , , [( q da- \
lw ~r (<r - r }dz = k 2 -^ -=- + <r ww' dz,
J dz\ dz) J\n 2 dy )
or, on integration by parts between two finite or infinite limits for which
w, w' vanish,
fo- ^^dz+k 2 faww'dz + k 2 ^ (^ww'dz = ...... (14)
J dz dz J n 2 J dz
In this equation w and w' may be interchanged if n' 2 be written for n 2 .
Hence
(16)
If now n 2 could be complex, there would be two solutions of the form
a + i/3, w' = a t/3, and equation (15) would become
which cannot be true if, as we suppose, a is everywhere positive.
Again, suppose in (14) that w = w. Thus
from which it is evident that, if d<r/dz be of one sign throughout, n 2 can only
be of the opposite sign.
100] OF AS INCOMPRESSIBLE HEAVY FLUID OF VARIABLE DESSTTY. 203
These conclusions are limited to the cases for which every mode of
disturbance is stable, or every mode unstable ; bat we know that if da dz be
anytchfre positive, instability must ensue: To see this from equation (9),
we may regard it as the condition (according to the methods of die Cal-
culus of Variations) that f(da dz)te*dz is a maximum or minimum, while
ftr {(diojjdif + &n?l dz is given, vf gk* being the then value of the ratio of
the integrals. If dc dz be anywhere positive, the first integral admits of a
positive value, and therefore of a positive maximum, so that one value at
least of it* is negative, and one mode of disturbance is unstable.
The simplest case of a variable density which we can consider is that
obtained by supposing a~*d<r ds to be constant, equal say to /3. or, on
integration,
<r=<r **; ................................. (18)
so that all strata of equal thickness are similarly constituted, differing only
in absolute density. In this case, with omission of k' as before, (10) becomes
If TO,, m, be the roots of
^+j9n-t I (l+^ii-^) = 0, ..................... (20)
the general solution of (19) is
w = Af** + Be-*, .............................. (21)
A and B being arbitrary constants.
Let us now suppose that the fluid is bounded by impenetrable horizontal
planes at z = and at z= 1. Since IP vanishes with z, B = - A, so that (21)
becomes
w = A(4* -*"*> ........................... (22)
Again, since w vanishes when z = l, *M-e-M= 0, or e^-^'= 1, whence
(m 1 - TO2 )/=2V, ........................... (23)
a being an integer. Thus (22) may be written
w = AJ***-** {*---** - -*-.-vj. = A'tr&* sin (* I), ...... (24)
by (20), (23X A' being a new arbitrary constant. The values of w corre-
sponding to the various values of a are obtained by comparison of (20) and
(23). From the former
( 1 -w^P = ^+4^(l+^i.-X ............... (25)
so that
aV, .................. (26)
204 INVESTIGATION OF THE CHARACTER OF THE EQUILIBRIUM [100
From (27) we see that the disturbances are all stable if (3 is negative, that is,
if the density diminishes upwards, and that in the contrary case they are all
unstable. The smallest admissible value of a is unity, and this corresponds
to the greatest numerical value of n 2 . Contrary to what is met with in most
vibrating systems, there is (in the case of stability) a limit on the side of
rapidity of vibration, but none on the side of slowness. In the case of insta-
bility we are principally interested in the mode for which the instability is
greatest, and this also corresponds to the unit value of a. When a is greater
than unity, there are internal nodal planes, as appears from (24).
If I, k, and a are given, n? is numerically greatest when /3 is such that
If I, a, ft be regarded as given, n 2 increases numerically from zero when k is
zero, up to a finite limit when k is infinite ; or, in the case of stability, as the
wave-length diminishes from oo to 0, the frequency of vibration rises from
to a finite value, given by
7i 2 = -<7/3, ................................. (28)
which is independent both of a and of I. These vibrations are isochronous
with the vibrations of a pendulum whose length is equal to the distance
between two strata of which the densities are as e : 1.
If the disturbance be not limited to two dimensions, we must write
^(k 2 + k' 2 ) for k 2 . The completely expressed value of w, corresponding to one
normal mode of disturbance, is then
cosn(t-t ) ...... (29)
A,
We will now apply the solution to the investigation of the case in which
the density for all values of z less than is a~ l , and for all values of z greater
than I is <7 2 , the transition from the one density to the other being in accord-
ance with the law <r = v* so that
(30)
When z > I, w oc e~ kz , so that for z = I, dw/wdz = k ; similarly for the lower
fluid, when z = 0, dw/wdz = + k. Thus, by (21), the boundary conditions are
= +k(A+B),
whence, by elimination of A : B,
100] OF AX INCOMPRESSIBLE HEAVY FLUID OF TAMABLE
This, in connection with (20v determines the admissible values of JL It may
be written
or
Bj (20) this mar be put into the form
or 7 if for brevity we write for (in., M,) I,
'>
This equation determines 0; and then, by
^=l( s -OP=l{( a + 1 )-4 1 B }=/9^+^(l+^ii^) I
......... 1 32>
giving n in terms of ft
Before going farther we may verify these results by applying them t> the
case of a sudden transition, for which / vanishes, while &l remains finite.
The principal solution of (31) gives 0* = /^^ approximately, so than
Using this in (33), we get
g
whence
' =
as before,
Other solutions of (31) are obtained by supposing 0~* tanh \& to vanish,
whence = i . a. . 2r, a being an integer other than zero. These are of no
importance, as the corresponding values of K vanish.
When the layer of transition is of finite thickness, the genera] solution
expressed by (31) T (32) is rather complicated. A simplification, which does
not involve much loss of interest, may be effected by supposing that the
whole change of density is small, so that (31X (32) become
- (33)
206 INVESTIGATION OF THE CHARACTER OF THE EQUILIBRIUM [100
From (33),
whence
A 9JL7
-tanh^-.or ......................... (35)
Equation (35) cannot be satisfied by any real value of 6. If we write 6 = i<f>,
we get in place of it,
I ,_ _ <f> Zkl ,ggv
in 59 - ^ , or ,
and in place of (34),
1. ....(37)
The series of admissible values of <f>, given by (36), extends to infinity, but
the higher roots correspond to small values of n 2 , which are of little interest.
Whether the equilibrium be stable or unstable, the most important root is
the smallest. It lies in the first quadrant, and is given by the second alter-
native of (36). The progress of n 2 as a function of kl is easily traced. When
kl is small, < 2 = 8kl, and g@/n? = - 2/kl, which leads to n 2 = - gk (<r 2 - oi)/2<r,
the known result for a rapid transition. As kl increases, < ranges from
to TT, and < 2 /4& 2 Z 2 or cot 2 </> ranges from infinity to zero. Thus the numeri-
cal value of n 2 continually increases, until for an infinitely small wave-length
it approaches the finite limit g{3, beyond which it cannot pass. The princi-
pal result of the substitution of a gradual for an abrupt transition is to arrest
the further increase of ?i 2 , after the wave-length has diminished so far as to
become comparable in magnitude with the thickness of the layer of transition.
In the case of the limiting value of n 2 , the length of the equivalent pen-
dulum is
I -r- (log o- 2 - log o-j).
If, for example, the extreme difference of densities amounted to one per
cent., the length of the equivalent pendulum would be 100 times the thick-
ness of the layer of transition.
For actual calculation (36), (37) may advantageously be written
|JW = i</> x tani, ........................... (38)
<l>, .................. (39)
the right-hand member of (39) being equal to unity, when kl is small.
Ascribing arbitrary values to \<j>, we can readily calculate corresponding
values of kl and &//sin 2 <, and thus exhibit the effect upon the equilibrium
100] OF AN INCOMPRESSIBLE HEAVY FLUID OF VARIABLE DENSITY. 207
of a gradual increase in the thickness of the layer of transition, the extreme
densities (determined by f) and the ware-length being given.
to
u
JH/rii>i*
o p .
kl
HKX)
10F
06155
1-021
20 3
2541
1-086
30=
046
1-209
4QP
1-172
1-418
5QF
2^)80 1-T72
60
3-628
2-418
TOP
6T13
S-801
SOP
15-S3.S
8-165
90
a
iW
101.
ON THE VIBRATIONS OF A CYLINDRICAL VESSEL
CONTAINING LIQUID.
[Philosophical Magazine, xv. pp. 385389, 1883.]
THE problem of a uniform cylinder vibrating in two dimensions is con-
sidered in my book on the Theory of Sound, 233. If the displacements at
any point a, 6 of the circumference be Br, a$0, then for a single component
Sr=aA n cosn&, 80 = n~ l A n sin nO ................ (1)
If d be the thickness and a- the volume-density of the material, the kinetic
energy of the motion for a length z measured parallel to the axis is
(2)
The corresponding potential energy is
in which B is a constant depending upon the material and upon the thickness.
As a function of thickness B oc d 3 ; so that we may write B = B d 3 , in which
BQ depends upon the material only. Thus
(3)
If the cylinder be empty, these expressions suffice to determine the periods
of vibration. Thus, if A n <xcosp t,
Po
showing that for a given material the frequency is proportional to the thick-
ness and inversely as the square of the radius.
101] ON THE VIBRATIONS OF A CYLINDRICAL VESSEL CONTAINING LIQUID. 209
If the cylinder contain frictionless fluid, the motion of the fluid will
depend upon a velocity-potential <f> which satisfies the equation
in which
k = p(a', .................................... (6)
a being the velocity of propagation of sound within the fluid. If the fluid
can be treated as incompressible, we may put k = 0. For the present
purpose we will retain k, but we will assume that the motion is strictly in
two dimensions. Introducing the further assumption that x cos n0, we get
in place of (5),
of which the solution is
J n (kr) ......................... (8)
The relation between a,, and A u of (1) is readily found by equating the value
of dfydr, when r = a, to d&r/dt, both of which represent the normal velocity
at the circumference. We get
". ...O)
The kinetic energy of the fluid motion is given by
. J n (ka) J n '(ka) + # jV.'(*r) rdr}. ...(10)
For the potential energy of the liquid, if compressible, we have
...... (11)
The sum of the potential and kinetic energies for the solid and liquid
together must be independent of the time. The unintegrated terms in (10)
and (11) cancel, and we find
<>
In the application of (12) ka is a small quantity. From the ascending series
for J* (ka) we find
^+ .....
u
R. II.
910 ON THE VIBRATIONS OF A [101
so that approximately
(H)
If p be the value of p when p = 0,
** (15)
From (14) or (15) we see that the effect of a finite as compared with an
infinitely small compressibility is to increase the depression of pitch due to
the fluid. As the velocity of sound is greater in liquids than in air, it would
seem that -^ k 2 a? would generally be negligible. In this case, for the prin-
cipal mode of vibration corresponding to n = 2, (15) becomes simply
In Auerbach's recent paper upon this subject* various observations upon
the depression of pitch due to the action of liquid are given. In his notation
p /p = 0, From (15) we see that if G be the value of G for water, the same
vessel being used in both cases,
.(17)
if s denote the specific gravity of the liquid, referred as usual to water as a
standard. Auerbach's observations are fairly accordant with (17); and there
seems to be scarcely sufficient warrant for attributing the discrepancies to
the influence of compressibility.
In observations with different vessels of the same material and filled with
the same fluid, difficulty was experienced in obtaining by direct measure-
ment a sufficiently accurate value of d. To meet this, d was determined
indirectly from the pitch. By (4) we have
from which it appears that G 2 - 1 is inversely proportional to the pitch
(before filling), as well as inversely proportional to the radius of the cylinder.
In Auerbach's notation a constant C is employed, whose value for the case
?i=2 would be by (18)
In actual experiment the two-dimensional character of the fluid motion
is disturbed by the existence of a free surface at which a special -condition
* Wied. Ann. Bd. xvn. p. 964.
101] CTLOTDRICAL VESSEL OOSTAIXISG UQTID. 211
most be satisfied. Hence arises a vertical motion of the surface, which is tfce
proximate cause of the " crispatious " usually to be observed under these
circumstances. In considering this question we may leave the force of
gravity out of account, inasmuch as the period of free waves of length com-
parable with the diameter of the cylinder is much greater than that of the
actual motion.
In accordance with <5i. if the fluid be treated as incompressible, we mav
:--.>.T
6 = w?pt cos r* + l t cos/rf 006110?-** J.(4r), ...... (20)
in which r is measured downwards from the surface, and I* denotes a root of
0. ................................ (21)
The coefficients A t are to be determined by the condition at the surface
which is simply ^ = 0. Thus for each value of i-
J m *(kr)rdr = Q. . ...<22
.'
Xow (see Theory of Sound, 203. 332)
so that
To calculate the kinetic energy we have to integrate <bd<j> dr< over the
whole boundary of the fluid. Xow at the free surface <f> = 0. and ai a great
depth the motion becomes two-dimensionaL We have therefore only to
consider the cylindrical surface. By supposition J m '(ka) = Q, and thus
cos/rf cos m$.
We get therefore
"The value of T is less than if the motion were strictly two-dimensional by a
quantity corresponding to the length
For M = 2, the values of ta from (21) are 3O54, 6 705, 9"965, 13 1, 16 3, Ac. :
and thus (25) becomes -2674o.
102.
ON THE CRISPATIONS OF FLUID RESTING UPON A
VIBRATING SUPPORT.
[Philosophical Magazine, xvi. pp. 5058, 1883.]
IF a glass plate, held horizontally and made to vibrate as for the produc-
tion of Chladni's figures, be covered with a thin layer of water or other
mobile liquid, the phenomena in question may be readily observed. Over
those parts of the plate which vibrate sensibly the surface of the liquid is
ruffled by minute waves, the degree of fineness increasing with the frequency
of vibration. Similar crispations are observed on the surface of liquid in a
large wine-glass or finger-glass which is caused to vibrate in the usual
manner by carrying the moistened finger round the circumference. All that
is essential to the production of crispations is that a body of liquid with a
free surface be constrained to execute a vertical vibration. It is indifferent
whether the origin of the motion be at the bottom, as in the first case, or, as
in the second, be due to the alternate advance and retreat of a lateral
boundary, to accommodate itself to which the neighbouring surface must rise
and fall.
More than fifty years ago the nature of these vibrations was examined by
Faraday with great ingenuity and success. His results are recorded in an
Appendix to a paper on a Peculiar Class of Acoustical Figures*, headed "On
the Forms and States assumed by Fluids in Contact with Vibrating Elastic
Surfaces." In more recent times Dr L. Matthiessen has travelled over the
same ground f, and on one very important point has recorded an opinion in
opposition to that of Faraday. In order more completely to satisfy myself, I
have lately repeated most of Faraday's experiments, in some cases with
improved appliances, and have been able to add some further observations in
support of the views adopted.
* Phil. Trans. 1831.
t Fogg. Ann. t. cxxxiv. 1868; t. CXLI. 1870,
102] THE CRISPATIOXS OP FLUID RESTDfG UPON A VIBRATING SUPPORT. 213
The phenomenon to be examined is evidently presented in its simplest
form when the motion of the vibrating horizontal plate on which the liquid
is spread is a simple up-and-down motion without rotation. To secure this,
Faraday attached the plate to the centre of a strip of glass or lath of deal,
supported at the nodes, and caused to vibrate by friction. In my experi-
ments an iron bar was used about 1 metre long and "0064 metre thick (in
the plane of vibration). The bar was supported horizontally at the nodes ;
and to its centre a glass plate was attached by gutta-percha and carefully
levelled. The vibrations of the bar were maintained electromagnetically, as
in tuning-fork interrupters, with the aid of an electromagnet placed under
the centre, the circuit being made and broken at a mercury-cup by a dipper
carried at one end of the bar. By calculation from the dimensions*, and
without allowance for the load at the centre, the frequency of (complete)
vibration is 33. Comparisons with a standard tuning-fork gave more accu-
rately for the actually loaded bar a frequency of 31.
The vibrating liquid standing upon the plate presents appearances which
at first are rather difficult to interpret, and which vary a good deal with the
nature of the liquid in respect of transparency or opacity, and with the
incidence of the light. The vibrations of the liquid, whether at the rate of
31 per second, or, as in feet, at the rate of 15 per second, are too quick to
be followed by the eye ; and thus the effect observed is an average, due to
the superposition of an indefinite number of components corresponding to the
various phases of vibration.
The motion of the liquid consists of two sets of stationary vibrations
superposed, the ridges and furrows of the two sets being perpendicular to
one another, and usually parallel to the edges of the (rectangular) plate.
Confining our attention for the moment to one set of stationary waves, let us
consider what appearance it may be expected to present. At one moment
the ridges form a set of parallel and equidistant lines, the interval being the
wave-length. Midway between these are the lines which represent at that
moment the position of the farrows. After the lapse of \ period : the surface
is flat; after another ^ period, the ridges and furrows are again at their
maximum development, but the positions are exchanged. Now, since only an
average effect can be perceived, it is clear that no distinction can be recog-
nized between the ridges and the farrows, and that the observed effect must
be periodic within a distance equal to half* wave-length of the real motion.
If the liquid on the plate be rendered moderately opaque by addition of
aniline blue, and be seen by diffused transmitted light, the lines of ridge and
furrow will appear bright in comparison with the intermediate nodal lines
where the normal depth is preserved throughout the vibration. The gain of
light when the thickness is small will, in accordance with the law of absorp-
* Theory of Sown*, ITL
214
ON THE CRISPATIONS OF FLUID
[102
tion, outweigh the loss of light which occurs half a period later when the
furrow is replaced by a ridge.
The actual phenomenon is more complicated in consequence of the co-
existence of the two sets of ridges and furrows in perpendicular directions
(x, y). In the adjoining figure the thick lines represent the ridges, and the
thin lines the furrows, of the two systems at a moment of maximum excur-
sion. One quarter period later the surface is flat, and one half a period later
the ridges and furrows are interchanged. The places of maximum elevation
and depression are the intersections of the thick lines and of the thin lines,
not distinguishable by ordinary vision ; and these regions svill appear like
holes in the sheet of colour. The nodal lines, where the normal depth of
colour is preserved, are shown dotted ; they are inclined at 45, and pass
through the intersections of the thick lines with the thin lines. The pattern
is recurrent in the directions of both x and y, and in each case with an
interval equal to the real wave-length (X). The distance between the bright
spots measured parallel to x or y is thus X ; but the shortest distance between
these spots is in directions inclined at 45, and is equal to V2 . X.
In order to determine the relation of the frequency of the liquid vibrations
to that of the bar, an apparatus was fitted up capable of giving an inter-
mittent view of the vibrating system. This consisted of a blackened paper
disk pierced with three. sets of holes, mounted upon an axle, and maintained
in rotation by a small electromagnetic engine of Apps's construction. The
whole was fastened to one base-board, and could be moved about freely, the
leading wires from the battery being flexible. The current was somewhat in
excess; so that the desired speed could be attained by the application of
moderate friction. At a certain speed of rotation the appearances were as
102] RESTING UPON A VIBRATING SUPPORT. 215
follows. Through the set of four holes (giving four views for each rotation of
the disk) the bar was seen double. Through the set of two holes the bar was
seen single, and the water-waves were seen double. Through the single hole
the bar was seen single, and the waves also were seen single. From this it
follows that the water vibrations are not, as Matthiessen contends, synchro-
nous with those of the bar, but that there are two complete vibrations of the
support for each complete vibration of the water, in accordance with Faraday's
original statement.
An attempt was made to calculate the frequency of liquid vibration from
measurements of the wave-length and of the depth. The depth (h), deduced
from the area of the plate and the whole quantity of liquid, was '0681
centini. ; and by direct measurement A, = - 848 centim. Sir W. Thomson's
formula connecting the velocity of propagation with the wave-length, when
the effect of surface-tension is included, is
where a = 2irh/\. With the above data we find for the frequency (r~ l ) of
vibration 20'8. This should have been 15'5 ; and the discrepancy is probably
to be attributed to friction, whose influence must be to diminish the efficient
depth, and may easily rise to importance when the total depth is so small.
Another method by which I succeeded in determining the frequency of
these waves requires a little preliminary explanation. If n = ZTT/T, and
k = 2-7T/X, the stationary waves parallel to y may be expressed as the resultant
of opposite progressive waves in the form
cos (kx + nt) + cos (kx nt) = 2. cos kx cos nt ................ (1 )
This represents the state of things referred to an origin fixed in space.
But now let us refer it to an origin moving forward with the velocity (n/k) of
the progressive waves, so as to obtain the appearance that would be presented
to the eye, or to the photographic camera, carried forward in this manner.
Writing kx' + nt for kx, we get
cos (kx' + 2nt) + cos kx' ............................ (2)
Now the average effect of the first term is independent of x', so that what is
seen is simply that set of progressive waves which moves with the eye. In
this way a kind of resolution of the stationary wave into its progressive com-
ponents may be effected.
In the actual experiment two sets of stationary waves are combined ; and
the analytical expression is
cos (kx + nt) + cos (kx - nt) + cos (ky + nt) + cos (ky - nt), ...... (3)
which is equal to
2 cos kx cos nt + 'Zcosky cosnt, ..................... ()
216 ON THE CRISPATIONS OF FLUID [102
or to
4cos {&(+?)} cos {%k(x-y}} cosnt ................... (5)
If, as before, we write kx + nt for kx, we get
cos (leaf + 2nt) + cos kx' + 2 cos ky cos nt ................... (6)
The eye, travelling forward with the velocity n/k, sees mainly the corre-
sponding progressive waves, whose appearance, however, usually varies with
y, i.e. along the length of a ridge or furrow. If the effect could be supposed
to depend upon the mean elevation only, this complication would disappear,
as we should be left with the term cos kx' standing alone. With the semi-
opaque coloured water the variation along y is evident enough ; but the
experiment may be modified in such a manner that the ridges and furrows
appear sensibly uniform. For this purpose the coloured water may be
replaced by milk, lighted from above, but very obliquely. The appearance
of a set of (uniform) ridges and furrows varies greatly with the direction of
the light. If the light fall upon the plate in a direction nearly parallel to
the ridges, the disturbance of the surface becomes almost invisible ; but if, on
the other hand, the incidence be perpendicular to the line of ridges, the
disturbance is brought into strong relief. The application of this principle
to the case before us shows that, when the eye is travelling parallel to x, the
ridges and furrows will look nearly uniform if the incidence of the light be
also nearly parallel to x ; but if the incidence of the light be nearly parallel
to y, the ridges will show marked variations along their length, and in fact
be resolved into a series of detached humps. The former condition of things
is the simplest, and the most suitable as the subject of measurement.
In order to see the progressive waves it is not necessary to move the
head as a whole, but only to turn the eye as when we look at an ordinary
object in motion. To do this without assistance is not at first very easy,
especially if the area of the plate be somewhat small. By moving a pointer
at various speeds until the right one is found, the eye may be guided to do
what is required of it ; and after a few successes repetition becomes easy. If
we wish not merely to see the progressive waves, but to measure the velocity
of propagation with some approach to accuracy, further assistance is required.
In my experiments an endless string, passing over pulleys and driven by a
small water-engine, travelled at a small distance above the plate so that its
length was in the direction of wave-propagation. A piece of wire was held
at one end by the fingers, and at the other rested upon the travelling
string and was carried forward with it. In this way, by adjusting the water
su Pply > tne speed of the string could be made equal to that of wave-propaga-
tion ; and the former could easily be determined from the whole length of
the string, and from the time required by a knot upon it to make a complete
circuit. Thus (on February 7) the velocity of propagation was found to be
5'4 inches per second. At the same time, by measurement of the pattern as
102] RESTING UPOX A VIBRATING SUPPORT. 217
seen by ordinary vision, 1-4X = 4f inches. Hence frequency = 5'4/X = 15'5
per second ; exactly one half the observed frequency of the bar, vi^ 31.
In addition to the phantoms which may be considered to represent the
four component progressive waves, others may be observed travelling in
directions; inclined at 4-5 . If we take coordinates f , if in these directions, (5)
may be written
4 cos (If V2) coB(*f V2> ***; .................... .(7)
in which if we put
(He. if we suppose the eye to travel with velocity .</%> . /), we get
2cos(Ir?y2) cos(VV2) + terms in in/-
The non-periodic part may be supposed roughly to represent the phe-
nomenon.
In order if possible to settle the question beyond dispute, I made yet
another comparison of the frequencies of vibration of the fluid and of the
support, using a plan not very different from that originally employed by
Faraday. A long plank was supported on trestles at the nodes, and could be
tuned within pretty wide limits by shifting weights which rested upon it
near the middle and ends. At the centre was placed a beaker \ inches in
diameter, and containing a little mercury. The plank was set into vibration
by properly timed impulses with the hand, and the weights were adjusted
until the period corresponded to one mode of free vibration of the pool of
mercury. When the adjustment is complete., a very small vibration of the
plank throws the mercury into great commotion, and unless the vessel is
deep there is risk of the fluid being thrown out. The question now to be
decided is whether, or not, the vibrations of the mercury are executed in the
same time as those of the plank.
On March 18 the plank was adjusted so as to excite that mode of vibra-
tion <of the mercury in which there are two nodal diameters. Two other
diameters bisecting the angles between these give the places of maximum
vertical motion. At one moment the mercury is eHevated at beth ends of one
diameter and depressed at both ends of the perpendicular diameter ; half a
period later the case is reversed. The frequency of the fluid vibrations could
be counted by inspection, and was found to be 30 (complete) vibrations in
15 seconds, or exactly two vibrations per second. The vibrations of the plank
wore counted by allowing it to tap slightly against a pencil held in the hand.
In five seconds there were 21 complete vibrations, Le. 4 vibrations per
second, almost exactly twice as many as was found for the mercury. The
were repeated several times; and the general result is beyond
218 ON THE CR1SPATIONS OF FLUID [102
On another occasion the mode of fluid vibration was that in which there
is but one nodal diameter, the fluid being most raised at one end of the
perpendicular diameter and most depressed at the other end. The frequency
of fluid vibration was 30/22 = 1'36 ; while that of the plank was 27/10 = 27.
Here again the fluid vibrations are proved to be only half as quick as those
of the support.
The mechanics of the question are considered in a communication to the
Philosophical Magazine for April, 1883*, to which reference must be made.
Merely to observe the phenomenon, it is sufficient to take a porcelain
evaporating-dish containing a shallow pool of mercury 2 or 3 inches in
diameter, and, holding it firmly with both hands, to impose upon it a vertical
vibratory motion. After a few trials of various speeds it is possible to excite
various modes of vibration, including those referred to in connexion with the
plank. The first (with two nodal diameters) is more interesting in itself, and
is more certainly due to a vertical as opposed to a horizontal vibration of the
support. The gradually shelving bank presented by the dish adds to the
beauty of the experiment by its tendency to prevent splashing.
Dr Matthiessen, in the papers referred to, records a long series of
measurements of the wave-lengths of crispations corresponding to various
frequencies of vibration, not only in the case of water, but also of mercury,
alcohol, and other liquids. He remarks that the nature of the liquid affects
the relation in a marked manner, contrary to the theoretical ideas of the
time, which recognized gravity only as a "motive" for the vibrations. In
the following year Sir W. Thomson gave the complete theory of wave-propa-
gationf, in which it is shown that in the case of wave-lengths so short as
most of those experimented upon by Matthiessen, the influence of cohesion,
or capillary tension, far outweighs that of gravity. In general, if T be the
tension, k = 2-Tr/A, the velocity of propagation (y) is given by
v = J(Tk + g/k); .............................. (8)
or, when X is small enough,
v = </(Tk) .................................. (9)
Since X = vr, the relation between T and \ is, by (9),
27rZV = X 3 ; .............................. (10)
or, if N be the frequency of vibration,
constant ............................... (11)
Dr Matthiessen's results agree pretty well with (11), much better in fact
than with the formula proposed by himself.
There is another point of some interest on which the views expressed by
Matthiessen call for correction. It was observed by Lissajous some years
* [Art. 97, vol. ii. p. 190.]
t Phil. Mag. Nov. 1871.
L RESTING UFOST A YIBRATIXG STPFOBT. 219
ago, that if two vibrating tuning-folks of slightly different pitch are made to
touch the surface of water, the nearly stationary wares formed midway
between the sources of disturbance travel slowly towards the graver. We
may take as the expression for the two progressive waves
cos (kx itf ) + cos (Ir'j- + m't),
or, which is the same,
2 cue j[4 ( + *)* + 4 (*-)'} x cos {(#- l^.r -I- (X + )*).
The potation at any time of the crests of the nearly stationary waves is
given bv
here m is an integer. The velocity of displacement V is thus
............................... '
from which it appears that in every case the shirting is in the direction of
propagation of waves of higher pitch, or towards the source of graver pitoh.
According to Matthiessen, the shifting takes place with a velocity qna3
to hah 7 the difference of velocities of the component trains, u*.
2F= I-~F
and in the direction of that component train which moves with greatest
velocity- So far as regards die direction merely, the two rules come to the
same thing for the range of pitch used by Uasajjore aod Maubiessen, since
over this range the velocity increases with pitch. If. however, we have to
deal with waves longer than the critical value <1"7 centim. for water I the
two rules are at issue, since now the velocity increases as the pitch diminishes.
The following are a few corresponding values, in C.G.& measure, of wave-
length, Telocity, and frequency of vibration calculated by Thomson's for-
mula (A>
-5 1-0 1-7 3-5 3^ 5^
;.- 34-T5 13^W 9-57
I have examined the matter experimentally with the aid of vibrators
making from 12 to 7 complete vibrations per second, and therefore well below
the critical point, with the result that the transference is towards the source
of graver pitch, although this is the direction of propagation of the compo-
nent which travels with the smaller velocity. I reserve for the present a
more detailed description of the apparatus, as I propose to apply it to the
general verification of Thomson's law of velocities.
103.
ON POROUS BODIES IN RELATION TO SOUND.
[Philosophical Magazine, xvi. pp. 181186, 1883.]
IN Acoustics we have sometimes to consider the incidence of aerial waves
upon porous bodies, in whose interstices some sort of aerial continuity is
preserved. Tyndall has shown that in many cases sound penetrates such
bodies, e.g. thick pieces of felt, more freely than would have been expected,
though it is reflected from quite thin layers of continuous solid matter. On
the other hand, a hay-stack seems to form a very perfect obstacle. It is
probable that porous walls give a diminished reflection, so that within a
building so bounded resonance is less prolonged than would otherwise be
the case.
When we inquire into the matter mechanically, it is evident that sound
is not destroyed by obstacles as such. In the absence of dissipative forces,
what is not transmitted must be reflected. Destruction depends upon vis-
cosity and upon conduction of heat ; but the influence of these is enormously
augmented by the contact of solid matter exposing a large surface. At such
a surface the tangential as well as the normal motion is hindered, and a
passage of heat to and fro takes place, as the neighbouring air is heated
and cooled during its condensations and rarefactions. With such rapidity
of alternations as we are concerned with in the case of audible sounds, these
influences extend to only a very thin layer of the air and of the solid, and
are thus greatly favoured by attenuation of the masses.
I have thought that it might be interesting to consider a little more
definitely a problem sufficiently representative of that of a porous wall, in
order to get a better idea of the magnitudes of the effects to be expected.
We may conceive an otherwise continuous wall, presenting a flat face, to be
perforated by a great number of similar small channels, uniformly distri-
buted, and bounded by surfaces everywhere perpendicular to the face. If
103] ON POROUS BODIES IN RELATION TO SOUND. 221
the channels be sufficiently numerous, the transition from simple plane
waves outside to the state of aerial vibration corresponding to the interior
of a channel of infinite length, occupies a space which is small relative to the
wave-length of the vibration, and then the connexion between the condition
of things inside and outside admits of simple expression.
Considering first the interior of one of the channels, and taking the axis
of x parallel to the axis of the channel, we suppose that as functions of x
the velocity-components u, v, w, and the condensation s are proportional to
e ikx , while as functions of t everything is proportional to & nt , n being real.
The relationship between k and n depends on the nature of the gas and
upon the size and form of the channel, and must be found in each case by
a special investigation. Supposing it known for the present, we will go on to
show how the problem of reflection is to be dealt with.
For this purpose consider the equation of continuity as integrated over
the cross section of the channel a. Since the walls are impenetrable,
so that
rf f f
<r = (1)
This result is applicable at points distant from the open end more than
several diameters of the channel.
Taking now the origin of x at the face of the wall, we have to form
corresponding expressions for the waves outside ; and we may here neglect
the effects of friction and heat-conduction. If a be the velocity of sound in
the open, and k = w/a, we may write
8 = (+<?** + Be-** x )e int , (2)
u = a(-<P** + B -***)*': (3)
so that the incident wave is
s = e i(nt ^ } , (4)
or, on throwing away the imaginary part,
s = cos ( n t + k x) ( 5 )
These expressions are applicable when x exceeds a moderate multiple of the
distance between the channels. Close up to the face the motion will be
more complicated; but we have no need to investigate it in detail. The
ratio of u and s at a place near the wall is given with sufficient accuracy by
putting x = in (2) and (3),
222 ON POROUS BODIES IN RELATION TO SOUND. [103
We now assume that a region about a? = 0, on one side of which (6) is
applicable and on the other side of which (1) is applicable, may be taken
so small relatively to the wave-length that the mean pressures are sensibly
the same at the two boundaries, and that the flow into the region at the
one boundary is sensibly equal to the flow out of the region at the other
boundary. The equality of flow does not imply an equality of mean velo-
cities, since the areas concerned are different. The mean velocities will
be inversely proportional to the corresponding areas that is, in the ratio
a : a + a-', if cr' denote the area of the unperforated part of the wall corre-
sponding to each channel. By (1) and (6) the connexion between the inside
and outside motion is expressed by
We will denote the ratio of the unperforated to the perforated parts of the
wall by g, so that g = a-' far. Thus,
17? If
K
If g = 0, k = k , there is no reflection; if there are no perforations, g=oc,
and then 5=1, signifying a complete reflection. In place of (7) we may
write
. ......... . ....................
k (1 + g) + k
which is the solution of the problem proposed. It is understood that
waves which have once entered the wall do not return. When dissipative
forces act, this condition may always be satisfied by supposing the channels
long enough. The necessary length of channel, or thickness of wall, will
depend upon the properties of the gas and upon the size and shape of the
channels.
Even in the absence of dissipative forces there must be reflection, except
in the extreme case g = 0. Putting k = k in (8), we have
If g = l (that is, if half the wall be cut away), B = , & = %, so that the
reflection is but small. If the channels be circular, and arranged in square
order as close as possible to each other, g = (4< 7r)/7r, whence = '121,
.B 2 = '015, nearly all the motion being transmitted.
It remains to consider the value of k. The problem of the propagation
of sound in a circular tube, having regard to the influence of viscosity and
heat-conduction, has been solved analytically by Kirchhoff *, on the suppo-
* Pogfl. Ann. cxxxiv. 1868.
103] OX POROUS BODIES IX RELATIOX TO SOUND. 223
sitions that the tangential velocity and the temperature-variation vanish
at the walla In discussing the solution, Kirchhoff takes the case in
which the dimensions of the tube are such that the immediate effects
of the dissipative forces are confined to a relatively thin stratum in the
neighbourhood of the walls. In the present application interest attaches
rather to the opposite extreme, viz. when the diameter is so small that the
frictional layer pretty well fills the tube. Nothing practically is lost by
another simplification which it is convenient to make (following Kirchhoff)
that the velocity of propagation of viscous and thermal effects is negligible
in comparison with that of sound.
One result of the investigation may be foreseen. When the diameter
of the tube is very small, the conduction of heat from the centre to the
circumference of the column of air becomes more and more free. In the
limit the temperature of the solid walls controls that of the included
gas, and the expansions and rarefactions take place isothennally. Under
these circumstances there is no dissipation due to conduction, and every-
thing is the same as if no heat were developed at all. Consequently the
coefficient of heat-conduction will not appear in the result, which will
involve, moreover, the Newtonian value of the velocity of sound (6) and
not that of Laplace (a).
Starting from Kirchhoff s formulae, we find as the value of jfc* applicable
when the diameter (2r) is very small,
p being the kinematic coefficient of viscosity. The wave propagated into
the channels is thus proportional to
eP*cos(nt + px + e), ........................... (11)
where
= =
* 1 - 1 br
7 being the ratio of the specific heats, equal to 1'41. In the derivation
of (10), nr*/(8i>), v being the therniometric coefficient of conductivity, is
assumed to be small.
To take a numerical example, suppose that the pitch is 256 (middle c
of the scale), so that n = 2ir x 256. The value of /*' for air is '16 C.GJS.
(Maxwell), and that of v is '256. If we take r = 1T ^ centim., we find
nr>/8v equal to about ^^. If r were 10 times as great, the approximation
would perhaps still be sufficient.
224 ON POROUS BODIES IN RELATION TO SOUND. [108
From (12), if n = 2?r x 256,
P- 1 ^^; (13)
so that if r= id l ()() , p = I'!5. In this case the amplitude is reduced in
ratio e : 1 in passing over the distance p~ l that is, about one centimetre.
The distance penetrated is proportional to the radius of the channel.
The amplitude of the reflected wave is, by (8),
or, as we may write it,
p
-'
where
p' = (l+g)p/k (15)
If / be the intensity of the reflected sound, that of the incident sound
being unity,
The intensity of the intromitted sound is given by
/' = l-/ = -^'
By (12), (15),
p' =
If we suppose r = -^^ centim., and g = 1 , we shall have a wall of pretty
close texture. In this case, by (18), jp' = 47'4, and /' = '0412. A four-per-
cent, loss may not appear to be much; but we must remember that in
prolonged resonance we are concerned with the accumulated effects of a
large number of reflections, so that rather a small loss in a single reflec-
tion may well be material. The thickness of the porous layer necessary to
produce this effect is less than one centimetre.
Again, suppose r = ^ centim., g = l. We find ^' = 4-74, /' = '342, and
the necessary thickness would be less than 10 centimetres.
If r be much greater than -^ centim., the exchange of heat between
the air and the walls of the channels is no longer sufficiently free for the
expansions to be treated as isothermal. When r is so great that the
thermal and viscous effects extend only through a small fraction of it, we
have the case discussed by Kirchhoff. If we suppose for simplicity g =
103] OX POBOUS BODIES Df RELATION TO SOCSD. 225
(a state of tilings, it is tree, not strictly consistent with channels of circular
*), we have
in which
y= V/+ ( - *) ^ ........................ (20)
The incident sound is absorbed more and more completely as the diameter
of the channels increases; bnt at the same time a greater thickness becomes
necessary in order to prevent a return from the farther side. If g = t there
is no theoretical limit to the absorption ; and, as we hare seen, a moderate
value of ff does not by itself entail more than a comparatively small reflec-
tion. A loosely compacted hay- or straw-stack would seem to be as effective
an absorbent of sound as anything tikely to be met with.
In large spaces bounded by non-porous walls, roof, and floor, and with
few windows, a prolonged resonance seems inevitable. The mitigating in-
fluence of thick carpets in such cases is well known. The application of
similar material to the walls, or to the roof, appears to offer the best chance
of further improvement
. U.
II
104
SUGGESTIONS FOR FACILITATING THE USE OF
A DELICATE BALANCE.
[British Association Report, pp. 401, 402, 1883.]
IN some experiments with which I have lately been occupied a coil of
insulated wire, traversed by an electric current, was suspended in the balance,
and it was a matter of necessity to be able quickly to check the oscillation
of the beam, so as to bring the coil into a standard position corresponding to
the zero of the pointer. A very simple addition to the apparatus allowed this
to be done. The current from a Leclanche cell is led into an auxiliary coil
of wire, coaxal with the other, and is controlled by a key. When the contact
is made, a vertical force acts upon the suspended coil, but ceases as soon as
the contact is broken. After a little practice the beam may be brought to
rest at zero at the first or second application of the retarding force.
This control over the oscillations has been found so convenient that I
have applied a similar contrivance in the case of ordinary weighings, and my
object in the present note is to induce chemists and others experienced in such
operations to give it a trial. Two magnets of steel wire, three or four inches
long, are attached vertically to the scale-pans, and underneath one of them is
fixed a coil of insulated wire of perhaps 50 or 100 turns, and of 4 or 5 inches
in diameter. The best place for the coil is immediately underneath the bottom
of the balance-case. It is then pretty near the lower pole of the magnet, and
is yet out of the way. The circuit is completed through a Leclanche' cell
and a common spring contact-key, placed in any convenient position. The
only precaution required is not to bring other magnets into the neigh-
bourhood of the balance, or at any rate not to move them during a set of
weighings.
The other point as to which I wish to make a suggestion relates to the
time of vibration of the beam. I think that, with the view of obtaining a
104] SUGGESTIONS FOR FACILITATING THE FSB OF A DELICATE BALANCE. 227
high degree of sensitiveness, the vibrations are often made too slow. Now
the limit of accuracy depends more upon the smallness of the force which can
be relied upon to displace the beam in a definite manner than upon the
magnitude of the displacement so produced- As in other instruments whose
operation depends upon similar principles, e.g. galvanometers, it is useless to
endeavour to increase the sensitiveness by too near an approach to instability,
because the effect of casual disturbances is augmented in the same proportion
as that of the forces to be estimated. If the time of vibration be halved, the
displacement due to a small excess of weight is indeed reduced in the ratio
of four to one, but it is not necessarily rendered any more uncertain. The
mere diminution in the amount of displacement may be compensated by
lengthening the pointer, or by optical magnification of its motions. By the
method of mirror-reading such magnification may be pushed to almost any
extent, but I am dealing at present only with an arrangement adapted for
ordinary use.
In the balance (by Oertling) that I am now using, the scale-divisions are
finer than usual, and the motion of the pointer is magnified four or five times
without the slightest inconvenience by a lens fixed in the proper position.
The pointer being in the same plane as the scale-divisions, there is no sensible
parallax. In this way the advantage of quick vibrations is combined with
easy visibility of the motion due to the smallest weights appreciable by the
balance.
To illuminate the scale the image of a small and distant gas-flame is
thrown upon it by means of a large plate-glass lens. This artificial illumina-
tion is found to be very convenient, as the instrument stands at some distance
from a window, but it is not at all called for in consequence of the use of the
magnifying lens.
152
105.
ON THE IMPERFECTION OF THE GALVANOMETER AS A TEST
OF THE EVANESCENCE OF A TRANSIENT CURRENT.
[British Association Report, pp. 444, 445, 1883.]
IN certain electrical measurements a galvanometer is used to indicate
whether or not the integral value of a current of short duration is zero. For
example, in the method given in Maxwell's Electricity, 755, for comparing
the coefficients of mutual induction, M, of two pairs of coils, the evanescence
of the integral current through the galvanometer is made the test of the
fulfilment of a certain relation between the coefficients of induction and the
resistances. The two primary coils are joined up in simple circuit with a
battery. The two secondaries are also connected together in such a way that
the inductive electro-motive forces conspire, and two points P, Q, one on each
connector, are brought into contact with the galvanometer terminals. In
special cases, as for instance when the two pairs of coils are similar, there is
no current through the galvanometer, whatever may happen in the primary
circuit ; but in general the establishment or interruption of the primary
current will cause a deflection of the galvanometer indicative of the integral
value of the current passing. The method consists in adding inductionless
resistance coils to one or other of the secondaries until this current vanishes.
The required conditions are most readily obtained by supposing the
galvanometer circuit broken, and inquiring into the value of the electro-
motive force E between the points P and Q. The same current y flows in
both secondaries, and if x be the primary current, the equations are :
105] OX THE MPERFECTIOX OF THE GALVANOMETER, ETC.
if,, Jfj, are the induction coefficients to be compared; R, S, the
of the two secondaries (with associated resistance coils) ; N lf N t , their co-
efficients of self-induction. Thus
Since y begins from and ends at 0, the integral electro-motive force
vanishes if
If this condition is satisfied, there is no integral current through the galvano-
meter, and then the ratio of induction coefficients is known by the ratio of
resistances.
In general, however, the evanescence of the integral current is obtained
by the opposition of consecutive positive and negative parts, and even although
the whole duration of the effect be but a small fraction of the time of vibration,
the needle of the galvanometer will be disturbed in such a manner as to make
it difficult to saj whether or not the whole impulse acting upon it be zero.
To obtain a satisfactory measurement it is necessary to secure at least an
approximate fulfilment of the second condition required in order that the
current may be zero throughout, viz.
In this there is no difficulty, as we can easily increase the defective self-
induction by the addition of other coils, placed at a sufficient distance. The
most convenient plan is to include two coils by the variation of the relative
situation of which the self-induction can be adjusted. With moderate care the
initial impulsive electro-motive force, caused by a sudden variation of the
primary current, and dependent only upon the induction coefficients, may be
made so small that the needle shows no uneasiness when the other adjust-
ment relative to the resistances is complete.
In March 1881 I attempted, in conjunction with Messrs Glazebrook and
Dodds, to carry out the plan above suggested for the comparison of two co-
efficients of mutual induction. No satisfactory result could be obtained in
the ordinary method of working, the needle showing uneasiness whatever
resistances were employed, so that it was impossible to fix upon any particular
value as corresponding to a zero integral current. The addition of other
coils to increase the self-induction of one of the secondaries was so far success-
ful that the needle could be reduced to quietness, but calculation showed that
the additional self-induction found to be necessary in experiment was much
in excess of what the above theory would indicate. The explanation which
afterwards suggested itself to me was that the anomalous effect was due to
the conducting rings upon which some of the coils were wound, and whose
230 ON THE IMPERFECTION OF THE GALVANOMETER, ETC. [105
presence complicates the otherwise simple theory. We verified this view by
bringing a coil of wire into the neighbourhood of one of the principal coils,
the behaviour of the galvanometer needle being very sensibly different
according as the auxiliary coil was open or closed.
The kind of embarrassment to which measurements of this kind are
subject is well illustrated by placing the galvanometer in a tertiary circuit,
not directly influenced at all by the battery current in the primary. A pair
of coils with double wires, such as are often used for large electro-magnets, is
suitable for the experiment. One wire of the first coil is connected with the
battery, and forms the primary circuit. The second wire of the first coil and
the first wire of the second coil are connected, and constitute together the
secondary circuit. The second wire of the second coil and the galvanometer
form the tertiary circuit. The apparatus must be so adjusted that no effect
is perceived at the galvanometer when the secondary is broken, whatever
may happen in the primary. When this adjustment is complete the secondary
is closed, and the effect is observed of opening or closing the primary. If the
contacts are properly made, the integral current through the galvanometer
at each operation is rigorously zero, but in the experiments that I have made
no one could infer the fact from the behaviour of the galvanometer needle.
The effect may be exaggerated by the insertion of a few iron wires into the
induction coils.
106.
ON LAPLACE'S THEORY OF CAPILLARITY.
[PkOosopkical Magazine, XVL pp. 309315. 1883.]
FKOM the hypothesis of forces sensible only at insensible distances
Laplace*, it is well known, arrived at the conclusion that the pressure
within a sphere of liquid of radius 6 may be expressed by
m
H is the constant on which capillary phenomena depend, and the effect of
the second term may be represented by the fiction of a constant tension in
the superficial layer. According to Laplace's theory, however, the first term
K is enormously the greater; only, being the same at all points in the interior
of the fluid, whatever may be the form of the boundary, it necessarily escapes
direct observation.
When two liquids are in contact the difference of pressures within them
will stall be of the form (1), but the values of K and H will depend upon the
properties of both kinds of matter.
The existence of an intense molecular pressure K is a necessary part of
Laplace's, and probably of any similar^ theory of these phenomena ; but it
has not met with universal acceptance^. The difficulty which has been felt
appears to depend upon an omission in the theory as hitherto presented.
Before we can speak of K as a molecular pressure proper to the liquid, it is
Mfj^ipnjMji to show that the change, which we may denote by K^. experienced
in paging the surface dividing liquid L from liquid TIT, is identical with the
sum of the changes denoted by K-& and K* : so that it makes no difference
whether we pass from L to UL directly or by way of IL That this should
be the case upon Laplace's principles will be shown further on. The point,
* Xeaudfme Celeste, Supplement to Tenth Book.
t Qmneke, Pon- J*- 1*70- Abo Bfley, PkiL Jfe*. JUrefa 1*B-
232 ON LAPLACE'S THEORY OF CAPILLARITY. [106
however, is so important that I propose to give in addition a proof of much
wider generality, by which the relation is placed upon a sound basis. The
existence of an intense internal pressure is probable for many reasons ; and
it is hoped that no further difficulty need be felt in admitting it as a legitimate
hypothesis.
Let us imagine different kinds of liquids, varying continuously or discon-
tinuously, to be arranged in plaue strata, and let us
examine the difference of pressure, due to the attracting
p forces, at two points A and B, round each of which the
fluid is uniform to a distance exceeding the range of
the forces. The difference of pressure in crossing any
B infinitely thin stratum at P is due to the forces
operative between P and all the other strata. The
force between one of the interior strata Q and P will depend upon the
thicknesses of the strata, upon the nature and condition of the fluids composing
them, and upon the distance PQ. But whatever may be the law of the action
in these respects, the force exerted by Q upon P must be absolutely the same
as the force exerted by P upon Q. Now, as we pass downwards from A to B,
every pair of elements between A and B comes into consideration twice. In
passing through P we find an increase of pressure due to the action of Q
upon P, but in passing through Q we have an equal diminution of pressure
due to the action of P upon Q. Along the whole path from A to B the only
elements which can contribute to a final difference of pressure are those which
lie outside, i.e. in the fluid above A and below B. By hypothesis the action
of the fluid above A on the strata traversed in going towards B ceases within
the limits of the uniform fluid about A ; and consequently the whole difference
of pressure due, according to this way of treating the matter, to the fluid
above A depends only upon the properties of A. In like manner the
difference due to the fluid below B depends only upon the properties of B ;
and we conclude that the whole difference of pressure due to the action of
the forces along the path AB depends upon the properties of the fluids at
A and B, and not upon the manner in which the transition between the two
is made. In particular the difference is the same whether we pass direct
from one to the other, or through an intermediate fluid of any properties
whatsoever.
It is evident that the enormous pressure which Laplace's theory indicates
as prevalent in the interior of liquids cannot be submitted to any direct test.
Capillary observations can neither prove nor disprove it. But it seems to
have been thought that the relation
.............................. (2)
implies a corresponding relation between the capillary constants
................................ (3)
106] ox LAPLACE'S THEORY OF CAPILLARITY. 233
and the fact that (3) is inconsistent with observation is supposed to throw
doubts upon (2). Indeed Mr Riley *, in his interesting remarks upon Capillary
Phenomena, goes the length of asserting that, according to Laplace, K is a
function of H. It is thus important to show that Laplace's principles, even
in their most restricted form, are consistent with the violation of (3).
In attempting calculations of this kind we must make some assumption
as to the forces in operation when more than one kind of fluid is concerned.
The simplest supposition is that the law of force between any two elements
is always the same, <(r), as a function of the distance, and that the difference
between one fluid and another shows itself only in the intensity of the action.
The coefficient proper to each fluid may be called the "density,'' without
meaning to imply that it has any relation to inertia or weight. The force
between two elements (of unit volume) of fluids I. and II. may thus be
denoted by p^^r): that between two elements of the same fluid by
pf&^r), or p/^(r), as the case may be,
We will first examine the forces operative in a fluid whose density varies
slowly, that is to say undergoes only a small change in distances of the order
of the range of the forces, supposing, for simplicity, that the strata are surface-
of revolution round the axis of z. The first step will be to form an expression
for the force at any point on the axis.
The direction of this force is evidently along z. and its magnitude depends
upon the variation of density in the neighbourhood of 0. If the density
were constant, there would be no force. We may write
dp
* p = fc Z
or in polar coordinates,
5 dp d^pr^co^ff (For* sin 1
-J- - --
-j 5 -*- 5 + terms in r. ....... (4)
dz ciz* 2 dj? z
For the attraction of the shell of radius r and thickness dr we have
and for the complete attraction,
ir^l f**(r)dr + terms in T r 5 i> (r) dr.
6 cLzjQ Jo
The difference of pressure corresponding to a displacement dz is found by
multiplying this by pdz. Thus
* Loe. cit. p. 193.
234 ON LAPLACE'S THEORY OF CAPILLARITY. [106
and
PI ~ P-2 = -- (pi 1 pz} \ r 3 </> (r) dr + terms in I i*d>(r)dr ....... (5)
<J Jo Jo
Laplace employs a function ty, such that
(6)
and he finds that in the case of a uniform fluid in contact with air the
principal term, K, depends upon/i/r (r) dr, and the second, H, upon fr-^r (r) dr.
For the continuously varying fluid here considered, we see from (5) that
(7)
and that there is no term of the order of the capillary force. Equation (7)
agrees with our general result that the difference of pressures required to
equilibrate the forces operating between two points depends only upon the
nature of the fluid at the final points ; and it shows further that, under the
more special suppositions upon which the present calculation proceeds, the
molecular pressure at any point is to be regarded as proportional to the square
of the density.
But what is more particularly to be noticed is that, in spite of the curva-
ture of the strata, there is no variation of pressure of the nature of the
capillary force ; from which we may infer that the existence of a capillary
force is connected with suddenness of transition from one medium to another,
and that it may disappear altogether when the transition is sufficiently
gradual.
For the further elucidation of this question we will now consider the
problem of an abrupt transition. It does not appear that Laplace has any-
where investigated the forces operative at the common surface of two fluids
of finite density, but the results given by him for a single fluid are easily
extended.
Let OA (equal to a) be the radius of a spherical mass of liquid of "density"
p 2 , surrounded by an indefinite quantity of other
fluid of density p ls and let us consider the varia-
g tion of pressure along a line from a point (say 0)
removed from the surface on one side to a point
B also removed from the surface on the other side.
The difference of pressure corresponding to each element of the path OB is
found by multiplying the length of the element by the local density of the
fluid and by the resultant attraction at the point.
The attraction of the whole mass of fluid may be regarded as due to an
uninterrupted mass of infinite extent of density p lt and to a spherical mass
106] ON LAPLACE'S THEORY OF CAPILLARITY. 235
OA of density (p PI). Since the first part can produce no effect at any
part of OB, we have to deal merely with the attraction of the sphere OA.
Laplace has shown that if OA were of unit density, its action along the
line OA would be
where
r 1 r*
^(r)rfr; (8)
while along AB its action would be
*-*.
a
The loss of pressure in going outwards from to A is thus
and from A to B.
Accordingly the whole difference of pressure between and B \&
(pi-fhY ......................... (9)
Thus, in addition to the former result that the difference of pressure inde-
pendent of curvature varies as (pf p/), we see that the capillary pressure,
proportional to the curvature, varies as (p 2 p^.
The reasoning just given is in fact little more than an expansion of that
of Young*. If the effect depends only upon the difference of densities, it
cannot fail to be proportional to (/>, p^.
Writing H^ = H(p^ p^f, we see that there is no reason whatever for
supposing that the capillary constants of three liquids should be subject to
the relation
On the contrary, the relation to be expected, if the suppositions at the basis
of the present calculations agree with reality, is
JH v = JH u + 4H n ............................. (10)
In (10) the three radicals are supposed to be positive, and H u is the greatest.
If we suppose that the third fluid is air, and put />, - 0, we have
Vfl^Vtfi-Vtf,, ........................... (11)
* Encyc. Brit. 1816. Young's Work, vol. i. p. 463.
236 ON LAPLACE'S THEORY OF CAPILLARITY. [106
in which H, > H 2 . From (11)
so that
H^<(H,-H,} ............................... (12)
The reason why the capillary force should disappear when the transition
between two liquids is sufficiently gradual will now be evident. Suppose
that the transition from to p is made in two equal steps, the thickness of
the intermediate layer of density \p being large compared to the range of
the molecular forces, but small in comparison with the radius of curvature.
At each step the difference of capillary pressure is only ^ of that due to the
sudden transition from to p, and thus altogether half the effect is lost by
the interposition of the layer. If there were three equal steps, the effect
would be reduced to one-third, and so on. When the number of steps is
infinite, the capillary pressure disappears altogether.
Although the relation (12) is given by Quincke* as the result of experi-
ment, the numerical values found by him do not agree with (11). In most
cases the tension at the common surface of two liquids exceeds that calculated
from the separate tensions in contact with air. This result, which must be
considered to disprove the applicability of our special hypotheses, need not
much surprise us. There was really no ground for the assumption that the
law of force is always the same with the exception of a constant multiplier.
The action of one fluid upon another might follow an altogether different law
from its action upon itself. Besides this we are not entitled to assume that
a fluid retains its properties close to the surface of contact with another fluid.
Even if the hypothesis, which would refer everything to a difference of
"densities," were correct, its application would be rendered uncertain by
any modifications which the contiguous layers of different liquids might
impose upon one another. As we have seen, if this modification were of the
nature of making the transition less abrupt, the capillary forces would be
thereby diminished.
[1899. Reference may be made to further papers "On the Theory of
Surface Forces," Phil. Mag. xxx. pp. 285, 456, 1890; xxxm. p. 209, 1892;
xxxiii. p. 468, 1892.]
* Loc. cit. pp. 27, 87.
107.
OX THE MEASUREMENT OF ELECTRIC CURRENTS.
[Proceeditgs of ike Cambridge Philosophical Society, v. pp. 5052, 1SS3.]
PERHAPS the simplest way of measuring a current of moderate mteasiirr,
when once the electro-chemical equivalent of silver is known, is to determine
the quantity of metal thrown down by the current in a given time in a silver
voltameter. According to Kohlrausch the electro-chemical equivalent of
silver is in C.G.& measure 1*136 x 10~*, and according to Maseart 1 124 x 10-*.
Experiments conducted in the Cavendish Laboratory during the past year
by a method of current weighing described in the British Association Rejrr
for 1888* Jhave led to a lower number, viz. 1-119 x 10-*. At this rate tine
silver deposited per ampere per h&xr is 4-028 grams, and the method of
measurement founded upon this number may be used with good effect when
the strength of the current ranges from ^ ampere to perhaps 4 amperes. It
rouuirett however a pretty good balance, and some experience in chemical
manipulation. [See ArL 11?]
Another method which gives good results and requires only apparatus
fir"' 1 "**' to die electrician, depends upon the use of a standard galvanic celL
The current from this cell is passed through a high resistance, such as
10,000 ohms, and a known fraction of the electro-motive force is taken by
touching this circuit at definite points. The current to be measured is caused
to flow along a strip of sheet German silver, from which two tongues project.
The difference of potential at these tongues is the product of the resistance
included between them and of the current to be measured, and it is balanced
by a fraction of the known electro-motive force of the standard cell (fig. 1>
With a, amaiiiie galvanometer the balance may be adjusted to about ^fa.
The German silver strip must be large enough to avoid heating. The
resistance lmiuut the tongues may be ^ ohm, and may be determined by a
method similar to that of if^^r and Hockin (Maxwells Electricity,
* [Ait. 88, ioL n. p. Ut]
238 ON THE MEASUREMENT OF ELECTRIC CURRENTS. [107
352). The proportions above mentioned are suitable for the measurement
of such currents as 10 amperes.
Fig. l.
Another method, available with the strong currents which are now
common, depends upon Faraday's discovery of the rotation of the plane of
polarization by magnetic force. Gordon found 15* as the rotation due to
the reversal of a current of 4 amperes circulating about 1000 times round
a column of bisulphide of carbon. With heavy glass, which is more
convenient in ordinary use, the rotation is somewhat greater. With a coil
of 100 windings we should obtain 15 with a current of 40 amperes; and
this rotation may easily be tripled by causing the light to traverse the
column three times, or what is desirable with so strong a current, the thick-
ness of the wire may be increased and the number of windings reduced.
With the best optical arrangements the rotation can be determined to one or
two minutes, but in an instrument intended for practical use such a degree
of delicacy is not available. One difficulty arises from the depolarizing
properties of most specimens of heavy glass. Arrangements are in progress
for a redetermination of the rotation in bisulphide of carbon. [See Art. 118.]
* [Jan. 1884. In a note recently communicated to the Royal Society (Proceedings, Nov. 15,
1883) Mr Gordon points out that, owing to an error in reduction, the number given by him for
the value of Verdet's constant is twice as great as it should be. The rotations above mentioned
must therefore be halved, a correction which diminishes materially the prospect of constructing
a useful instrument upon this principle.]
108.
ON THE CIRCULATION OF AIR OBSERVED IN KUNDT'S
TUBES, AND ON SOME ALLIED ACOUSTICAL PROB-
LEMS.
[Philosophical Transactions, CLXXV. pp. 121, 1883.]
EXPERIMENTERS in Acoustics have discovered more than one set of
phenomena, apparently depending for their explanation upon the existence
of regular currents of air resulting from vibratory motion, of which theory
has as yet rendered no account. This is not, perhaps, a matter for surprise,
when we consider that such currents, involving as they do circulation of the
fluid, could not arise in the absence of friction, however great the extent of
vibration. And even when we are prepared to include in our investigations
the influence of friction, by which the motion of fluid in the neighbourhood
of solid bodies may be greatly modified, we have no chance of reaching an
explanation, if, as is usual, we limit ourselves to the supposition of infinitely
small motion and neglect the squares and higher powers of the mathematical
symbols by which it is expressed.
In the present paper three problems of this kind are considered, two
of which are illustrative of phenomena observed by Faraday*. In these
problems the fluid may be treated as incompressible. The more important
of them relates to the currents generated over a vibrating plate, arranged as
in Chladni's experiments. It was discovered by Savart that very fine powder
does not collect itself at the nodal lines, as does sand in the production of
Chladni's figures, but gathers itself into a cloud which, after hovering for a
time, settles itself over the places of maximum vibration. This was traced
by Faraday to the action of currents of air, rising from the plate at the
places of maximum vibration, and falling back to it at the nodes. In a
* " On a Peculiar Class of Acoustical Figures; and on certain Forms assumed by groups of
particles upon Vibrating Elastic Surfaces," Phil. Iran*. 1831, p. 299,
240 ON THE CIRCULATION OF AIR OBSERVED IN KUNDT'S TUBES, [108
vacuum the phenomena observed by Savart do not take place, all kinds of
powder collecting at the nodes. In the investigation of this, as of the other
problems, the motion is supposed to take place in two dimensions.
It is probable that the colour phenomena observed by Sedley Taylor* on
liquid films under the action of sonorous vibrations are to be referred to the
operation of the aerial vortices here investigated. In a memoir on the
colours of the soap-bubble f, Brewster has described the peculiar arrange-
ments of colour, accompanied by whirling motions, caused by the impact of a
gentle current of air. In Mr Taylor's experiments the film probably divides
itself into vibrating sections, associated with which will be aerial vortices
reacting laterally upon the film.
The third problem relates to the air currents observed by Dvorak in a
Kundt's tube, to which is apparently due the formation of the dust figures.
In this case we are obliged to take into account the compressibility of the
fluid.
My best thanks are due to Mr W. M. Hicks, who has been good enough
to examine the mathematical work of the paper. The results are thus put
forward with greater confidence than I could otherwise have felt.
1. In the usual notation the equations of motion in two dimensions
are
1 dp du - _, du du }
- -f- = - -r + v V 2 u -u-, v -7-
p dx dt dx dy
1 dp _ dv _ 2 dv dv j
p dx dt dx dy J
and since the fluid is incompressible,
_ -f- _ = o (2)
In virtue of (2) we may write
u = d-^fdy, v = d^rfdx (3)
Eliminating p between equations (1), we get
v V2 (^L _ *^\ _ ^_ (^ _ ^\ _ d f du du\ d f dv dv
Now
^ J f" ^ ~j~ = 2 7 ~t~ ^ I ~r T~ I
dx dy dx \dy dxj
dv dv _ . d (
dx dy ~ * dy
* Proc. Roy. Soc. 1878.
t Edinburgh Transactions, 1866 67.
10\ AXD OX SOME ALLIED ACOUSTICAL PROBLEMS. 241
so that
etr
For the first approximation we neglect the right-hand member of (4) as
being of the second order in the velocities, and take simply
-;*- ........................... < 5 >
The solution of (5) may be written*
................................. (6)
where
We will now introduce the suppositions that the motion is periodic with
respect to JT, and also (to a first approximation) with respect to t. We thus
assume that ^r, and ^r a are proportional to cos L-JT. and also to ***. The wave-
length (X) along x is 2'i, and the period T is 2*; a. The equations (T ) now
become
(--*)*- ............. "
by which ^r, and ^r* are to be determined as functions of y. If we write
................................. 9i
we have as the most general solutions of (8)
(10)
(11)
With respect to the value of k\ we see from (9) that it is complex. If
we write
^ = ^0620, np = P i sin22.
then k' = P cos a + iP sin a.
In all the applications that we shall have occasion to make, an approxi-
mate value of Zr' is admissible. On account of the smallness of w. n 9 is very
large in comparison with *, that is to say, the thickness of the stratum
through which the tangential motion can be propagated in time T is very
small relatively to the wave-length X We may therefore neglect Zr 1 in the
equation
P*
and take simply
P S = */V.
* Stakes, "On Pendulums," Co*. PJkO. !>*. *oL n_ 1350.
E. IL 16
242 ON THE CIRCULATION OF AIR OBSERVED IN KUNDT's TUBES, [108
Again (sin a - cos a) 2 = 1 - sin 2a = |&V/n 2 ,
so that the difference between cos a and sin a may be neglected. We will
therefore write
tf = /8(l+t), ................................. (12)
where
= V(n/2iO .................................. (13)
We must now distinguish the cases which we have to investigate. In the
first we suppose that a wave motion is in progress in a vessel whose hori-
zontal bottom occupies a fixed plane y = 0. We may conceive the fluid to be
water vibrating in stationary waves under the action of gravity, the question
being to examine the influence of the bottom upon the motion. If there are
no other solids in the neighbourhood of the bottom, we may put D = 0,
y being measured upwards, and /3 being taken positive.
The conditions to be satisfied at y = are that u and v should there
vanish. Thus
A + B + C = 0, -kA + kB-k'C=Q,
so that tfr = G{- cosh ky + (k'/k) sinh ky + e~ k 'y} ,
and 11 = G { k sinh ky + k' cosh ky k'e~ k ' y ] .
At a short distance from the bottom, u, k'C. If we denote by u the
maximum value of u near the bottom, we have
k'G = MO e int cos kx,
and then
7 ( cosh ku sinh ky
kx -- -* + +_ , ............ (14)
( k }
u = u e int cos kx \ - 77 sinh ky + cosh ky - e~ k 'v ( ........ (15)
I * J
f k k }
v M e int sin kx \ -j-, cosh ky + sinh ky + Y, e~ Ky \ ....... (16)
( K K }
These are the symbolical values. If we throw away the imaginary parts,
we have as the solution in real quantities by (12),
. , ( cosh ky . , sinh ku
^ = u cos kx | -- -|-^* cos (nt - JTT) + -^ " cos nt
e -Pv ]
+ pj 2 cos (nt - ITT - J3y) j, ...... (17)
, f k sinh ky
u = MO cos kx j -- 7T/2 C S ^ nt> ~ ^ 7r ) + COS ^ ty cos nt
-e~^cos(nt-^, ...... (18)
v = MO sin kx I - C ~ cos ( n t ~ I T) + sinh ky cos nt
108] AND ON SOME ALLIED ACOUSTICAL PROBLEMS. 243
This is the solution to a first approximation. At a very small distance
from the bottom the terms in e~ fty become insensible.
Although the values of u and v in (18) and (19) are strictly periodic, it is
proper to notice that the same property does not attach to the motions
thereby defined of the particles of the fluid. In our notation u is not the
velocity of any particular particle of the fluid, but of the particle, whichever
it may be, that at the moment under consideration occupies the point x, y.
If x + , y + ij be the actual position at time t of the particle whose mean
position during several vibrations is x, y, then the real velocities of the
particle at time t are not u, v, but
du .. du dv dv
and thus the mean velocity parallel to ar is not necessarily zero, but is equal
to the mean value of
du du
C f + V
in which again
=fudt, r,=jvdt.
From the general form of u, viz., cos kx F(y, t), it follows readily that
For the second term we must calculate from the actual values as given in
(18), (19). Thus
- J3y)
of which the two first terms may be neglected relatively to the third (con-
taining the large factor ). The product of T] and du/dy will consist of two
parts, the first independent of t, and the second harmonic functions of Znt.
It is with the first only that we are here concerned. The mean value of the
velocity parallel to x is thus
162
244 ON THE CIRCULATION OF AIR OBSERVED IN KUNDT'S TUBES, [108
On account of the factor e~*, this quantity is insensible except when ky
is extremely small. We may therefore write it
(2Q)
V (equal to k/n) being the velocity of propagation of waves corresponding to
k and n.
The only approximation employed in the derivation of (15) and (16) is
the neglect of the right-hand member of (4), and the corresponding real
values of u and v could, if necessary, be readily exhibited without the use of
a merely approximate value of k'. To proceed further we must calculate the
value of
..............................
v ax v ay
in (4), for which it will be sufficient to take the values given by the first
approximation. Thus
V 2 -/r = V 2 ^ 2 = i
and by (17)
dfa nu cos kx e~to
from which we find as the value of (21),
k /3\/2
sinh ky sin fty \/2 cosh ky cos fty + V2 e~?
+ terms in 2nt.
On account of the factor e~ fty this quantity is sensible only when y is very
small. We may write it with sufficient approximation
nku 2 sin 2kx e~^ y ( fi . R _ | ,.
The terms in Znt, corresponding to motions of half the original period, are
not required for our purpose, which is to investigate the non-periodic motion
of the second order. The equation with which we have to proceed is found
by equating (22) to V 4 \|r. The solution will consist of two parts, one resulting
from the direct integration of (22) and involving the factor e~ fty , the second a
complementary function with arbitrary coefficients satisfying VSfr = 0. In
the calculation of the first part we may identify V 4 with d^/dy 4 , on account of
the smallness of k relatively to . In this way our equation becomes
, .-.(23)
of which the solution is
If cos fty + $ sin fty + tfy sin fty + ft e~* y \ . . . .(24)
108] AND OX SOME ALLIED ACOUSTICAL PROBLEMS. 245
The complementary function, being proportional to sin 2fcr, may be
written
If the fluid be uninterrupted by a free surface, or otherwise, within
distances for which ky is sensible, we must suppose ( A' + Ky) = 0, so that by
(13) the complementary function may be written
a, 5 sin 2JUr . .
The condition that F (equal to d^ dx) must vanish when y = 0, gives
A = jf . For the velocity parallel to x we have
{B-2k(A+By\].
In order that u should vanish when y = 0, we must have
approximately. Thus
To obtain the mean velocity parallel to x of a particle, we must add to
(25), the terms previously investigated and expressed by (20). If we call the
total ', we have
u' = ^^[e^{-sin^-|e-^} + |e^[l-2^}]. ...... (27)
At a short distance from the bottom ~** becomes insensible, and we have
simply
(28)
(29)
_
pv
The steady motion expressed by (28) and (29) is of a very simple character.
It consists of a series of vortices periodic with respect to x in a distance $X,
For a given x the horizontal motion is of one sign near the bottom, and of
the opposite sign at a distance from it, the place of transition being at
y = (2*)- 1 = V 4 *"- The horizontal motion of the first order near the bottom
246 ON THE CIRCULATION OF AIR OBSERVED IN KUNDT'S TUBES, [108
being by (18) u = u cos kx cos nt, we see that it is a maximum when
kx = 0, TT, 2-7T, ... If we call these places loops, and the places of minimum
velocity nodes, (29) shows that v is negative and a maximum at the loops,
positive and a maximum at the nodes. The fluid therefore rises from the
bottom over the nodes and falls back again over the loops, the horizontal
motion near the bottom being thus directed towards the nodes and from the
loops. The maximum horizontal motion is simply gU^/V, and is independent
of the value of v. We cannot, therefore, avoid considering this motion by
supposing the coefficient of viscosity to be very small, the maintenance of the
vortices becoming easier in the same proportion as the forces tending to
produce the vortical motion diminish.
To ascertain the character of the motion quite close to the bottom, we
must include the terms in e~^ y . When y is extremely small
- 1 sin 2kx -
.(30)
so that the motion is here in the opposite direction to that which prevails
when e~P y can be neglected.
A few corresponding values of fiy and of (sin (By + ^e~ ?y ) e~& y + f are
annexed, in order to show the distribution of velocities within the thin
frictional layer.
ft
Py
7T
16
-038
Sir
+ 055
TT
~8
-054
7T
2
+ 151
Sir
16
-049
7T
+ 374
7T
I
-025
8
a
+ 384
It appears that (sin 2fec being positive) the velocity is negative from the
plate outwards until fiy somewhat exceeds |TT, after which it is positive, until
reversed by the factor (1 - 2%). The greatest negative velocity in the layer
is about f of that which is found at a little distance outside the layer.
Faraday found that fine sand, scattered over the bottom, tends to collect
at the loops. This is in agreement with what the present calculation would
lead us to expect, provided that we can suppose that the sand is controlled
by the layer at the bottom whose motion is negative. The exceeding thinness
of the layer, however, presents itself as a difficulty. The subject requires
further experimental investigation ; but in the meantime the following data
may be worth notice, though in some respects, e.g., the shallowness of the
108] AND OX SOME ALLIED ACOUSTICAL PROBLEMS. 247
liquid in relation to the wave-length, the circumstances differed materially
from those assumed in the theoretical investigation.
The liquid was water (> = -014c.G.s.), and the period of vibration was ^,
so that K = 2-B- x 15. The thickness of the layer
= l^r vX2r 11) = -0135 centim.
Measurements of the diameters of the particles of sand gave about
02 centim., so that the grains would be almost wholly immersed in the
negative layer, even if isolated. It seems therefore that the observed motion
to the loops gives rise in this case to no difficulty. But it is possible that
the behaviour of the sand is materially influenced by the vertical motion of
the vessel by which in these experiments the liquid vibrations are main-
tained*.
| 2. In the problem to which we now proceed the motion will be sup-
posed to have its origin in the assumed motion of a flexible plate situated
when in equilibrium at y = 0. Thus for a first approximation we take
H == 0, r = fy sin kr *"*, when y = 0, and the question is to investigate the
resulting motion of the fluid in contact with the plate.
The solution to a first approximation is readily obtained. As in 10 n 11 .
we have
in which we may take as before
By the condition at y = 0,
so that
J = -^C,
i^ccsfcr Z
k-L-' \ k
-(33)
lfr*-lfr*n <34)
k-k- r
In passing to real quantities it will be convenient to write
FTP' 1611 - (35 >
Thus throwing away the imaginary parts of (33) r (34), we get
....(38)
See a l-pe* " On the Crispatwns of Fluid noting upon Yibfmtiiis Support," PWI. Jf-f-
Jafy, 1883. [Art. 108, wL n. p. 212.]
248 ON THE CIRCULATION OF AIR OBSERVED IN KUNDT's TUBES, [108
From (32), (35), the approximate value of H is - v //8>/2, and that of e is
^TT. More exact values will however be required later. We find
77- ___ o <> i , /QQ\
+ '
(40)
The values of u and v above expressed give u = 0, v = V Q sin kx cos nt, when
y = 0. This is sufficient for a first approximation, but in proceeding further
we must remember that these prescribed velocities apply in strictness not to
y = 0, but to
y = sin kx sin nt.
y n
Substituting the latter value of y in the expressions (37), and (38), we find
u = \f2. pH cos kx { ky cos (nt + e + ITT) + >/2 . fiy cos (nt + e + %TT)}
sin Zkx sin nt - - cos (nt + e + |TT) + cos (nt + e + Jw
sin 2kx < sin (e + ^TT) sin (e + |-TT)|- + terms in Znt.
The first term within the bracket is of the second order in k/ft relatively
to the latter term, and may be omitted. Thus
The terms in 2nt we need not further examine. From (39), (40),
H cos e = Vo/2/3 very approximately, so that we may write
(41)
To the same degree of approximation, v = v sin kx cos nt, simply.
We have next, as in the first problem, to consider the complete equation
(42)
in the right-hand member of which we use the approximate values given by
(36), (37), (38). Thus
d^lr
~TI = nH cos kx e~P y sin (nt + e /3t/),
and (42) becomes
nk/3H 2 sin 2Jfor e~^ ( ,. /2 .
v ^ = ~ - e - sm & - sm to - cos ^
108] AXD OX SOME ALLIED ACOUSTICAL PROBLEMS. 249
It will be found presently that the term divided by k disappears from the
final result, and thus we have to pursue the approximation further than might
at first appear necessary. We may however neglect terms of order i*//?, in
comparison with the principal term. Thus V* may be identified with tf/rfy*,
and the equation becomes
whence
And
............ (46)
To obtain the value of M at the surface of the plate it will be sufficient to
put y=0in(46). Thus
By (.32), (39)
if as before we put F for kjn. Thus in (47)
...(48)
To obtain the complete value of u at the surface of the plate, corresponding
to (37), (46), we have to add to (48) that given in (41). The term of lowest
order disappears, and we are left simply with
(49)
In like manner we find for the complete value of F at the surface of the
plate corresponding to (38), (45),
The values of u and r expressed in (49) and the second part of (50) must
be cancelled by a suitable choice of the complementary function, satisfying
V*^r = 0, so that to the second order of approximation the fluid in contact
with the plate may have no relative motion.
250 ON THE CIRCULATION OF AIR OBSERVED IN KUNDT'S TUBES, [108
The complementary function is
TJr = (A+ By) e-^y sin Ikx,
whence
u = [B - 2k (A + By)} e~ 2k v sin 2km,
v = -2k(A + By) e~^ cos 2kas.
Determining the constants as indicated above, we get
q,. 2
u = |f(l~ 2%) e-*y sin 2kx, ...................... (51)
(52)
The velocities given by (51), (52) are the only part of the motion of the
second order which is sensible beyond a very small distance from the vibrating
plate. The nodes of the plate (where sand would collect) are at the points
given by kx = 0, TT, 2?r... , and the loops at the points kx = ^ir, |TT. .. At the
former points v is negative, and at the latter positive. For kx = ^TT, u is
positive, and for kx = f TT, u is negative.
Jr TT |TT
node loop node loop
The magnitude of the vortical motion is independent of the coefficient of
friction.
The complete value of u to the second order of approximation (except the
terms in 2n<) is obtained by adding together (37), (46), and (51), and it will
contain the term divided by k in (46), whose appearance, however, is mis-
leading. The objectionable term will be got rid of, if we express the mean
velocity of a particle, instead of as in (46), the mean velocity at a point. For
this purpose we are to add to (46), (51), the mean value of
..du du
***%'
as calculated from the first approximation, where
As in the former problem the mean value of ^dujdx is zero.
Multiplying together du/dy, and $vdt as found from (37), (38), and re-
jecting the terms in 2nt, we get with omission of k*,
in which we may write
108] AND ON SOME ALLIED ACOUSTICAL PROBLEMS. 251
Combining (53), (46), and (51), we get finally
which expresses the mean particle velocity.
When J3y is very small, (54) gives
, ......... (54)
.--) (55)
. r
from which it appears that quite close to the plate the mean velocity is in
the opposite direction to that which is found outside the frictional layer.
3. In the third problem, relating to Kundt's tubes, the fluid must be
treated as compressible, as the motion is supposed to be approximately in one
dimension, parallel (say) to x. The solution to a first approximation is merely
an adaptation to two dimensions of the corresponding solution for a tube of
revolution by Kirchhoff*, simplified by the neglect of the terms relating to
the development and conduction of heat. It is probable that the solution to
the second order would be practicable also for a tube of revolution, but for
the sake of simplicity I have adhered to the case of two dimensions. The
most important point in which the two problems are likely to differ can be
investigated very simply, without a complete solution.
If we suppose p = a*p, and write <r for log p log/? 9 , the fundamental
equations are
% da __ du du du _ , d fdu dv'
dx dt dx dy djc \dx dy) '
with a corresponding equation for v, and the equation of continuity,
du dv dtr d<r dtr _
Whatever may be the actual values of u and r, we may write
dx dy ' dy dx '
in which
From (56), (57),
,d\d<r du du du , d
'.E + 'rfy ....... < 60
d\dtr dv _, dv dv ,d d<r d<r
Pogg. Atoi. i. CZXZIY. 1868.
252 ON THE CIRCULATION OF AIR OBSERVED IN KUNDT's TUBES, [108
Again, from (60), (61),
/ d f d\-. d 2 a d f da da\ . ,. _ / da da\
( a * +V dt +V dt) V<r -M" = dt( U dx + *dy)- (v + v}V ( U dx + V dy)
-*(to + to\d,tod\
dx\ dx dy) dy\ dx dyj
For the first approximation the terms of the second order in u, v,
and o- are to be omitted. If we assume that as functions of t, all the
periodic quantities are proportional to e int , and write q for
(62) becomes
qV 2 a + n?a =
Now by (57), (59),
so that
and
u = -^- + -, v= -, ---
n dx dy n dy dx
iqdcr dty iq da d\lr
-^- + ^-, v= -, --- ................. (64)
Substituting in (60), (61), with omission of terms of the second order, we
get in view of (63),
( V 2 - in) -f- = 0, (z/V 2 -m)--^ = 0,
' dy ' dx
whence
( V 2 in) ty = (65)
If we eliminate a directly from the fundamental equations (56), we get
d _A _ d / du du\ d f dv dv\ d . __ . d
dt J dy \ dx dy/ dx \ dx dy) dy dx
du dv\ a , rfV 2 ^|r dV^-Jr
j + j- V 2 i|r + u s-t + v r^ (66)
dx dyj dx dy
If we now assume that as functions of x the quantities a, ty, &c., are
proportional to e ikx , equations (63), (65) may be written
(dtldtf - k"*) a = 0, where k" 2 = k* - n*/q, (67)
(dtldtf - k' 2 ) -f = 0, where A;' 2 = 7c 2 + in/v (68)
If the origin for y be in the middle between the two parallel boundaries,
a must be an even function of y, and ty must be an odd function. Thus we
may write
a = A cosh k"y . e int e ikx , ^ = B sinh k'y . e int e ikx , .... (69)
- -2 A cosh k"y + k'B sinh k'y] e int e ikx
^ A sinh k"y - ikB sinh k'y] e ir '
* It is unnecessary to add a complementary function </>', satisfying y 2 0' = 0, as the motion
corresponding thereto may be regarded as covered by \j/.
108] AXD OX SOME ALLIED ACOUSTICAL PROBLEMS. 253
If the fixed walls are situated at y = y,, M and r most vanish for these
values of y. Eliminating from (70) the ratio of A to 1?, we get as the
equation for determining 1%
** tanh i-'y^i-t" tanh k"^, ..................... (71)
in which k', Jt" are given as functions of by (67), (68). We now introduce
further approximations dependent upon the assumption that the direct
influence of friction extends through a layer whose thickness is a small fraction
only of / 3 . On this supposition k' is large, and k" is small, so that we may
put tanh L~'y l = 1, tanh k"y l = k"y t . Equation (71) then becomes
** = *-*>,, ................................ (72)
or if we introduce the values of it', k" from (67), (68),
I* = (t* - < 9 ) y, V(* + /">
Since in "9 is great, Jtr = ir q = 11* a s approximately.
Thus
n* ** =1
~ + ~
and
If we write k= fr, + it*
fc = r
2y, }* * + %i
which agrees with the result given in 347 (11) of my book on the Theory of
Sound.
In taking approximate forms for (70), we must distinguish which half of
the symmetrical motion we contemplate. If we choose that for which y is
negative, we replace coshfr'y and sinh fry by 4~~*X For eoshfr"y we may
write unity, and for sinh fr"y simply fr"y. If we change the arbitrary multi-
plier so that the maximum value of K is unity, we have
...... (75)
in which, of course, u and r vanish when y = y l .
If in (75) we change k into L; and then take the mean, we obtain
(76)
Although k is not absolutely a real quantity, we may consider it to be
so with sufficient approximation for our purpose. If we write as before
254 ON THE CIRCULATION OF AIR OBSERVED IN KUNDT'S TUBES, [108
we get from (76) in terms of real quantities
u = cos kx [- cos nt + e~^ ( y + ^ cos {nt - (y + yj}]
v = - 5^5 sin kx \%- cos (nt - \TT) + erW+yJ cos {nt - ITT - (y + y,}]
It will shorten the expressions with which we have to deal if we measure
y from the wall (on the negative side) instead of as hitherto from the plane
of symmetry, for which purpose we must write y for y + y lf Thus
u = cos kx { cos nt + e~^ y cos (nt /3y)}
V
From (78) approximately
. cos kx e-M sin (nt -ir- fty), .............. (79)
-^ + -2- = k am kas cos nt, .. ...(80)
dx dy
u + v = i k/3 sin 2kx erM (- cos 0y + e~^} + terms in 2nt, (81)
+ V'-f = - p/9 sin 2kx e~^ (sin ^y + cos p + terms in 2nt. (82)
As in former problems the periodic terms in 2nt will be omitted. For the
non-periodic part of ^ of the second order, we have from (66)
V 4 -^ = - sin ZkxerM {sin/3y + 3 cos/% - 2e~^} ......... (83)
In this we identify V 4 with d 4 /dy*, so that
t = k ^ 16^3' (8^ /gy + 3 ^s fly + frr*}. ............ (84)
to which must be added a complementary function, satisfying V 4 o/r = 0, of the
form
t = ^f (^ sinh 2A; (y, - y) + 5 (y t - y) cosh 2A; (y, - y)J, . . .(85)
or as we may take it approximately, if y 1 be small compared with the wave-
length X,
y) 3} ............... (86)
The value of o- to a second approximation would have to be investigated
by means of (62). It will be composed of two parts, the first independent of
t, the second a harmonic function of Znt. In calculating the part of d$\dx
independent of t from
_. da- da da-
V 2 <1> = -- = -- U ^ -- V -y- ,
dt dx dy
108] AND OX SOMZ ALLIED ACOUSTICAL PROBLEMS. 255
we shall obtain nothing from d<r'dt. In the remaining terms on the right-
hand side it will be sufficient to employ the values of , p, a of the first
approximation. From
d<r du dv
in conjunction with (80), we get
G- = (u t a) sin kx sin nt,
whence
It is easily seen from this that the part of u resulting from d<f> djc is of
order X 4 ^ in comparison with the part (87) resulting from:^, and may be
omitted.
Accordingly by (S4\ with introduction of the value of $ and (in order to
restore homogeneity) of u 3
,87)
and from (86)
y) 5 }, (89)
When y = 0, the complete values of M and r. as given by the four last
equations, must vanish. Determining in this way the arbitrary constants J.'
and R, we get as the complete values at any point,
c = ~ ^aa* {^(sin y + 3 cos;
(92)
Outside the thin film of air immediately influenced by the friction we may
put r* = 0, and then
256 ON THE CIRCULATION OF AIR OBSERVED IN KUNDT's TUBES, [108
From (93) we see that u changes sign as we pass from the boundary y =
to the plane of symmetry y = y\, the critical value of y being ^(1 VsX
or -423^.
The principal motion being u = u cos kx cos nt, the loops correspond to
kx = 0, TT, 2-7T,..., and the nodes correspond to TT, |TT, ____ Thus v is positive
at the nodes and negative at the loops, vanishing of course in either case
both at the wall y = 0, and at the plane of symmetry y = y\.
Plane of symmetry
,
IT TT |T
loop node loop node
To obtain the mean velocities of the particles parallel to x, we must make
an addition to u, as in the former problems.
In the present case the mean value of
du du w 2 sin
so that
When fty is small,
Inside the frictional layer the motion is in the same direction as just
beyond it.
We have seen that the width of the direct current along the wall is
423 y lt and that of the return current (measured up to the plane of symmetry)
is '577 ^.so that the direct current is distinctly narrower than the return
current. This will be still more the case in a tube of circular section. The
point under consideration depends only upon a complementary function
analogous to (86), and is so simple that it may be worth while to investigate
it.
The equation for -fr is
(* !_4*.Y t -o, (97)
Var 2 rdr J T
but if we suppose that the radius of the tube is small in comparison with X,
k? may be omitted. The general solution is
f = {A + r> + Rr 3 log r + Cr*} sin 2kx, (98)
108] AXD ON SOME ALLIED ACOUSTICAL PROBLEMS. 257
so that
u = i ^ = (2B + K (2 log r + 1 ) + 4CV} sin 2fcr,
whence R = 0, by the condition at r = 0. Again,
v = - - = - 2 .4 r- 1 + 5 r + Cr* cos 2X-a-,
r dx
whence ^1=0.
We may take therefore
= (25 + 4CV) sin 2A-or, v = - 2 (5 r + Cr! cos 2*.r. . . .(99)
If v = 0, when r = R, B + C'^? 2 = 0, and
M = 2C(2^- JP)sin2fcr. ....................... (100)
Thus u vanishes, when
r = ^ 6 = -707 J2, tf - r = 293 R.
Vz
The direct current is thus limited to an annulus of thickness '29-3 R.
the return current occupying the whole interior, and having therefore a
diameter of
2 x -707 12 = 1-414 .R.
109.
THE FORM OF STANDING WAVES ON THE SURFACE OF
RUNNING WATER.
[Proceedings of the London Mathematical Society, xv. pp. 69 78, 1883.]
THE present investigation had its origin in an attempt to explain
more fully some interesting phenomena described by Scott Russell* and
Thomson f, and figured by the former. When a small obstacle, such as a
fishing line, is moved forward slowly through still water, or (which of
course comes to the same thing) is held stationary in moving water, the
surface is covered with a beautiful wave-pattern, fixed relatively to the
obstacle. On the up-stream side the wave-length is short, and, as Thomson
has shown, the force governing the vibrations is principally cohesion. On
the down-stream side the waves are longer, and are governed principally
by gravity. Both sets of waves move with the same velocity relatively to
the water ; namely, that required in order that they may maintain a fixed
position relatively to the obstacle. The same condition governs the velocity,
and therefore the wave-length, of those parts of the wave-pattern where the
fronts are oblique to the direction of motion. If the angle between this
direction and the normal to the wave-front be called 6, the velocity of propa-
gation of the waves must be equal to v cos 0, where v represents the velocity
of the water relatively to the (fixed) obstacle.
Thomson has shown that, whatever the wave-length may be, the velocity
of propagation of waves on the surface of water cannot be less than about
23 centims. per second. The water must run somewhat faster than this
in order that the wave-pattern may be formed. Even then the angle is
.subject to a limit defined by v fi cos = 23, and the curved wave-front has a
corresponding asymptote.
The immersed portion of the obstacle disturbs the flow of the liquid
* Brit. Assoc. Report for 1844.
+ Phil. Mag. Nov. 1871.
109] THE FORM OF STANDING WAVES, ETC. 259
independently of the deformation of the surface, and renders the problem
in its original form one of great difficulty. We may, however, without
altering the essence of the matter, suppose that the disturbance is pro-
duced by the application to one point of the surface of a slightly abnormal
pressure, such as might be produced by electrical attraction, or by the
impact of a small jet of air. Indeed, either of these methods the latter
especially gives very beautiful wave-patterns.
Even with this simplification, the difficulties remain considerable. It
would appear to be a necessary first step to solve the problem in two
dimensions ; that is, to find the standing wave-form produced in running
water by the impact of a sheet of wind, which strikes the surface along a
straight line. Of this I have succeeded in obtaining the solution, and it
accounts satisfactorily for one of the leading features of the phenomenon,
the existence of the waves of small wave-length only on the up-stream side,
and of the waves of greater wave-length only on the down-stream side of
the place of disturbance. In terms of this solution, that of the original
problem is analytically expressible, since we may imagine the pressure
localised round a point to be the result of the superposition of an infinite
system of linear pressures, whose lines of action pass through the point,
and are distributed equally in every direction. But the expression in terms
of an integral is not readily interpretable, and it is even doubtful see (23)
whether it has a definite limit when the viscosity of the liquid is supposed
to be infinitely small. In fact, that element of the integral which represents
a system of parallel waves, travelling (perpendicularly to their own fronts)
with the minimum velocity, has an infinite coefficient, as might perhaps have
been expected from the corresponding problem for sound, where all waves
travel with the same velocity. The prominence of this part of the system is
a marked feature of the observed wave-pattern.
But, without an exact solution, it is possible to determine the form of
the curved wave-fronts, considered as the envelope of a system of straight
lines, and thus to obtain from theory a pretty good general idea of the
phenomenon as a whole. In fig. 3 this construction is carried out for the
particular case in which the asymptotes include a right angle.
Let us suppose that deep water, originally in motion with uniform
velocity c parallel to the horizontal coordinate x, is disturbed slightly in
two dimensions. If <f> and ifr be the potential and stream functions, we
may take
<f> = cx + 2,ae- tz sin(kx + \ Tfr = cz 2ae- k2 cos(kx + e) (1)
In (1) z is measured downwards from the undisturbed surface, the wave-
length is Zir/k, and, for each value of k, a and e are arbitrary. For the
velocity at any point, we have, from (1),
e) (2)
172
260 THE FORM OF STANDING WAVES [109
In calculating the pressure, we will suppose that the motion of each
element is opposed by a retarding force proportional to the velocity*, of
which therefore the components parallel to the axes may be denoted by
hit, hv, h being positive. This (Theory of Sound, 239) is not incon-
sistent with the existence of a velocity potential, but we must imagine a
bodily force to act throughout the fluid sufficient to maintain the velocity c.
The only other force acting within the fluid is gravity. Hence, on the
supposition that the motion is steady, the equation for the pressure takes
the form
pfp = const. + gz h (<f> coo) %U 2
= const. + gz ASe~ fo sin(A; + e) - c'koie~ kz cos (&# + e) .......... (3)
The equation of the surface, found from (1) by putting -^ = 0, is
cz = 2 a cos (kx + e) ........................... (4)
Thus, for the variable part of the pressure just below the surface, we get
Bp/p = Sa (gc~ l - kc) cos (kx + e) - A2o sin (kx + e) .......... (5)
In passing from (5) to the expression for the pressure -BT which must act
externally upon the surface, we must include the effect of the capillary
tension T. The curvature of the surface is
(r 1 ^ al? cos (fee -M),
and thus
cn/p = ^a(g + T'k* - Arc 2 ) cos (kx + e) - AcSa sin (kx + e), ...... (6)
in which T' is written for T/p.
If we introduce a new angle e', defined by
- (7 >
(6) may be written
2 } cos (kx + e + e') .......... (8)
In the problem before us, we are to regard OT as given, and thence deter-
mine the form of the surface. If we suppose
vr/p = 2/8 cos (kx + e), ........................... (9)
where (3 and e are given for each value of k, then e + e' = e, and
c/3 = a V {(g + T'k* - kc-) 2
Accordingly, by (4),
_ /? cos (kx -f e e')
T'k* - Arc 2 ) cos (kx + e) + /3hc sin (kx + e)
gives the equation to the surface corresponding to the applied pressures (9).
* January, 1884. The dissipative forces here introduced are ultimately supposed to vanish,
but without them it did not seem easy to interpret the analytical expressions to which we are led.
109] ON THE SURFACE OF RUNNING WATER. 261
If we suppose that h is small, and limit ourselves to the case of a
single train of waves, t.e., to a single value of k, we see that the phases of
v and z are in general coincident or opposite, according as (g + T'tf IT)
is positive or negative. The first case arises when the wave-length is
either very great or very small, and then the pressure is in excess over
the troughs and in defect over the crests of the waves. The actual velocity
of the waves relatively to the water (c) is here less than that of free waves
of the given wave-length, i.e.,
</{ffik+T'k}.
But when the actual velocity c is greater than that of free waves of the
given wave-length, (g + T'& kc*) is negative, and then the excess of pres-
sure is to be found over the crests, and the defect of pressure over the
troughs of the waves. In the case of transition, when c coincides with the
velocity of the free waves, the term in h must be retained, and it shows
that the place of maximum pressure is now at that shoulder of the wave
where the water in its forward motion is falling.
In general, when the pressure along the surface is arbitrary, we must
have recourse to Fourier's theorem. Thus
(11)
p TJo J _ z
which is of the form (9).
We now suppose that the abnormal pressure is confined to a very narrow
strip at x = 0, so that ^(p) = 0, except when v is very small. In this case
(11) may be written
^ = 1 (A(r)rfer (~ dk cos kx = - 3> T dk cos kx, ...(12)
P TJ JO T Jo
if we put 4> for I (r) dv.
The corresponding value of z, from (10), is
** rg(9+r* -**)<**** + ic**kx.
- Jo (g+rp-k*y + h*<?
and this gives the form of surface assumed by the running water when
subjected to a small excess of pressure acting over a narrow strip at the
origin.
Before entering upon the general integration and interpretation of (13),
it may be well to point out its application in the case where the water is
originally at rest (c = 0). The formula (13) then reduces to
262 THE FORM OF STANDING WAVES [109
the upper sign being taken when x is positive, and the lower when x is
negative.
This solution of the statical problem may of course be obtained inde-
pendently from the differential equation
In the subsequent treatment of (13) it will conduce to brevity if we put
unity for c and T', symbols which can always be restored when desirable from
considerations of dimensions. We have, then, to consider
f (g-k + k?) cos kx + h sin kx
Jo* (<7-fc + &7 + A*
and it will assume different forms according as the roots of
are real or imaginary. For the present, we will take the former alter-
native, which is equivalent to supposing that the velocity of the water
exceeds the minimum velocity of propagation of free waves. We assume,
accordingly, that
g-k + fr^fa-fyfa-k),
where
kik 2 = g, k 1 + k 2 =l.
The quantities k lt & 2 are positive, and we will suppose them to be in
ascending order of magnitude. We thus replace (14) by
x &! k} (& 2 k) cos kx + h sin kx . .
and of this integral we shall require only the limiting form when h = 0, as
we do not propose to consider in general the effect of finite dissipative
forces. On this understanding, the first part of the integral may be
replaced at once by
coskx 1 f r coskxdk r cos kx dk\
k 2 -k)-k. 2 -k l \! k,-k -Jo k^nr]"
The integrals which make up (16) are even functions of x, i.e., they take
the same arithmetical values whether x be positive or negative. For dis-
tinctness, we will suppose that x is positive. Now
["coskxdk f t t*coe(fc 1 -tt)cZ f h ^ x cosudu ^^Kinudu
j - * - = = cos k^x -- h sin k,x ,
Jo ki-k J_oo u J-30 u J-, u
r*!* cosudu f 00 cosudu . ., .
- = - = ci(k 1 x),
J-co U J klX U
in which
sin u du r*i x sin u du
109] OS TEE SURFACE OF RU5XIXG WATER. 263
The functions ci and a may be regarded as known functions, and hare been
fully tabulated by Gfeusher*. Thus
r = cos k^ ci (,) + sin k^x ||w + si (!*)$, (17)
*i *
md
- - ': ':: 1 cos k^x ci k^jt 4- si
/ (k^ k)(k^ ky ktktlcoRkaXciksBsa
--(IS)
When *ijr=x.
tt-y = 0, ci t^x = X T a jr = 0.
In the Easter case the limiting form for ci (i\jr) is
...: ............ (19)
so thai, when x = 0,
,.;
When , = j, (18j changes its form and is replaced by
- r fees fcr ci kf + wt fa (T + a fa);,
that is,
We have now to consider tike second part of (15), that k. the Bimniit
when A=0 of
With respect to this, it is evident that the only elements of tine
which contribute to the limiting value are those for which the denominator
vanishes with A, i*, those lying in the immediate neighbourhood of the
roots L\ and k*. Thus, as k passes through Jt x we may pmt ij^ k) equal to
(L'. L\\ and as k passes thr>>ogh kywe may put (fr a 1-) equal to (l" a JLju
Hence the limit of (21) is the same as the limit of
kmkxdk
or the same as the limit of
k'mtxdk
of the Hnerieal Tabs of tbe
2(34 THE FORM OF STANDING WAVES [109
in which, k z being greater than k lt h' and h" are positive, and are supposed
ultimately to vanish. Now
r lisa*. lex dk [ Jfl h'cosuxdu , [ kl h'sinuxdu
I (I- Z-VT />'2 = SU1 lX I 2 ,i/ 2 COR KitS \ ^2 i frf* '
and
f fc > A'ainiwcdw ,. f + A' sin uxdu _
f fc > h'cosuxdu ,. [ +0 h'cosuxdu
= lim -/_l COS l^" = 1
Accordingly, the limit of (21) is
TT sin &j a; + TT sin & 2 #
and retains the same form whether x be positive or negative.
It is evident that in the case of equal roots (22), unlike (18), becomes
infinite, so that the retention of h is necessary for a practical result. It is
not difficult to show that, when h is very small,
r h sin kx dk
Jo (h-ky + h?
TT sin
which therefore represents for this case the leading term of the complete
expression (15).
Combining (18) and (22), we see that, when x is large and positive, the
value of (15) is
2-7r sin k^x ,~ .,
k z ki
and that, when x is large and negative, the value of (15) is
27T sin k^x /s>~\
i _ , (25)
On both sides of the place of disturbance, the surface is covered with
waves whose free velocity is that of the water. On the down-stream side
(x positive) the wave-length is the greater of the two which satisfy the
condition ft < k. 2 ) ; on the up-stream side it is the smaller. In the imme-
diate neighbourhood of the place of disturbance the form is a little more
complicated, and is best understood from a drawing.
When the roots of g k + k 2 = are imaginary, which happens when
the velocity of the water is less than that of any free wave, the analytical
expressions change their form. The second part of (14), written separately
109]
ON THE SURFACE OF RUNNING WATER.
265
in (21), vanishes when h = Q, the denominator being always finite. For the
first part we have, in place of (16),
r cos
Jo g-
coskxdk
cos kx dk
f x cos nx du . , f sin ux du
i# T -sinia; -. ....... (26)
J-.u"- + - ]-j,i? + g-l
in which g is positive.
So far as I have been able to learn, the integrals in (26), or others
equivalent to them, have not been tabulated. On this side, therefore, the
solution of our problem is incomplete, but fortunately this is not the case
to which the most interest attaches. It is probable that the disturbance
is limited to the immediate neighbourhood of the origin.
For the numerical calculation, it will be convenient to write (17) in
the form
cos L\x ci k\x + sin k\x (si k\x ^TT), .................. (27)
+ TT sin k\x,
of which the part (27) vanishes when x is great enough. The value of (27)
as a function of kx is shown by curve A (fig. 1). It is negative throughout,
and infinite when kx = 0.
The form of the standing wave produced by the local application of
to the surface depends upon the velocity of the water. To take
266 THE FORM OF STANDING WAVES [109
a case, we will suppose that this is such that the wave-lengths before and
behind are in the ratio of 1 : 2, so that k z =Zk 1 . The value of
cos k^x ci ktX + sin ^x (si k^ \TT}
cos l^x ci 2&!# sin 2k : x (si 2^ TT) ...... (28)
is shown by curve B (fig. 1), and the ordinates are to have the same value
when x is negative as when x is positive. The part near the origin is filled
in from the approximate analytical value
log e 2
The wave-form is now easily deduced, and is shown in fig. 2. On the
positive side we are to add to (28) 2?r sin kx, and on the negative side
we are to add 27rsin2& 1 #*.
We now pass to the consideration of the effect of a pressure localized
near a point, instead of distributed along a line. The wave-form is to be
found by the superposition of an infinite series of systems similar to (24), (25),
at various degrees of obliquity (0), and of such wave-lengths that
v cos 6 = v,
v being the velocity perpendicular to the wave-front in each case, and v the
velocity of the water (previously denoted by c). Now
thus the relation between k and 6 is
v ( ?QQ&e = kT' +g/k ......................... (29)
By (23) and (24), we see that the crests of the component trains are
situated at distances from the origin equal to (ra + |) X, where m is an
integer. The various wave-fronts thus form a system of similar and simi-
larly situated curves, whose shape is defined as the envelope of a system
of straight lines, the perpendicular on which from the origin is equal to p
and is inclined at an angle 6 to the direction of the stream, the relation
between p and being
P + z* p + r. ......................... (80)
j y
In the case of water, we have in C.G.S. measure T'/g = '073, so that
pt+ ^e pJr . 073=0 ...................... (81)
The roots are equal when
(32)
* [1899. Two sentences, vitiated by an error pointed out to me by Lord Kelvin, are here
omitted.]
109]
ON THE SURFACE OF RUNNING WATER.
267
The case proposed for consideration is that in which the asymptotes include
an angle of 90 3 , so that the maximum value of is 45. Substituting this
in (32), we find, vj[g = I'OSl, and thus
p = -5405 cos 1 0V{-5405 cos 4 0--0730} (33)
From this equation we may calculate any number of corresponding values
of p and 0, and thus draw the tangents of which the required curves are the
envelopes. The annexed table contains a few such pairs of values, sufficient
for an indication of the forms of the curves :
o-
1-0086 -0724
1-0000
1-0086
0724
9 C
9800 -0744
9877
9922
0753
18
8964 -0814
9511
9425
0856
27=
7625 -0957
8910
8557
1074
36
5823
1253
8090
7196
1549
45 3
2702
2702
7071
3821
3821
|
The two last columns give the intercepts on the axis, by means of which
and the value of cos 6 the lines are more readily drawn than from the
perpendiculars themselves. The result is shown in fig. 3.
110.
ACOUSTICAL OBSERVATIONS. V.
[Philosophical Magazine, xvn. pp. 188 194, 1884.]
Smoke-jets by Intermittent Vision.
IN the second series of these observations (Phil. Mag. 1879 [vol. I.
p. 406]) I proved that when stationary sonorous waves occupy the region
surrounding a sensitive flame, the action of sound in causing the flame to
flare manifests itself when the burner is situated at a loop, but not when
the burner is situated at a node; from which we infer that the effects
are due to a lateral disturbance causing the issuing jet to bend from its
course. During the same year I made a stroboscopic examination of a jet
of phosphorus-smoke issuing from a drawn-out glass nozzle, and disturbed
by the neighbourhood of a vibrating tuning-fork of pitch 256. So much
light is necessarily lost in this method of observation that some precau-
tion is required in illuminating the jet. Two points should be especially
attended to. In the first place, the eye must be so situated that the
scattered light by which the jet is seen is but slightly deflected from its
original course; and, secondly, the background must be thoroughly dark.
By carrying out adequately this system of illumination, and by so choosing
the revolving disk that the apertures bore a not too small proportion to
the entire circumference, I was able to see tolerably well by the light of a
good gas-flame. When the coincidence of periods was nearly approached,
the serpentine motion of the jet previous to rupture was clearly observ-
able. By placing the nozzle exactly iii the plane of symmetry between
the prongs of the fork, the law above stated could be confirmed. In this
position there was but little effect ; but the slightest displacement caused
an early rupture.
1 1 ACOCSmCAX, QBSEBYATIOXS. 269
Smwbc-jets amd Resonator*.
In order to exalt the sensitiveness of jets to notes of moderate pitch,
I found the use of resonators advantageous. These may be of Helmholtx's
pattern; but suitably selected wide-mouth bottles answer the purpose.
What is essential is that the jet should issue from the nozzle in the
region of rapid reciprocating: motion at the mouth of the resonator, and
in a transverse direction. I usually placed the resonator's month uppermost,
so that the jets were horizontal
Good results were obtained at a pitch of 256. When two forks of
about this pitch, and slightly out of tune with one another, were allowed
to sound simultaneously., the evolutions of the smoke-jet in correspondence
with the audible beats were very remarkable. By gradually raising the
presore at which the smoke is supplied, in the manner usual in these
experiments, a high degree of sensitiveness may be attained, either with
a drawn-out glass nozzle or with the steatite pin-hole burner used by
TyndalL In some ewes (even at pitch 2.56) the combination of jet and
resonator proved almost as sensitive to sound as the ear itself,
The behaviour of the sensitive jet does not depend upon the so:>k<r-
particlesy whose office is merely to render the effects more easily visible.
I have repeated these observations without smoke by simply causing air-jet^
from the same nozzles to impinge upon the flame of a candle placed at a
suitable distance. In such cases, as has been pointed out by Tyndali the
flame acts merely as an indicator of the condition of the otherwise invisible
jet. Even without a resonator the sensitiveness of such jets to hissing
sounds may be taken advantage of to form a pretty experiment.
The combination of jet, resonator, and flame shows sometimes a ten-
dency to speak on its own account; but I did not succeed in getting a
well-sustained sound. Such as it is, the effect probably corresponds to
one observed by Savart and Plateau with water-jets breaking up under the
operation of the capillary tension, and when resolved into drops impinging
upon a solid obstacle, such as the bottom of a sink in mechanical connexion
with the nozzle from which the jet originally issues. In virtue of the con-
nexion, any regular cycle in the mode of disintegration is able, as it were, to
propagate itself.
The increased and more discriminating sensitiveness obtainable by use
of resonators is turned to account in the arrangement of flame described in
the Prwxedmg* of the Cambridge Philosophical Society for November &
1880. (ToL L p. 500.]
In this case the resonator takes the form of a tube, one of whose ends
opens in the gas-chamber dose to the nozzle. The other end is closed by
270 ACOUSTICAL OBSERVATIONS. [110
a cork, whose position can be adjusted so as to vary the pitch. I see
from my note-book that, on the evening of Dec. 4, 1879, I found the
flame nearly as sensitive as the ear to vibrations of frequency 512; but
I have not always been equally successful in subsequent attempts to recover
this degree of delicacy.
With the very acute sounds, to which alone the high-pressure gas-flame
(lighted at the burner) is sensitive, little can be expected from the use of
resonators.
Jets of Coloured Liquid.
In the hope of being able to make better observations upon the trans-
formations of unstable jets, I next had recourse to coloured water issuing
under water. In this form the experiment is more manageable than in
the case of smoke-jets, which are difficult to light, and liable to be dis-
turbed by the slightest draught. Permanganate of potash was preferred as
a colouring agent, and the colour may be discharged by mixing with the
general mass of liquid a little acid ferrous sulphate. The jets were usually
projected downwards into a large beaker or tank of glass, and were lighted
from behind through a piece of ground glass.
The notes of maximum sensitiveness of these liquid jets were found to
be far graver than for smoke-jets or for flames. Forks vibrating from 20
to 50 times per second appeared to produce the maximum effect, to observe
which it is only necessary to bring the stalk of the fork into contact with
the table supporting the apparatus. The general behaviour of the jet
could be observed without stroboscopic appliances by causing the liquid
in the beaker to vibrate from side to side under the action of gravity.
The line of colour proceeding from the nozzle is seen to become gradually
more and more sinuous, and a little further down presents the appearance
of a rope bent backwards and forwards upon itself. I have followed the
process of disintegration with gradually increasing frequencies of vibra-
tional disturbance from 1 or 2 per second up to about 24 per second,
using electro-magnetic interruptors to send intermittent currents through
an electro-magnet which acted upon a soft-iron armature attached to the
nozzle. At each stage the pressure at which the jet is supplied should be
adjusted so as to give the right degree of sensitiveness. If the pressure
be too great, the jet flares independently of the imposed vibration, and
the transformations become irregular: in the contrary case the phenomena,
though usually observable, are not so well marked as when a suitable ad-
justment is made. After a little practice it is possible to interpret pretty
well what is seen directly; but in order to have before the eye an image
of what is really going on, we must have recourse to intermittent vision.
110] ACOUSTICAL OBSERVATIONS. 271
The best results are obtained with two forks slightly oat of tune, one of
which is used to effect the disintegration of the jet, and the other (by
means of perforated plates attached to its prongs) to give an intermittent
view. The difference of frequencies should be about one per second. When
the means of obtaining uniform rotation are at hand, a stroboscopic disk
may be substituted for the second fork. It was, in fact, with the use
of such a disk, driven by a water-engine, that the drawing (fig. 1) was
made by Mrs Sidgwick in August 1880. It is hardly necessary to say
that these appearances are difficult to reproduce in drawings, and that the
result must be regarded merely as giving a general idea of what is actually
observed. The upper part of the jet is seen sufficiently steadily to be pretty
accurately copied ; but further down true periodicity is lost, and no steady
impression is produced upon the eye.
The carrying out of these observations, especially when it is desired
to make a drawing, is difficult unless we can control the plane of the
bendings. In order to see the phases properly it is necessary that the
plane of bendings should be perpendicular to the line of vision : but with
a symmetrical nozzle this would occur only by accident. The difficulty
may be got over by slightly nicking the end of the drawn-out glass nozzle
at two opposite points. In this way the plane of bending is usually ren-
dered determinate, being that which includes the nicks, so that by turning
the nozzle round its axis the sinuosities of the jet may be properly presented
to the eye.
Occasionally the jet appears to divide itself into two parts imperfectly
connected by a sort of sheet This appears to correspond to the duplica-
272
ACOUSTICAL OBSERVATIONS.
[110
tion of flames and smoke-jets under powerful sonorous action, and to be
due to what we may regard as the broken waves taking alternately different
courses.
Fish-tail Burners.
"Experiments upon jets from fish-tail burners*. As with gas, so with
smoke and coloured water, these are sensitive, and when much excited
throw out tall streamers in the perpendicular plane. I have not yet fully
succeeded in tracing the genesis of these, but believe them due to the
rupture or collision of the sinuosities which are formed in the quickly-
moving part of the sheet. When the sheet, seen broadways on, is excited
by slow vibrations, a line of deepened colour is seen to descend, and presently
becomes very deep. This means that the sheet is so far bent over as to be
seen tangentially."
Even with the best arrangements as to sensitiveness and intermittent
vision, the appearances presented by these jets are somewhat difficult to
interpret and to reproduce in a drawing. The jets shown in figs. 2 5
issued from flattened glass nozzles, and are of the same character as those
given by fish-tail burners. In fig. 2 the flat side is presented to the
Fig. 3.
Fig. 2.
Fig. 4.
observer; in fig. 3 the sheet (if undisturbed) would be seen edgeways.
The complication arises, partly at any rate, from the different degrees of
sensitiveness of different parts of the sheet, from which it results that one
part reaches disruption arid loses its periodicity, while another is yet in
the earlier stages of the transformation. In figs. 2 and 3 the jet is under
the influence of a vibration sufficiently powerful to cause it to flare in a
regular manner; in figs. 4 and 5 the vibration is less powerful, and the
transformations stop short of the final stage.
Laboratory Note-book, Dec. 12, 1879.
110] ACOUSTICAL OBSERVATIONS. 273
Influence of Viscosity.
It has already been noticed that the notes appropriate to water-jets are
far graver than for air-jets from the same nozzles. Moreover, the velocities
suitable in the former case are much less than in the latter. This difference
relates not. as might perhaps be at first supposed, to the greater density,
but to the smaller viscosity of the water, measured of course kinematically.
It is not difficult to see that the density, presumed to be the same for
the jet and surrounding fluid, is immaterial, except of course in so far as
a denser fluid requires a greater pressure to give it an assigned velocity.
The influence of fluid viscosity upon these phenomena is explained in a
former paper on the Stability or Instability of certain Fluid Motions* ;
and the laws of dynamical similarity with regard to fluid friction, laid
down by Prof. Stokes^f, allow us to compare the behaviour of one fluid
with another. The dimensions of the kinematic coefficient of viscosity are
those of an area divided by a time. If we use the same nozzle in both
cases, we must keep the same standard of length : and thus the times
must be taken inversely, and the velocities directly, as the coeflicients of
viscosity. In passing from air to water the pitch and velocity are to be
reduced some ten times. But, in spite of the smaller velocity, the water-jet
will require the greater pressure behind it, inasmuch as the densities differ
in a ratio exceeding 100 : 1.
Guided by these considerations, I made experiments to try whether the
jets would behave differently in warm and cold water. At temperatures
respectively about 130 : F. and 52 = F., the difference was found to be
extremely well marked. " With a drawn-out glass nozzle, a pressure of
li inch was enough with hot water to cause flaring, whereas perhaps
3| inches were necessary with the cold water. At one inch the jet in
cold water was dead, but in hot water was still quite active^/
These experiments were resumed at Cambridge in April and May 1SSO
bv Mrs Sidgwick, with use not only of hot and cold water but also of mix-
tures of alcohol and water, whose viscosity is known to be much greater
than that of water alone. In order to retard cooling, and thus to diminish
convection-currents, the experimental beaker was placed within a larger
one. and supported at the rim only, so as to be surrounded by a jacket
of warm air. The liquid intended to form the jet was placed in a narrow
* Math. Soe. PHK. Feb. 12, 1880. [Vol. L P . 474.]
t Camb. Phil. Tra*t. ia50, On the Effect of Internal Friction of Fluids on the Motion of
Pendulums," 5. See also Helmholtz, Witd. Ann. Bd. vn. p. 337 (1879). or Reprint, voL t
p. 891.
* Laboratory Note-boot, Jan. 30, 1880. Prof. Osborne Reynolds has availed himself of
differences of temperature in oider to vary the viscosity, in some recent important observations
upon the cognate subject of the flow of water in tubes, Proe. Boy. Soc. March 15, IMS.
R. ii. 18
274
ACOUSTICAL OBSERVATIONS.
[110
glass jar about 10 inches high, and the head was adjusted by raising or
lowering the jar and by varying the amount of liquid. The communica-
tion between the two vessels was by a glass syphon, whose lower end
was drawn out so as to form a suitable nozzle of about -fo inch diameter
(fig. 6). The transparent tube was advantageous on account of the more
ready detection of air-bubbles, the presence of which, especially near the
nozzle, is a source of disturbance. The apparatus stood in front of a
window, supported on a stone table carried by the walls of the building,
and the sensitiveness of the jet was usually tested by dropping upon the
table a large nail through a height of about 2 inches. Observations were
made of the greatest pressure that the jet would bear, in the absence of
Fig. 6.
external disturbance, without flaring before reaching the bottom of the
beaker, and also of the least pressure at which the jet was sensitive. In
the case of the mixture of alcohol and water in equal parts, a modified
arrangement was necessary in order to obtain sufficient head.
With plain water the colour was given by permanganate of potash, and
was discharged, as soon as the jet was broken up, by ferrous sulphate pre-
viously added to the liquid in the beaker. In some of the more delicate
experiments it was found necessary to bring the densities to a more exact
equality by the addition of indifferent saline material to the jet, but in
most cases this precaution is superfluous. For the jets of methylated alcohol
and water, permanganate was found unsuitable, and was replaced by soluble
aniline blue.
The following table will give an idea of the results of a large number of
observations carefully repeated on different occasions :
110] AOoramcAL OBSERTATIOXSL 275
Beroes n
Aboat 13 inches
- -
V_l:
I.-"- "--.-
^
It will be seen that the effect of Tarring the viscosity is very
In the extreme eases a jet which will not hear a pressure of more than
inch without flaring when the liquid is water nearly at the boiling-poms,
requires from 20 to 30 inches to make it flare when we pass to the mixture
of alcohol and water in equal proportions. It is probable that with the hoc
water the results are somewhat disturbed by the impoe&biKty of getting the
liquid perfectly quiet owing to the formation of convection-currents ; but the
differences are so great that the general conclusion cannot be disturbed
by this circumstance. The comparisons between the cold water and the
alcoholic mixtures, which were all used at the temperature of the room, are
entirely independent of this source of error.
111.
ON THE MEASUREMENT OF THE ELECTRICAL RESISTANCE
BETWEEN TWO NEIGHBOURING POINTS ON A CON-
DUCTOR.
[Camb. Phil. Soc. Proc., v. pp. 133, 134, 1884]
As an alternative to the method of Matthiessen and Hockin, the fol-
lowing process may be used with good effect for the above purpose. It is
founded upon the combination of resistances described in a paper upon the
determination of the ohm by Lorenz's method (Phil. Trans. 1883 [Art. 94]),
by which a small effective resistance is obtained from elements of moderate
and accurately measurable value.
The main current 7 passes principally through a shunt of resistance a,
but partly through a small resistance 6 and a large resistance c arranged in
series. The terminals of a galvanometer of resistance g are connected to
the extremities of b. If g were infinite, the difference of potentials at its
electrodes would be
ab
a + b + c^'
so that ab/(a + b + c) is the effective resistance of the combination. For
example, if a= 1, b = 1, c = 98, the effective resistance is ^, and notwith-
standing its smallness is susceptible of accurate determination. Suppose now
that the main current traverses also a German silver strip (Proc., Nov. 26,
1883 [Art. 107]) provided with tongues between which we require to know
the resistance. It is evident that by adjustment of c the combination may be
made to give the same effect upon the galvanometer as the German silver
strip, so that the required result would be readily obtained from the above
formula. If c is taken from a resistance-box, we may find the effects, one
greater and one less than that of the strip, corresponding to resistances c
Ill] OX THE MEASUREMENT OF ELECTRICAL RESISTANCE, ETC. l77
and e+l, whence the value that would give exactly the same effect is
deduced by interpolation. In order to guard against disturbance from
thermo-electricity the readings should be taken by reversal of the battery,
and to eliminate the effects of varying current the combination and the
strip should be interchanged as rapidly as possible,
In practice the resistance of the galvanometer could not usually be
treated as infinite, and the interpretation of the results is a little more
complicated. In the case of the combination it may be shewn that the
current through the galvanometer is
aby
) '
By putting a infinite, or otherwise, we see that the corresponding current for
the strip is xy (g + x), if x be the required resistance between the tongues.
Equating these, we find
This method has recently been tested in the Cavendish Laboratory by
Messrs Shackle and Ward, and the results appear to shew that even with
so moderate a main current as "2 ampere, the sensitiveness is sufficient, the
mean of a few readings being probably correct to
112.
ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER, AND
ON THE ABSOLUTE ELECTROMOTIVE FORCE OF CLARK
CELLS. BY LORD RAYLEIGH, D.C.L., F.R.S., AND MRS H. SIDGWICK.
[Philosophical Transactions, 175, pp. 411460, 1884.]
1. IN former communications* to the Royal Society we have in-
vestigated the absolute unit of electrical resistance, and have expressed
it in terms of the B.A. unit and of a column of mercury at of known
dimensions. The complete solution of the problem of absolute electrical
measurement involves, however, a second determination, similar in kind,
but quite independent of the first. In addition to resistance, we require
to know some other electrical quantity, such as current or electromotive
force. So far as we are aware, all the methods employed for this purpose
define, in the first instance, an electrical current; but as a current cannot,
like a resistance, be embodied in any material standard for future use,
the result of the measurement must be recorded in terms of some effect.
Thus, several observers have determined the quantity of silver deposited,
or the quantity of water decomposed, by the passage of a known current
for a known time. In this case the definition relates not so much to
electric current as to electric quantity. A more direct definition of the
unit current, and one which may perhaps be of practical service for the
measurement of strong currents of 50 amperes or more, would be in terms
of the rotation of the plane of polarisation of sodium light, which traverses
a long column of bisulphide of carbon enveloped by the current a given
number of times f.
Other observers have expressed their results as a measurement of the
electromotive force of a standard galvanic cell. In this case it is neces-
* Proceedings, April 12, 1881 [vol. n. p. 1]; Phil. Tram. 1882, Part II. [vol. n. p. 38]
and 1883, Part I. [vol. n. p. 155].
t See Camb. Phil. Proc. Nov. 26, 1883 [vol. n. p. 237].
OX THE ELECTRO-CHEMICAI. BQCTVALEST OF SILVER. i~
sary to assume a knowledge of resistances. The known current in passing
a known resistance gives rise to a known electromotive force, which is
compared with that of the celL
In the present communication are detailed the experiments that we
have made to determine the electro-chemical equivalent of silver, and the
electromotive force of standard Clark cells. As regards the choice of filter
there is not much room for a difference of opinion. The difficulties to he
overcome in the use of a water voltameter are much greater. Copper
is, indeed, employed in ordinary laboratory practice and for commercial
purposes ; bat it is decidedly inferior to silver, both on account of its
tendency to oxidise when heated in the air, and aim because it changes
weight in contact with copper sulphate solution without the passage of an
electric current. Dr Gore* has made observations upon this subject, and
our own experience has shown that no constancy of weight is to be found
under these circumstances. Silver, on the other hand, seems to be entirely
unaffected by contact with neutral solution of the nitrate.
2. The readiest method of measuring currents is, perhaps, that fol-
lowed by Kohlransch, both in his earlier 4 ' and in his recent ^ work up:n.
this subject, viz.. to refer the current to the earth's horizontal magnetic
intensity (H) with an absolute galvanometer. The constant of the gal-
vanometer is readily found from the data of construction with the neces-
sary accuracy, and there is no doubt that in a well-equipped magnetic
observatory the method is satisfactory- But the determination of H is
no such easy matter, and its continual fluctuations must be registered by
an auxiliary instrument. Many of the results obtained in past years do
not appear to be very trustworthy, though Kohlrauseh and Wild, who Las
discussed the sources of error in an elaborate manner, are of opinion that
a high degree of accuracy is attainable. When, however, a current deter-
mination is the only object, the exclusion of this element seems to be
desirable, except for rough purposes, when a sufficiently accurate value of
H can be assigned without special experiment.
3. Of the arrangements which may be adopted for measuring the
mechanical action between a fixed and a mobile conductor conveying the
same current, the one that is best known is Weber's electio-djnamo-
meter. Two fixed coils may be arranged on Helmholtzs principle, so as
to give at the centre a very uniform field of force, in which the movable
coil is suspended bifilarly. In the equilibrium position the planes of the
coils are perpendicular, but under the influence of the current they tend
* Xctart, Feb. 1. 1*53 ; Feb. 15. 1333.
t P*9- J- Bd. COM*. & 170, 1*73.
* Ber. *er Pkyt.Xe*. Ge*. at WmrAay* MB1.
| Maxwell's Elettneity. $ 72*.
280 ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER, [112
to become parallel, and the deflection produced may be taken as a mea-
sure of the square of the current. The constant of the instrument, so
far as dependent upon the dimensions of the large coils, can be readily
determined ; the difficulty is to measure with sufficient accuracy the di-
mensions of the small coil, and to determine the force of restitution
corresponding to a given rotation. The latter element is usually obtained
indirectly from the moment of inertia of the suspended parts and from
the time of vibration. If the small coil contain a large number of turns
in several layers, its constant is very difficult to determine by direct
measurement. If, indeed, we could trust to the inextensibility of the
wire, as some experimenters have thought themselves able to do, the
mean radius could be accurately deduced from the total length of wire,
and from the number of turns; but actual trial has convinced us that
fine wire stretches very appreciably under the tension necessary for
winding a coil satisfactorily. It is possible that the difficulty might
be satisfactorily met by an electrical determination of the area of the
windings after the method given by Maxwell*, or that employed in the
present investigation.
4. In the researches of Joule and Cazin the electromagnetic action
is a simple attraction or repulsion, and can be evaluated directly by
balancing it against known weights. This method has been followed by
Mascart in his recent important work upon this subject -f. A long solenoid
is suspended vertically in the balance, and is acted upon by a flat coaxal
coil of much larger radius, whose mean plane coincides with that of the
lower extremity of the solenoid. If the solenoid is uniformly wound, it is
equivalent to a simple magnet, whose poles are condensed at the terminal
faces. The electromagnetic action then depends upon (M M ), where M
is the coefficient of mutual induction between the fixed coil and the
lowest winding of the solenoid, and M the corresponding, much smaller,
quantity for the uppermost winding.
This arrangement, though simple in conception, does not appear to us
to be the one best adapted to secure precise results. It is evident that a
large part of the solenoid is really ineffective ; those turns which lie nearly
in the plane of the flat coil being but little attracted, as well as those
which lie towards the further extremity. The result calculated from the
total length of wire (even if this could be trusted), the length of the
solenoid, and the number of turns, has an appearance of accuracy which
is illusory, unless it can be assumed that, the distribution of the wire
over the length is strictly uniform. In order to save weight, it would
appear that all the turns of the suspended coil should operate as much as
* Electricity, 754. McKichan, Phil. Trans. 1873, p. 425. See also Kohlrausch, Wied. Ann.
Ed. xvm. 1883.
t Journal de Physique, March, 1882.
112] AND OX THE ABSOLUTE ELECTROMOTIVE FORCE OF CLARK CELL&. 881
possible, that is. that the suspended coil should be compact and should be
placed in the position of maximum effect*.
5. Neglecting for the time the small corrections of the second order
rendered necessary by the sensible dimensions of the sections, let us con-
sider the attraction between two coaxal coils of mean radii A and a,
situated at distance x. If M be the coefficient of mutual induction for
the central turns, n, n', the number of windings in the two coils, i the
current which passes through both, the attraction is
(1)
In this expression i 5 is already of the dimensions of a force, and If is
linear. Accordingly dN dx, though a function of A T a, and JT, is itself
a pure number, and independent of the absolute dimensions of the
system. Its value is a question only of the ration a A, r".A. If we
write dM dx = rf(A t a, x), and consider the variation of / as a function
of the three linear quantities., the coefficients in the equation
df dA da dx
7 =x x + ^ + *ir ...... .................. < 2 '
are subject to the relation
If the coils are placed at such a distance apart that the attraction is
a maximum, r = 0, and the calculation is independent of small errors in
the value of x. Under these circumstances X+/* = 0, so that proportional
errors in A and a affect the result in the same degree and in opposite
directions. In other words, the attraction becomes practically a function
of the ratio a A only.
To this feature we attach great importance. The ratio of galvanometer
constants can be accurately determined by the purely electrical process of
BffigBfha without linear measurement of either, and from this ratio we
can pass to that of the mean radii by the introduction of certain small
corrections of the second order.
In this way all that is necessary for the absolute determination of
currents can be obtained without measurements of length, or of moments
of inertia, or even of absolute angles of deflection. The forces are. how-
ever, evaluated in gravitation measure, so that the final result requires a
knowledge of gravity at the place of observation : but except through this
quantity there is no reference to the units of space or time.
6. The final calculation of the attraction is best made with the use
of elliptic functions : but useful information, sufficient for a general idea of
die conditions and for the design of the apparatus, may be derived from the
series developed in Maxwell's Electricity, 699. If B t b be the distances
* Brit. Aaoc, Report, 1*82, p. 445 [roL n. p. 136).
282 ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER, [112
of two coaxal coils of radii A and a from a point on the axis taken as
origin, and (7 2 = A* + IP, we have
in which a, 6 are supposed to be small relatively to A, B. If we limit
ourselves to the first term, which we may do when a/A is small, we see
that so far as it depends upon the small coil the effect is proportional to
the area. The position of maximum effect for given coils is found [see
below] by making B/C 5 a maximum, which leads to B = ^A; so that to
obtain the greatest attraction the distance of the coils must be equal to
half the radius of the larger.
In the present measurements there were two equal fixed coils, one on
either side of the small coil. If we take the origin midway between, the
terms of odd order in 6 ultimately disappear in virtue of the symmetry, arid
we may write
There would be some advantage in a disposition of the coils such that
B*- f ^! 2 = 0, for then the attraction would be in a high degree independent
of the position of the suspended coil*f. In this case
(6)
* [1899. The equation for M, as well as additional terms in that for dM/db, is now inserted.]
t [1899. This was the arrangement adopted for the Board of Trade standard gauge. The
coefficient of I 2 in (5) is proportional to
_
If this vanishes, the first approximation for the ratio of B to A gives, as above, B--^A-0.
A second approximation is
w^-S)'
It is not unimportant to remark that independence of b' 1 carries with it a corresponding
112] AND OX THE ABSOLUTE ELECTROMOTIVE FORCE OF CLARK CELLS. 283
If, on the other hand, we take B* = %A*, we find from the first term
(7)
showing a not unimportant increase of effect To the second order of
approximation [see below] the distance between the fixed coils (25), cor-
responding to the maximum effect upon a small coil suspended at their
centre, is given by
so that when a* I A* is sensible the fixed coils should be somewhat closer
than when a- A 2 is negligible. For the actual apparatus used a-. A- is very
sensible, and the ideal state of things was only imperfectly approached.
The coils of the dynamometer used tor the "fixed coils" conform to the
relation B 2 =A 2 , and are not adjustable. It will be seen later that but
little is practically lost by the slight imperfection of the arrangements in
this respect.
Formula (7) is sufficient for the preliminary estimate of the attraction
to be expected, and from (5) we can form an idea of the exactitude neces-
sary in the adjustment of the suspended coil. Thus if b be not zero, the
correcting factor is, when B = ^A,
1-&2&/A* (9)
With the actual apparatus an error in 6 of one millimetre alters the
attraction by only 20 J) 00 -
[1899. It may be well to exhibit the approximate values of X, p, v
in (2). If we make 6 = 0, retaining the two first terms of (5), we see
that f may be considered to be proportional to
A z Ba- o(B--$A-)cr\
~~C*~ \ ~ ~2C* j '
in which C = A* + B*. Hence
df_ dA \'2&-3A* 5A-
f ~ A 1 P* "^ 9f~'
/ A. ( O /C
dB (A~ ^B 2 oB*o,* f
+ -B\C~* 2C^( 2 ~
da (. 5 (45= - 34) a s )
-f *2 ^ k
independence of lateral displacements. For if we consider the value of the attraction (parallel
to the axis) for a coil moved without rotation whose centre is at the point x, y, z, we recognise
that it satisfies Laplace's equation in these coordinates. It x, y, z be measured from the
central position, and the attraction be expanded in powers of these quantities, the terms of the
first order vanish by symmetry and those of the second order will be proportional to (2x* - y* - r*),
if x be the coordinate parallel to the axis. Independence of x 3 , viz. b 2 , involves accordingly
independence of y- and z 3 . The variable part of the attraction thus becomes a quantity of the
fourth order in the displacements.]
284 ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER, [112
According to the method adopted, a is not measured, but instead a/ A. If
this be called a, we have da/a = da/oi + dA/A; and
3;l 2 )a 2 ( {(LA dB\ fitf-A* 5B*a 2 (5A 2
< ) + \A~~B] { C* 2C
To secure independence of dA and dB, we have as a first approximation,
B 2 = ^A-. The second approximation is
4>B 2 -A 2 36a 2
~~&~ +
whence, as above,
If the relation be actually B = ^A, we have
7 = { 2 + 5Z" 2 |T + 25Z 2 {^~ ~B
and this agrees with values found below for the actual experiment in which
A =2-4,2 a.]
7. It may be convenient to carry through the rough theory so
as to show the dependence of the current upon the quantities actually
measured. Thus
Force of attraction = hnn'i 2 a 2 /A 2 ,
where h is written for 6?r 2 x '2862. If the ratio of the galvanometer con-
stants of the coils be /3, we have
whence
Force = h/3 2 i 2 ri s /n,
and
i = @- l h-lnlri~% (Force)* (10)
We may observe that an error in the number of windings, or, which
comes to the same thing, a defect of insulation, produces a more serious
effect in the case of the suspended than in the case of the fixed coils. The
error in the ratio of the galvanometer constants enters proportionately, but
the error in the weighings is halved.
Full details of the coils are given later. It will be sufficient here to
say that the radius of the large coils is about 25 centims., and that of the
suspended coil about 10 centims. The total number of windings on the
fixed coils is 450, and on the suspended coil 242. The current usually
employed was about ^ ampere, and the double attraction was about the
weight of one gram*.
* The actual apparatus was not adapted to the measurement of currents much exceeding
^ ampere. The flexible copper connexions of the suspended coil would take an ampere, but the
112] AND OX THE ABSOLUTE ELECTROMOTIVE FORCE OP CLARK CELLS. 285
8. The double attraction is spoken o inasmuch as the readings
were always taken by reversal of the current in the fixed coils, for which
purpose (fig. 1. E) a suitable key was provided. The difference of the
weights required to balance the suspended parts in the two cases gives
twice the force of attraction between the suspended coil and the fixed
coils, independently of the action upon the former of any other part of
the circuit, and of terrestrial or other permanent magnetism. The cur-
Fig, i.
rent was supplied from about 10 either Grove or secondary cells A, and
traversed in succession a rough tangent galvanometer D (convenient for
a preliminary test of the strength and direction of the current), two or
more silver voltameters in series C, the suspended coil G, and then (of
course, in opposite directions) the two fixed coils F. The weights neces-
sary for balance (in the same position of the key) alter somewhat, both
on account of variation in the electric current and also from the forma-
tion of air currents., due to a slight progressive warming of the suspended
coiL By recording the times of each weighing we can plot two curves
(| 24), from which we can find what would have been at any moment the
weighing in either position of the key. The difference of ordinates gives
us what we should have observed, were it possible to make both measure-
ments simultaneously. The whole duration of an experiment was from
three-quarters of an hour to two hours, measured by a chronometer, and
as a weighing could be taken about every five minutes there was ample
material for the construction of the curves. What we require for com-
parison with the deposited silver is the mean current, whereas what
we should obtain directly from the curves represents the square of the
current. The whole interval is divided into periods (usually of fifteen
minutes), and the difference of ordinates corresponding to the middle
itself is unduly heated by the passage of an ampere for more than a few minutes.
Had it been desirable to use stronger currents, it would, of course, have been possible to do so by
increasing the gauge of the wire. The grooves in which the wire is wound being given, it is
evident that a proportional increase of the current and of the section of the wire leave both
die heating and the electromagnetic effects unaltered. In this way the apparatus might easily be
modified, so as to take currents of 3 or 4 amperes, the only other change that would be required
being a multiplication of the flexible leading wires, several of which might be arranged in
parallel. But for the determination of the electro-chemical equivalent of sflver, the currents
were quite strong enough.
286 ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER, [112
of the periods is taken from the curves. The mean square root of the
numbers thus obtained gives us a result to which the rate of silver deposit
should be proportional.
9. The use of a balance for the measurement of electromagnetic
attraction involves some special arrangements. The suspended coil must
in every case be brought to rest in its proper position, corresponding to
the zero of the pointer of the balance. It was found desirable to give
the balance a shorter period of vibration than usual, and to obtain control
over the arc of vibration an auxiliary coil was introduced, through which,
with the aid of a key, the current from a Leclanche cell could be made
to pass. By this means a force tending to raise or to lower the suspended
parts could be brought into play at the will of the operator, who, after a
little practice, is able to stop the vibrations with very little delay*. The
weighings were recorded to milligrams only ; but the accuracy really ob-
tained was greater than might appear, since by anticipating somewhat the
change in progress it was possible to note the time at which the balance
demanded an integral number of milligrams.
The current was led into the suspended coil by means of fine flexible
copper wires. To diminish the force conveyed by these to the suspended
parts, they were bent so as to place themselves naturally in the required
positions before the final solderings were made. It is important, however,
to observe that no assumption is made as to the equality of these forces
before and during the passage of the current. Under its influence the
fine wires are no doubt sensibly warmed, but this effect and any conse-
quent alterations in the mechanical properties are the same in both sets
of readings, the only change relating to the direction of the current in
the fixed coils.
This point is the more important since the balance is not used in
these experiments in quite the normal manner. In ordinary weighings
there is no force in operation upon the pans but gravity, and this vertical
force is transferred to the beam. In the present application the " pan " is
not quite free and is subjected to forces which may have a small horizontal
component. In virtue of the freedom of rotation about the knife-edge
suspending the pan, these forces are transferred without change to the
beam. The horizontal component would, however, produce little effect in
any case, since in the horizontal position of the beam its direction would
pass very nearly through the knife-edge supporting the beam. The weights
in the other scale-pan give rise to a strictly vertical force. We shall thus
be doubly secured against error if we provide that the force to be mea-
sured (due to the reversal of the current in the fixed coils) is strictly
* See " Suggestions for Facilitating the Use of a Delicate Balance." Brit. Assoc. Report,
1883 [vol. ii. p. 226].
112] AXD OK THE ABSOLUTE ELECTROMOTITE FORCE OF CLARK CELLS. 287
vertical, and that the horizontal force, if sensible, remains unaltered in
passing- from one direction of the current to the other. These objects
are attained when die ceils are carefully levelled, and when the readings
are always taken for a definite position of the suspended coil conveying
a constant current.
10. The suspended coil is wound upon an ebonite ring <f 13 L and
is supported, by three screws upon a light brass triangle hanging in the
balance by a stout copper wire. The fixed coils are those of the dynamo-
meter, described in Maxwell's Electricity, \ 725. and in Larimer Clark's
paper (Pftif. Trvm*^ 1ST 4. Part It In setting up the apparatus die
ebonite coil is first suspended, and the dynamometer coils are levelled,
and adjusted laterally until concentric with it. This is tested by carrying
round a metal piece making fire contacts with the upper ring of the
dynamometer, and provided with a pointer just reaching inwards to the
circumference of the ebonite coiL The piece in question may be described
as a sort of three-legged stool, standing upon the upper horizontal face
of the dynamometer ring and carrying below two studs which are pressed
outwards into contact with the inner cylindrical face of the ricg A-
the piece is carried round the pointer describes a circle coaxal with ihe
dynamometer rings. To level the ebonite ring, the distance is calculated
by which its upper surface should be below the upper surfaoe of ihe
(upper) dynamometer ring, and a pointer attached to a straight role is
so adjusted that when tine rule is laid upon its edge along the upper
face of the dynamometer ring the pointer should just scrape the upper
face of the ebonite ring. By applying this test at three po-iiiis The
ebonite ring is brought to occupy the desired position. These aiius:-
ments were made in the first instance by our assistant, Mr G. Gordon.
and subsequently examined by ourselves. With a little care the neces-
sary accuracy is attained without difficulty, for, it is scarcely necessary to
say, all the errors due to maladjustment are of the second order. When
in use the suspended parts are protected from currents of air by a
suitable paper casing.
Examination showed that the insulation of the various parrs was satis-
factory. Twenty-five cells of a De la Rue's battery failed to show any
appreciable leakage between the wire and the rings of the dynamometer
coils, though the capacity of the co*dt<?r thus formed was very noticeable.
11. The test for leakage from winding to winding of a coil is a more
difficult matter. The ebonite ring was first wound on August 9, 1882,
and its galvanometer constant was compared with that of one coil of the
dynamometer by Mr J. M. Dodds. The result agreed very ill with the
measurements taken during the winding, and led to the suspicion that
several turns were short-circuited by a false contact. The matter was
288 ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER, [112
put to a further test in two ways. A second coil of the same dimen-
sions was wound with the same number of turns ; and the two coils
were placed co-axally close together, and so connected in series that a
current would circulate opposite ways. The circuit was completed by
a galvanometer of long period. Under these circumstances when one pole
of a very long steel magnet is thrust suddenly through the opening, there
should be no effect observable if the insulation is good ; but if any of
the turns of one of the coils are short circuited the other coil will of
course have the advantage, and the galvanometer will indicate a current
in the corresponding direction. It was found in fact that the second coil
preponderated, and that 13 extra turns had to be put upon the first coil
to obtain the balance. With proper precautions this method of testing
seems satisfactory, being approximately independent of the equality of
mean radii of the coils compared.
A second test was suggested and executed by Mr Glazebrook. The
two coils retaining a fixed position, the ratios of the self-inductions of
each to the mutual induction of the pair were determined by Maxwell's
method*. These ratios, which should have been nearly equal, were found
to differ considerably in the direction which showed a deficiency in the
self-induction of the ebonite coil.
After this it was no longer doubtful that the coil was defective. In
unwinding it more than one bad place was detected, although the original
winding had been carefully done under our own eyes. The ring was
rewound with fresh wire on Nov. 30, 1882 ; and we were so much im-
pressed with the necessity of a thorough check upon the insulation that
we devised a delicate test similar, as we afterwards found, to one that
had already been successfully used by Graham Bell-f*. Four similar coils
of fine wire, wound upon wood, and of the same mean diameter as the
ebonite coil, were arranged so as to form a Hughes induction balance.
The lower coils form a primary circuit, and are connected with a micro-
phone clock or other source of variable current. The upper coils and
associated telephone form a secondary current. The distance between the
upper and lower coils is such as to allow the insertion of the ebonite coil
between them, suitable support being provided for it to guard against
displacement of the principal coils. If the distances of the four coils
are adjusted by screw-motions to an exact balance, so that no sound is
audible in the telephone (held at some distance away), the introduction
of a tertiary circuit between one primary and secondary causes a revival
of sound whose intensity depends upon the conductivity, &c., of the
* Electricity and Magnetism, 756.
+ "Upon the Electrical Experiments to determine the Location of the Bullet in the Body of
the late President Garfield," &c. A paper read before the American Association for the Advance-
ment of Science, August, 1882.
112] AXD OS THE ABSOLUTE ELBCTBOMOniTE TORCH OF CLARK CELtS. 289
tertiary circuit. If the tertiary circuit consists of a single turn of wire,
such as that on the ebonite ring, the sound heard is quite loud, and
remains audible when a resistance of about 1 ohm is included. A single
circlet of copper wire ~OO4 inch diameter gives a very distinct sound.
When the ebonite coil, with ends unconnected, is introduced,, the sound
is audible, but much leas than that from the fine copper cirdet. Fart
of this effect may be attributed to its finite capacity as a condenser, in
virtue of which SWUM! might be heard in any case; but it is probable
that the insulation is in reality somewhat imperfect. The dosing of the
circuit through a megohm gives a distinct augmentation of sound ; and thus
it is evident that the insulation, if not perfect, is at any rate abundantly
sufficient for the purposes of the present investigation.
The current weighing apparatus was set up in February, 1883, and
worked satisfactorily from the first. Apart from errors in the ciMDtsfciM
of the instrument, the determination of the mean value of a cmrracit <cf
(say) half an hour's duration should easily be correct to
Hue jjLred coil*.
| 12. These are the coils of the dynamometer eeaisiraeted! by it foe-
Electrical Committee of the British Association (see 10). Hue mean
radii of the two coils and the- dimensions of the sectors are very E-tar! T
identical, and for our purpose it is unnecessary to note anything bet nibe
mean. The following are derived from the. dimensions recorded m Pr<>fessor
Maxwells handwriting in the laboratory note-book :
A = mean radius = 2*81016
2 = distance of mean planes = 2-51HQ
2& = radial dimension of section = 1-29
2* = axial = 150
tiie unit in each case being the centimetre.
The number of turns of wire on each coil is 225.
The above values are those employed in the calculations of the present
investigation, and they can be only partially verified without unwinding
the wire. Owing, however, to the final result being comparatively inde-
pendent of A and B, even a rough verification is not without value. The
distance parallel to the axis from outside to outside of the grooves in which
the wire is wound can be found pretty accurately with callipers, and was
determined to be 10-433 inches, From inside to inside of the grooves
the corresponding distance was 9-252 inches. The mean of these is the
distance of mean planes, which is thus &S425 inches, or 25*000 centims.
exactly. This element is, therefore, verified with abundant accuracy. The
19
290
ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER,
[112
half difference of the two numbers above given represents the axial
dimension of the section, and comes out 1'5024 centims., practically iden-
tical with 1'50 ceutims. The mean radius and the radial dimension of
the section are not now accessible to measurement, but the outside cir-
cumference agrees sufficiently well with that calculated from the recorded
dimensions to serve as a verification.
The number of turns has to be taken entirely upon trust ; but the
use of the method given in Maxwell's Electricity, 708, makes a mistake
in this respect very unlikely. Moreover, the electrical comparisons to be
detailed later ( 14) verify the equality of the number of windings on the
two coils.
The resistance of each coil is about 14^ B.A. units, and both coils are
well insulated from the frame on which they are wound.
The suspended coil.
13. This consisted of 242 turns of copper wire insulated with silk
saturated with paraffine wax, and was wound upon an ebonite ring supplied
by Messrs Elliotts. The weight of the ring was 135 grms., and its section
is shown full size in the adjoining figure (fig. 2). The weight of the wire
WM
V/////////////A
Fi
was 440 grms., so that the total weight to be carried in the balance was
about 575 grms. The mean diameter of the coil of wire, as determined
from the inside and outside circumferences, was 8*090 inches ; but it cannot
be so determined with sufficient accuracy, and the result is not used in the
calculation. It agrees perhaps about as well as could be expected with that
deduced electrically by comparison with the large coil.
The radial dimension of the section (2/t') = '9690 centim.
The axial (2k') = 1-3843 centims.
The difficulties experienced in respect of the insulation, and the tests
applied, have already been related ( 11).
112] AND ON THE ABSOLUTE ELECTROMOTIVE FORCE OF CLARK CELLS. 291
The electrical comparison of radii ( 14) gave for the ratio of the
dynamometer radius A to that of the suspended coil a
2-42113,
whence
a = 10-2473 centims.
The mean radius thus determined is not necessarily that corresponding
to the geometrical centre of the section, as it allows for any inequality in
the distribution of the windings.
The resistance of the coil is about 10 ohms.
Determination of mean radius of suspended coil.
14. This quantity cannot be determined advantageously by direct
measurement, but its ratio to that of the large coils can be deduced
from the ratio of the galvanometer-constants of the coils, and this ratio
can be accurately determined by the electrical method introduced by
Bosscha*
It may be shown^ that for all purposes we may take the mean radius
and mean plane of a coil to correspond with the circle passing through
the centre of density of the windings. If the windings are distributed
with absolute uniformity, this point coincides with the geometrical centre
of the section ; otherwise there may be an appreciable distinction. The
corrections of the second order, which in consequence of the fmiteuess of
the section must be introduced in calculating the effects of the coil, have
the same values as if the density of the windings were absolutely, instead
of merely approximately, uniform.
For example, the galvanometer-constant G^ is related to the mean
radius A (as above defined) and to the radial and axial dimensions of the
section, 2h, 2k, according to J
If, therefore, we can determine for two coils the ratio of galvanometer
constants, it is a simple matter to infer therefrom the ratio of mean radii.
In Bosscha's method the two coils to be compared are arranged approxi-
mately in the plane of the magnetic meridian, so that their axes and mean
planes coincide, and a very small magnet with attached mirror is delicately
suspended at the common centre. If the current from a battery be divided
between the coils, connected in such a manner that the magnetic effects
* Fogg. Ann. xcra. p. 402, 1854.
t Camb. Phil. Proc. Feb. 12, 1883 [vol. n. p. 184].
t See Maxwell's Electricity, 700.
192
292 ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER, [112
are opposed, it is possible by adding resistances to one or other of the
branches in multiple arc to annul the magnetic force at the centre, so
that the same reading is obtained whichever way the battery current may
circulate. The ratio of the galvanometer constants is then simply the
ratio of the resistances in multiple arc.
To obtain this ratio in an accurate manner, the two branches already
spoken of are combined with two standard resistances so as to form a
Wheatstone's balance. Of these resistances both must be accurately
known, and one at least must be adjustable. The electromagnetic balance
is first secured by variation of the resistance associated with one of the
given coils, which resistance does not require to be known. During this
operation the galvanometer of the Wheatstone's bridge is short-circuited.
Afterwards the galvanometer is brought into action, and the resistance
balance is adjusted. The ratio of the galvanometer constants is thus
equal to the ratio of the known resistances. The two adjustments may
be so rapidly alternated as to eliminate any error due to changes of
temperature in the copper wires.
The above comparison was carried out for each of the two coils of
the dynamometer, and the coil wound on the ebonite ring, called for
shortness the ebonite coil. On account of the smallness of the latter
some care is necessary in the adjustments, which, however, do not require
to be described in detail. It will be sufficient to refer to the description
of the adjustments when the ebonite coil was suspended, and to mention
that the errors arising from maladjustment (all of course of the second
order) could hardly affect the final ratio by more than 10> o 00 . The length
of the magnet was ^ inch, and the error due to neglecting it could not
exceed 10i ^ 00 . To the magnet was attached a light silvered glass mirror,
such as are employed in Thomson's galvanometers, and it was protected
from air currents by a glass cell. The readings were taken by observing
the motion of a spot of light thrown upon a scale in the usual way.
The electrical connexions are shown in the adjoining figure (fig. 3).
The current from a large Daniell cell A, after passing the reversing
key B, divides itself at C between the brass coil of the dynamometer D
and the ebonite coil E. The remaining terminals of these coils are led
into mercury cups F and H, into which also dip the terminals of the
bridge galvanometer g. With the ebonite coil is associated a resistance
box N. The other branches of the balance were (in one arrangement)
composed of a coil of 10 units in multiple arc with which was placed a
high resistance box K, and three coils combined in series whose values
were about 24, 1, 1 units, making together 26. All these . coils were
of the standard pattern, and their values had been already carefully
determined. From the cup L the current passed back to the key B.
112] ASD OX THE ABSOLUTE ELECTROMOTIVE FORCE OF CLARK CELLS. 293
The high resistance box K gives a fine adjustment by which the ratio
of resistances can be brought to the required value. The smallest re-
sistance actually used here was 4000 units. While the electromagnetic
Fig. 3.
balance was under observation a horse-shoe piece of stout copper rod P,
connected with the key as shown in the figure, was inserted in the
cups F, H. By this means these cups are brought accurately to the
same potential, and nearly all the current is diverted from the standard
resistance coils.
The determination of the electromagnetic balance is rendered more
troublesome by the fact that the first motion of the magnet on the
reversal of the current is influenced by induction, and cannot be used as
a test. No attempt was made actually to complete the adjustment, but
by preliminary trials resistances from N differing by about J- unit were
found, such that the effects observed were reversed in passing from one
to the other. From the magnitude of these effects the required result is
obtained by interpolation. At the beginning and end of a series the two
ratios of resistances were determined by use of K, the horse-shoe P being
of course withdrawn : and the mean of the initial and final values (which
usually differed extremely little) was employed in the reduction,
As an example, we may take some observations on Sept. 5, 1883, with
the coil of the dynamometer marked B. The difference of readings on re-
versal of the battery in a given manner was taken alternately with certain
resistances from N, which we may call a and 6. The results were
with a +7, + -3, +1-3, +1-0, mean + '8;
with b - 8-4, - 8-4, - 85, - 9'5, mean - 8'7.
Now with a the resistance from K, associated with the [10], and uecessary
for the resistance balance, had to be such that (at a standard temperature)
294 ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER, [112
the resultant resistance of this branch was 9'97772 ; while with b the re-
sultant resistance had to be 9'99182. The resistance that would have been
required here, if N had been accurately adjusted for the electromagnetic
balance, is thus
9-97772 + ^ x -01410 = 9*97890.
9'5
The resistance in the other branch was 25'95648, so that the ratio of
galvanometer constants is determined to be
25-95648/9-97890 = 2-60113.
It will be seen that even with a single cell the sensitiveness was such that
the errors of reading could scarcely exceed 7^^ ; indeed, the weakest part
of the arrangement is in the standard resistances.
With use of the above resistance coils the values obtained for coil B
on three occasions were
2-60087, 2-60098, 2-60113, mean 2-60099.
As a further check, the experiment was repeated with a different com-
bination of resistance coils. The 26 was replaced by 13, made up of three
singles and of the same [10], while the [10] was replaced by a [5]. Two
experiments gave
2-60046, 2-60026, mean 2-60036.
The mean result of the two arrangements is thus 2'60067. The difference
is about 475770, and would be explained by an error of ^^ in the value of
the [10]*.
For coil A of the dynamometer the ratio of galvanometer constants was
found in like manner to be 2-60072, the close agreement of which with
2'60067 is a verification of the winding and insulation of the coils. For
the further calculations we require only the mean, and we therefore take
as the ratio of galvanometer constants for the ebonite coil and a coil of
the dynamometer
2-60070.
The accuracy obtained in the above determinations is doubtless quite
sufficient for the purposes of the present investigation, but if it were
desired to push the power of the method to its limit it would be neces-
sary to design the coils so that the ratio should be (approximately)
expressible by very simple numbers. If in the present case, for example,
we were content to sacrifice one-fifth of the number of turns on the
ebonite coil, the ratio could be made to approach that of 2:1. The
* For the methods used to find the values of the [24], &c., reference must be made to former
papers.
112] AND OX THE ABSOLUTE ELECTROMOTIVE FORCE OF CLARK CELLS. 295
standard resistances might then be composed of three equal resistance
coils, which could be more accurately combined and tested than the more
complicated combinations that we were obliged to use. In such a case
the limit of accuracy could probably depend upon the difficulty of ad-
justing the coils under comparison and the suspended magnet to their
proper places. It is scarcely necessary to say that care must be exercised
in the disposition of the leading wires, and that the direct action of the
current in the principal coils upon the needle of the bridge galvanometer
must be tested, and, if necessary, allowed for.
We hare now to deduce the ratio of mean radii For the ebonite coil
the correcting factor is
1 + A"Va s - 4**/a*= 1 + 000741 - -002269.
For the dnamometer coil
1 + J A*/^ 1 * - W A * = ! + -000225 - -000457.
Thus
A/a = iff x 2^0070 x 1-001296 = 2-42113 :
and from this when A is known the value of a can be deduced ( 13").
Calculation of attraction.
15. The attraction between two coaxal circular currents of strength
unity, of which the radii are A, a, and distance of planes is B, is ( Maxwell,
701)
where F y and E y denote the complete elliptic integrals of the first and
second kind whose modulus is sin 7. The value of sin 7 itself is
The functions F y and E r were tabulated by Legendre. In an Appendix
[p. 327] will be found a table of
sin7l2F 7 -(l+sec--7) 7 ], ........................ (3)
calculated with seven-6gure logarithms from those of Legendre for the
purpose of the present and similar investigations. It has been carefully
checked, and it is hoped is free from error, except of course in the last
The value of (1), with omission of the factor -, is denoted by f(A, a, B\
and, as has already been explained, it is a function of no dimensions. To
calculate it for the central windings of the fixed and suspended coils, we
296 ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER, [112
have first to find 7 from (2). With the data already given 7 = 58 574',
whence with use of the table
/ (A, a, B) = 1-044576.
This multiplied by IT, by the product of the numbers of terms in the
two coils, and by the square of the strength of the current, gives very
nearly the force of attraction, but a correction is required for the finite
dimensions of the sections. The quadruple integration over the two areas
may be effected by suitably combining various values of f corresponding
to the central turn of one section and to the middle of one of the linear
boundaries of the other. (See Maxwell's Electricity, 2nd edition, 706,
Appendix II.) We find
f(A+h, a, B) = -992719)
f(A- h, a, B) = 1-098740} su
f(A, a + h', B) = 1-158576)
f(A,a-h',B) = -937866J S
f(A, a, B + k) =1-0246 1 2)
/(A, a,B-k) -1059526} sum 2 ' 084138
f(A, a, B+k') =1-026306}
The sum of the eight values is 8'356914. From this we subtract
2xf(A, a, B), viz., 2*089152, and divide by 6; whence for the mean
value of / applicable to the sections as a whole
/= 1-044627,
differing, as it turns out, extremely little from f (A, a, B).
From the values given we see that f increases very sensibly as B
diminishes, so that, as was expected, the distance between the fixed and
the suspended coil, or between the two fixed coils, is too great to realise
fully the advantageous condition of things described as the ideal, in
which f would be approximately independent of variations in B.
To express the actual variations of f as a function of A, a, B, we
write
df_ d,A da dB
and we obtain sufficiently accurate values of \, p,, v from those of / already
given. Thus
f(A+h,a,B)-f(A-h,a,B) 2h ,
f(A,a,B) -*A =
In like manner /z = + 2'23, v = - '28 ; so that
~' 28 ir
112] AND ON THE ABSOLUTE ELECTROMOTIVE FORCE OF CLARK CELLS. 297
In the present investigation, however, a is not measured directly, but
by comparison with A. If we write a/A=a, so that
da _ da _ cL4
a a A
and eliminate da/a, we have
*-MI* + *^-*.
/ a A B
which is the equation by which the suitability of the proportions is to be
judged. It will be seen that the stress is thrown upon the measurement
of or, and that the errors of A and B enter to the extent of only about
one quarter. If the proportions had been those described as ideal, the
coefficients of dA/A and dB/B would have been zero.
It must not be forgotten that the error of f itself is halved in the
final result, which thus involves the errors of A and B only after division
by 8.
If the current be t, and the number of turns in the fixed and suspended
coils, n, ri, the attraction or repulsion is measured by
irnri&f.
This is expressed in absolute units. To find the value in gravitation units
we must divide by g. If m be the observed difference of weights in air
necessary to counterpoise the suspended coil when the current is reversed
in the fixed coils,
7rnn'i 2 f= %mg x "99986,
the last factor representing the " correction to vacuum " rendered necessary
by the finite density of the brass weights.
The value of g at Cambridge is taken to be 981-2282. Introducing
this and the numerical values of n, n',f, already given, we find
where i = '0370484.
The silver voltameters.
16. The arrangement adopted for the voltameters is similar to that
recommended originally by Poggendorff. The deposits are formed upon
metallic basins (usually of platinum) charged with a neutral 15 per cent.
solution of pure silver nitrate. They are prepared by careful cleaning
with nitric acid and distilled water with subsequent ignition. After com-
plete cooling in a desiccator, they are weighed to ^ millignn. in a delicate
balance with trustworthy weights. The anode, by which the current enters
the voltameter, is formed of fine silver sheet, suspended by platinum wire
in a horizontal position near the top of the solution. In order to protect
298 ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER, [112
the cathode from disintegrated silver, which in our experience is invariably
formed upon the anode, the latter is wrapped round with pure filter paper,
secured at the back with a little sealing-wax. This arrangement appears
to us for several reasons preferable to the vertical suspension of the elec-
trodes in the form of flat plates. In the latter arrangement the deposited
metal usually aggregates itself upon the edges and corners of the kathode
with a tendency to looseness. Again the solution rapidly loses its uni-
formity. At the kathode the solution becomes impoverished and at the
anode it becomes concentrated. With vertical plates the strong solution
soon collects itself at the bottom, and the weak solution at the top, so
as to give rise to considerable variation of density. It is true that the
horizontal position of the electrodes necessitates the use of a porous
wrapping, which would increase the difficulty of determining the loss of
weight at the anode. M. Mascart appears to have succeeded in deter-
mining this loss, but the disintegration which we have always met with
rendered the attempt on our parts hopeless. It is possible that something
may depend upon the mechanical condition of the metal, but as to this
we cannot speak with confidence. The blackish powder left upon the
anode has at first the appearance of being due to chemical impurity, but
it occurs with anodes of the highest quality of silver, and is completely
soluble in nitric acid.
In our earlier trials, dating from October, 1882, we were much impressed
with the importance of obtaining sufficient coherence in the deposit to
guard against risk of loss in the washing and subsequent manipulations.
The addition of a very small proportion of acetate of silver was found to
be in this respect a great improvement, affording a deposit less crystalline
in appearance and of much closer texture ; and in consequence nearly
all our experiments during the first year were conducted with solutions
containing sensible quantities of acetate. In order to detect whether any-
thing depended upon the " density " of the current, two platinum basins
of different sizes were employed, the area of deposit being in about the
proportion of 2:1, but no distinct systematic difference was observed.
When the deposits were completed the basins were rinsed several times
with distilled water, and then allowed to soak over night. The next day
after more rinsings they were dried in a hot air closet at about 160 C.,
and after standing over another night in a desiccator were carefully
weighed. Repetition of these weighings after intervals of standing in
the desiccators showed that they were correct to ^ milligrm., so that as
the total weights of deposit amounted to 2 or 3 grms., a high degree of
accuracy in the final evaluation of the ratio of deposit to current was
expected. Discrepancies, however, presented themselves of an amount
much greater than we had been prepared for, and they were of such a
character as to show that the disturbing causes were to be sought in the
112] AND ON THE ABSOLUTE ELECTROMOTIVE FORCE OF CLARK CELLS. 299
behaviour of the voltameters ' and not in the current weighing apparatus.
Thus it was found that the numbers obtained on the same occasion from
the two voltameters in series, through which exactly the same quantity of
electricity had passed, were liable to as great a disagreement as the numbers
derived from experiments on different days.
17. At this stage the question presented itself as to whether the
deposits were really pure silver. Two or three gravimetric analyses by
conversion into chloride, conducted both by ourselves and by Mr Scott,
to whose advice and assistance we have been constantly indebted through-
out these investigations, having favoured the idea that the deposits were
not quite pure, we arranged for a systematic volumetric analysis of all
the deposits. The bulk of the metal after solution in pure nitric acid
having been thrown down with a known quantity of chloride of sodium
in strong solution, the titration was completed with weak (y^y) salt solu-
tion from a burette in the usual manner. The bottle containing the
solution was enclosed in a dark box and lighted in the manner recom-
mended by Stas, with a convergent beam of yellow light which bad
passed through a flask containing chromate of potash. Towards the close
of the operation the effect of the addition of two drops of solution (con-
taining Jjj milligrm. of salt) becomes difficult of observation unless the
liquid be very thoroughly cleared. At this stage we found it convenient
to filter off about half the liquid into another bottle, through a funnel
plugged with (purified) cotton wool. As soon as the pores are penetrated
by the chloride of silver the nitration is effective, and yet so rapid that
but little time is lost by the adoption of this procedure. The two drops
of chloride solution are added to the liquid thus filtered, and shaken up
so as to effect a complete mixture, and the bottle is then placed so that
the cone of light traverses the body of the liquid. After an interval
varying from a few seconds to several minutes the cloudiness develops
itself, and the delay gives an indication of how nearly the point is ap-
proached. Before each test the filtrate is of course returned to the stock
bottle and thoroughly shaken up. The operation is complete when the
last addition of two drops gives no effect after a quarter of an hour.
There is no difficulty in determining in this way the necessary quantity
of salt to Jj milligrm., and the point may be recovered any number of
times after addition of small known quantities of silver.
In the interpretation of the results we placed no trust in the purity
of the NaCl, nor depended upon any assumption as to the ratio of NaCl
to Ag, but made comparison with the numbers obtained from precisely
similar determinations with substitution for the electro-deposits of equal
weights of silver of the highest quality, supplied by Messrs Johnson and
Matthey. A large number of such comparisons showed that there was
300 ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER, [112
no difference that could be depended upon between the two kinds of
silver; there was, indeed, a slight indication of inferiority in the deposits,
to the extent of perhaps ^Vo> but not more than might plausibly be
attributed to the greater risk of loss in dissolving the deposits from off
the platinum basins. The standard silver was dissolved without trans-
ference in the bottle used for the subsequent analysis, and thus under
more advantageous conditions than were possible in the manipulation of
the deposits.
18. Table I. [p. 308] gives the results of a laborious series of
determinations made with solutions containing more or less acetate. It
will be seen that up to August 16 the numbers in the final column are
fairly concordant, and they rather narrowly escaped being accepted as
satisfactory. In the month of November, however, the experiments were
continued with a fresh stock of depositing solution (probably containing
less acetate), when a systematic change became apparent in the direction
of smaller deposits. From the first we had taken, as we thought, full
precautions to secure adequate washing out of the silver salt, and special
experiments had proved that the weights were not appreciably changed
by further washing with pure water, or by resoaking in the depositing
solution with a second washing and drying conducted like the first.
Nevertheless the appearance of the deposits under the microscope was
such as to suggest a doubt whether a complete elimination of the salt
from its pores was possible with any amount of washing, and the evidence
of the analyses was felt not to be decisive, inasmuch as the deficiency to
be found in this way would correspond to only about one-third of the
weight of salt actually present. According to this view the diminution
in the weight of the deposits after August 16 was due to a more com-
plete washing out of the salt, rendered possible by the more open texture
of the deposits, and we proceeded to test the behaviour of pure nitrate
solutions. The result was a further small, but distinct, diminution in the
weights, as shown in Table II., and we were now convinced that the use
of acetate had been a great mistake, costing us six months' almost fruitless
labour. When the deposits are taken upon the concave surface of a bowl,
they are coherent enough for convenient manipulation without the aid of
acetate. The danger of the retention of salt or other impurity is far
greater than of loss of metal, and this danger is aggravated by the
acetate. Indeed it would be scarcely too much to say that the danger
is converted into a certainty, for from the fine pores of these deposits it
seems almost impossible to remove the salt effectually.
It is evident that, in spite of the retention of a small quantity of salt,
a satisfactory conclusion might be reached were there any means of esti-
mating its amount. Theoretically the analysis for silver, as many times
112] AXD OX THE ABSOLUTE ELECTROMOTIVE FOBCE OF CLAKK CELLS. 301
effected, is adequate to this purpose, since the difference of the total
weight of the (impure) deposit, and of the metal found on analysis., would
represent the XO, of the salt. But the circumstances are so disadvan-
tageous that no satisfactory result could he looked for without an extra-
ordinary, and perhaps impossible, perfection of manipulation. A direct test
for nitric acid is not applicable: but at a sufficiently high temperature
the silver nitrate would be decomposed, so that the loss of weight incurred
on heating to redness (after previous thorough drying at, say, 160" C.)
would represent the NO S . Unfortunately this method is difficult to carry
out thoroughly without injury to the platinum basins, inasmuch as silver
and platinum begin to alloy at a red heat. But an exposure for five
minutes to a heat just short of redness does not seriously damage the
basins, and appears to be nearly, if not quite, sufficient to drive off the
last traces of NO,. With a pure nitrate depositing solution, and with
the treatment for elimination of the salt presently to be described, there
was sometimes no loss on heating (Table IL), but perhaps more often
the balance indicated a loss of one or two-tenths of a milligram. With
respect to the interpretation of this, it is difficult to say whether or not
it ought to be regarded as due to traces of salt retained in spite of
all the washings. If so, the true weight of deposit is smaller still by
nearly twice the apparent loss ; but it is very possible that there may
be traces of grease liable to be burnt off at a red heat, so that the loss
in question cannot with confidence be attributed to nitrate. On this
account the real amount, of the deposit remains somewhat uncertain to
nearly half a milligram.
With respect to the procedure best adapted to eliminate the salt from
the pores of the deposit, it is evident that the difficulty is to cause any
displacement of the liquid in the interior. It was thought that this object
might to some extent be attained by rapid alternations of temperature, and
for this purpose the basins were (after thorough rinsing) passed backwards
and forwards between cold and boiling distilled water. Recourse was had
also to soaking in alcohol,, somewhat diluted. Still wet with the alcohol,
the basins were plunged into boiling water with the idea of promoting
disturbance inside the cavities of the deposit. After a course of treatment
of this kind the basins were filled and allowed to stand over night so as
to give free play to diffusion. They were then rinsed a few times, and
placed in the air closet to be dried at 160 : C.
19. In order to meet the difficulty of the alloying of silver and
platinum at a temperature high enough to decompose with certainty the
last traces of silver nitrate, we made, at the suggestion of Professor Dewar,
several attempts to replace platinum by silver bowls. One evident objec-
tion to the silver is the impossibility of removing the deposit with nitric
302 ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER, [112
acid, so as to restore the original condition of the bowl. But a more
serious difficulty arises from the want of constancy in the weight of a
silver bowl (without deposit) when strongly heated. On more than one
occasion a gain of a milligram or two was observed after heating in a
porcelain basin over an alcohol flame. We have reason to believe that
this effect depends upon the presence of traces of copper. In order to
test the question we carefully cleaned and dried at 160 a piece of the
highest quality of silver, such as was used latterly for the anodes. The
weight was now 281628, and after heating to redness for a quarter of
an hour over a naked alcohol flame fell to 28'1619. On another occa-
sion a loss of 2 milligrms. was observed under similar circumstances. On
the other hand, a parallel experiment with a less pure sample of silver,
known to contain a small quantity of copper, gave after the first heating
to redness a gain of 3 milligrms., followed by a further gain of 2 milligrms.
after a second heating.
These changes are, however, insignificant compared to that observed
by Mr Scott, who heated one of our large silver basins in a porcelain
bowl for a long time over a Bunsen gas-flame. After two nights' treat-
ment the weight had risen from 57'3008 to 57'4521. Mr Scott traced
the increase in his case to the formation of silver sulphate, but it does
not appear possible that this can be the explanation of the changes
observed by us. The matter appears worthy of the further attention of
chemists ; but for our purposes the conclusion is that, for the present
at any rate, platinum is preferable to silver. With suitable precautions,
the platinum basins may be heated to redness without changing more
than Jg- milligrm.
20. In some of our later experiments (e.g., those on January 30,
April 2) we included a voltameter, charged with a higher proportion of
acetate, in order to exaggerate the errors that we had met with, in the
hope of better detecting their origin. When the nitrate solution is nearly
saturated with acetate, the deposit is of a beautiful snow-white appearance,
and almost always 5 or 7 milligrms. too heavy. On the second weighing,
after heating to the verge of redness, a loss revealed itself, whose amount
usually agreed fairly well with the view that the original excess of weight
was due to nitrate, reduced to metal by the second heating.
21. In the hope of obtaining better evidence as to the cause of the
anomalous weights, and also with the view of confirming our results by
the substitution for nitrate of some other salt of silver, we have made
several observations on deposits from chlorate of silver. The salt was
prepared for us by Mr Scott from chlorate of barium, and was found
to give as good deposits as the nitrate. The chlorate was used in a
112] AND ON THE ABSOLUTE ELECTROMOTIVE FORCE OF CLARK CELLS. 303
nearly saturated 10 per cent, solution*, and also in a 5 per cent, solu-
tion. Voltameters charged with nitrate were included in the same circuit,
so that the comparison was made under the most favourable conditions.
The results (Table II.) show an exceedingly good agreement, and con-
stitute perhaps the most accurate verification which Faraday's law of
electrolysis has as yet received.
But the second object which we had in view in using the chlorate has
not been attained. The idea was to get a too heavy deposit by addition
of acetate, and then after washing and weighing as usual, to dissolve up
the metal with nitric acid and test for chlorine. If chlorate were present,
and were the cause of the excessive weight, it should on strong heating
be resolved into chloride, whose presence might be detected. Preliminary
experiments showed that as little as ^ milligrm. of silver chloride could
be rendered evident. The deposits were dissolved in nitric acid, and
strongly supersaturated with pure ammonia. After standing for some
time with frequent stirring, the solution was diluted, and again rendered
acid with nitric acid. The deposits from chlorate, which we had reason
to regard as pure, stood the test almost perfectly, the amount of chloride
of silver present being less than ^ milligrm. If one drop of the dilute
NaCl (^ milligrm.) were added to the solution in its alkaline condition,
the cloud formed on acidification was perfectly evident after a minute or
two when examined in Stas' box. When a piece of fused silver chloride
weighing 3 milligrms. was added to the alkaline solution, it dissolved
after about half an hour, and gave a dense milkiness on addition of
nitric acid.
The application of this method to deposits from chlorate and acetate,
which the balance showed to be several milligrams too heavy, has given
the unlooked-for result that no corresponding quantity of chloride was
present. Something more than a mere trace was indeed detected, but of
amount probably not exceeding \ milligrm. The deposit from chlorate
and acetate of April 2, and another which does not appear in the table
as the current weighings were not taken successfully, in which the excess
was about 7 milligrms. were both treated in this way with similar results.
The loss of weight on strong heating appears to exclude the supposition
that though chlorate was present it escaped decomposition, and thus we
seem almost driven to the conclusion that the redundant matter is prin-
cipally acetate, although the proportion of acetate to chlorate in the
solution is a small one.
22. We have had occasion to examine another point relating to the
chemistry of electrolysis, of which the result may here be recorded. In
* The tendency to crystallise upon the anode is an objection to the nse of the strong solutions,
and probably makes itself the more felt in consequence of the paper wrapping, which impedes
the free circulation of the liquid.
304 ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER, [112
our earlier experiments we used anodes containing an appreciable quantity
of copper. The copper evidently tended to accumulate in the solution,
becoming after a time apparent by its colour even when neutral ; on
addition of ammonia a distinct blue was struck. We were desirous of
ascertaining whether under these circumstances there is danger of the
deposits becoming contaminated. A distinctly blue solution was prepared,
in which the proportion of copper to silver was considerable, and a de-
posit made. The texture was very much modified by the action of the
copper, and the appearance was such that it was difficult to believe that
the weight could be more than a small fraction of that of the simul-
taneous deposit from a pure silver solution. Some of the metal, which
adhered very loosely, was lost in the washing, but the weights agreed to
within a few milligrams. On dissolution in nitric acid and supersatura-
tion with ammonia the solution showed no trace of colour, although about
10< ooo f c PP er can thus be detected.
23. In the absolute measurements the determination of the interval
(never less than three-quarters of an hour) between the first passage of
the current through the voltameters and its final cessation could readily
be effected with sufficient accuracy (probably to 7^000)' but a slight cor ~
rection is called for in order to take account of the loss of time incurred
at each operation of the reversing key which controlled the direction of
the current in the fixed coils ( 8). To obtain the necessary data for this
correction the main current was led through a few turns of wire surrounding
a reflecting galvanometer. The resulting deflection is independent of the
position of the key, but at the moment of reversal the current is inter-
rupted, and the spot of light falls back towards zero. From a comparison
of the amount of this falling back with that of the steady deflection, in
conjunction with observations of the period of vibration, it is easy to
deduce the time of interruption. It proved to be less than T ^ second, and
was so nearly constant that after sufficient experience had been gained
further observations were judged to be unnecessary. The connexions for
this purpose are accordingly not shown in the diagram (fig. 1).
24. In order more fully to explain the procedure in taking a
deposit it will be advisable to give the details of one experiment. Thus
on March 10, 1884, the current, roughly regulated to the desired value
with the aid of the tangent galvanometer, was allowed to pass through
the coils of the current-weighing apparatus for about half an hour. The
electromotive force of the storage cells (when in good order) remains
almost perfectly constant during an experiment, but the gradual warming
of the copper conductors causes a slight falling off of current. On the
present occasion the preparatory current was a little stronger than that
ultimately used, so as to produce a slight overheating. During this time
112] AXD XS THE ABSOLUTE
OFCUU
tiie three platinum voltameters, previouslj cleaned and weighed,
charged with solution of silver nitrate: and the pure silver
wrapped in filter paper, were adjusted to their places at the top of the
liquid. As will be seen from Table H, two of the bowk were charged
with solution of normal strength (15 per een1L|, and the other with solu-
tion of double tins strength. When all was ready., the current, previously
running along a shunt, was caused to pass through the voltameters at
4* 17* by the chronometer. The weights required to bring the pointer
of the current-weighing balance to zero, with the
are given in Table HL In the second column the
TABLE TIT
4. m. *.
Ik. Tl. t.
i 1; ... -"I'.H:
* as : -..:
4 m 15 - -
* , a: **n
k ,. :: --v.-'.-
i :: :, *-3!
4 53 It '
4 3 39 ft'Tr
'.'.: -T?9
that at the moment in question the weight required to brnkm^
pended ca r as acted upon ekettromagnetiea% 9 was 71SS1 gpms.,
577-694 grmK, but the 570 grms. being never moved need n
In this position of the reversing key the eledtromagnesie
the apparent weight of the suspended cml Tne 4t5n^ir sru <otf
which the magnetic force tended to lift the ooil,, are givem ran nb
At 5* 2 the circuit was interrupted.
From the numbers above given two curves are eonsftrmdfced
what would have been observed in eMner positti'oa
n%. 4>L
rf the
306
ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER,
[112
key during the whole course of the experiment. To effect the integra-
tion of the current, the whole time, 45 m , is divided into nine periods of
5 m each, and the magnitude of the current at the middle of each period
is taken to represent its value throughout the period. A more elaborate
evaluation could easily have been applied, but was superfluous. The
difference of ordinates at the middles of the periods gives the difference
of weights in the second column of Table IV., and the mean of the square
roots of these differences, viz. '95171, is the square root of the difference
of weights corresponding to the mean current.
TABLE IV.
Time
Difference
of
weight
Square root
of
Difference
h. m. s.
4 19 30
897
9471
4 24 30
900
9487
4 29 30
904
9508
4 34 30
906
9518
4 39 30
908
9529
4 44 30
908
9529
4 49 30
909
9534
4 54 30
910
9539
4 59 30
910
9539
Mean
95171
The whole time of deposit was 2700 seconds, but from this a deduction
has to be made for the time lost in operating the reversing key. The
loss of time at each operation was found (by a process already described)
to be '083 second. Thus the actual time of passage of the current through
the voltameters is to be taken at
2700 - 7 x -083 = 2699'4 seconds.
After the deposits had been formed they were washed in the manner
already described with alcohol and hot and cold water, soaked over night,
then rinsed and set to dry at 160 C. In the first row of Table V. will
be found the weights of the bowls without deposits; in the second the
weights after the deposits had been dried at 160 C. ; in the third the
differences representing the weights of the deposits; in the fourth the
weights of the bowls after heating for about five minutes nearly to red-
ness over an alcohol flame ; and in the fifth the weights of the deposits
as determined from the previous row.
112] A5TD OST TEE ABSOOTTE EUBCTBQ*M<mVE FOfiCE Of CLARK CELLS. 307
TABLE T. Deposits of March 10, 1884.
xv. :
m-*m
m.-sa w-vsas ^i
GadbB UK*? iiBft3 :-..-.,-
i:": :-.-:ir -:!,M- 1^1 M^HT
fiaiia . ^ i .._ '
T> obtain numbers wfakh, thoogh of no absolute significance, allow of
idle comparison of experiments made on different occasions, we may divide
-95171 (the square root of the difference of the current weighing?) % ihe
amount of silver deposited per second. Thus Ihr March 10 we
951T1
The magnitude of the current was about '-I ampere, and the areas vf
depweit abiomt 37 sq. oentims. fix- the small bowls, and aJboTiT 75 sq.
for the large bowi
The whole resistance of the cnrrent-weighing appaiatus auod c>*f
Tohameteis is about 42 ohms,, so that sufficient current can be
from 10 small Grore cdK or from a rather less number of cells of a
secondsunr batterjr.
25. The tables in which axe embodied the lesuftte of these pr-Mrac-i-ed
experiments wfll not now require much expknatiDn. Tlbcee << Table L are
certainly erroneous on account of the presence of aceta&e | Is/, aiad no
weight is given to them in calculating a final result. Foe the same reasc^
those deposits in Table H. which were prepared foom s-jlratioios to which
acetate had been added for the purpose of investigating the nature of
the disturbance thereby produced, are of eotoise excluded. The weight*
adopted for the silver deposits are those found after strong heating (nearly
to redness) for about fire minutes, no distinction being made between the
from chlorate and from nitrate of sflrer. The final mean 241* 45
the square root of the difference of current weighings in grams
divided by the rate of silver deposit in grams per second.
If we consider separately the deposits foam chferate of sflver (without
addition of acetate]^ we get as the mean number corresponding to the
above 24143, in almost perfect agreement.
The deposits made on March 25 were f ** strongly heated with inter-
mediate weighing. Similar tests have been applied in other cases not
recorded in the tables.
302
308
ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER,
[112
i
OiOt~t--^CS>OCi
COOSOOCSrHOlrH
!
<M<N<MC<J<N<M<M<M
Mean square
root of double
attraction in
grams
(M g CJ t~ >O t- GO CO
2| $ r* t- a 35 P
Gp9 5 pO3rHiH'HrH
Duration in
seconds,
corrected
(M GO t^ CO
Ift-^CptpOpCpGO
aOGOGOaDOO5GOGO
rH rH O3 rH -^ O C<l d
C~t-COt--<*iaOOCO
li
II
3O O l O "O >O GO O
jg(M (M p^ e H rH <M rH
|i
OJGO WOOO t-(N >O
COCCOt> (COOO
C^ Ol C^C^Ol Cl "^ CO
Weight of
deposits in
grams
CD CO O5 C5 CD CO t- rH rH COCO OO O O
O5O5 (NCS C^<?1 (?Q C^ICQ CQO5 -rtl^ COCO
I
ag aa aa a ga aa aa as
as as as s ps sa aa as
aa flfl as o a c ^s s^ 1 so
"_G '43 '-J3 '-S "43 '-3 '-3 '-" ' '-S ",3 '-3 '_S V3
r2^ r2r2 rS^ r3 43-3 rSrS ,2^ ,*jS
ftft ft ft ftft ft ftft ftft ftft ftft
Solution
"o-SS-gj 3
5fl | rH
II^JI |
i*iil 1?
S A * ^ fe
s? > o'S "S o
^.-r^ O 3
tO 01 rH Cfl CS 13
S3
3
-2
cS
PJ
O CO OS
CD SH *"* ^
r-" -* GO C^ " "I
[>i tiD J>
^c3 ^ p o
112] AND OX THE ABSOLUTE ELECTROMOTIVE FORCE OF CLARK CELLS.
r
JUS' : 25
1111 I I ?
~ = - =s
'=-= -
s- ~
5 s
! ^
I i
yt 35
L a J|| -if- lsiiii!t?:!32-fi iiiimiii ?HH
^ = -zH -?f- II :????! ?l?l???^??^? ^^s^oi^f^^ fifirf
2 -i-S. S Era-
's 1
t I ||1-32|
s *;s?^t^=5
-2 :
zz zz zz zzzz z ;;
I !_.
1
|l || a . i : | .n .1 -51 .1 -gl - 1 - s - 1 -
J1JIJ ' I Ml '* "Ji "^ "Ji "5l 5 I " 51 J
310 ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER, [112
It should be stated that every determination since November, 1883, in
which the manipulations were successfully conducted, is included in the
table, and that nothing is excluded except in consequence of a decision
made before the result was known. In one or two cases the current was
too irregular to give good weighings of the suspended coil, and then the
observations were not reduced with the view of obtaining absolute results,
although the comparison of the silver deposits in different bowls might
still be of interest. This happened on an occasion already alluded to when
acetate and chlorate of silver were used in combination.
. The results of Table II. agree together about as well as could be
expected, the extreme difference from the mean being ^5^. It must be
remembered that apart from the difficulties of manipulating the silver
deposits errors may arise in the determination of the current, whose mean
value has to be deduced from observations relating to only a part of the
whole time involved. A small fluctuation in the strength of the current,
lasting for a short time only, may thus escape detection. There is also
an error involved in the determination of the time of electrolysis, which
may altogether amount to nearly half a second on a total in some cases
as low as 2700 seconds. When so many experiments are made we must
expect the cases to arise in which the small errors, due to various causes,
are accumulated in the result.
26. We may now calculate the results of our experiments in absolute
measure. In the notation of 15 we have, as the relation between the
current i and the difference of weighings observed in air m,
i = p*Jm, where p, = "037048.
If w be the electro-chemical equivalent of silver in C.G.s. measure, viz.,
the quantity of silver in grams deposited per second by the unit C.G.s.
current, then the rate of deposit by current i is w . i, or w . p . \/m. Now,
by the table this rate of deposit is \/w*./2414'45 ; so that
In terms of practical units we have as the quantity of silver in grams
deposited per ampere per hour
1-11794 x 10~ 3 x 3600 = 4-0246.
The number found by Kohlrausch in his recent experiments is
w = '01 1183,
while that found by Mascart* is
* Journal de Physique, March, 1882.
112] ASTD OX THE ABSOLUTE ELECTROMOTIVE FORCE OF CLAKK CELLS. 311
The agreement between Kohlransch and ourselves is |rrrlnt|n as good
a? could be expected, and would be diminished almost to nothing were
we to take in oar experiments the weights as found after drying at
160" <X vit, before the strong heating. The account hitherto published
by Kohlranseh is only an abstract, and does not explain how the deposits
were treated*.
27. Considering that the silver voltameter may now be used satis-
factorily for the standardising of current-measuring instruments, we have
made some experiments in order to ascertain the limits within which the
method is applicable. With regard to the strength of the nitrate solution
there is considerable latitude when the currents are weak, e^, not ex-
ceeding \ ampere. In such cases a 4 per cent, solution may be used
satisfactorily in our voltameters. However, for practical purposes at the
present time the object will usually be to measure stranger current*,, and
then it is advisable to keep the solution up to 15 or 30 per cent. If the
solution is too weak in relation to the density of current, the deposit has
a tendency to looseness, and is liable to grow up in an irregular manB<er,
so as to meet the anode. In a 5-inch platinum bowl such a solution will
allow of a current of about 1 ampere for a period of an hour. The
strongest current which we have been able to use with a single volta-
meter is about 2 amperes, and for this purpose we employed a solution
containing one part of salt to two parts of water. It is probable that
the deposit would have deteriorated if the current had been allowed to
flow for much longer than a quarter of an hour, but in that time an
ample amount (about 2 grms.) is obtained. The practical conclusion is
that currents not exceeding 1 ampere may be conveniently measured in
a 3-inch voltameter by using a strong solution, and by stopping the opera-
tion after about a quarter of an hour. A shorter time than this would
hardly allow of sufficiently precise measurement when a high degree of
accuracy is aimed at. For purposes where an error of | per cent is ad-
missible, a duration of five minutes <300 seconds) would be sufficient, and
under these circumstances a stranger current would be orabjectitioable.
It will be seen that the application of this method to the measure-
ment of such currents as are usually passed through incandescent lamps
presents no difficulty, and we hope that it may be generally adopted as
a control upon the indications of instruments depending for their trust-
worthiness upon the constancy of springs or of steel magneto The anodes
should be composed of fine silver sheet (about | inch thick), such as is
sold for five shillings per ounce, and should not approach the sides of the
bowl too closely. As there need be no waste of metal, the expense of
silver as compared with copper should not be allowed to stand in the
312 ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER, [112
way of its use. For practical purposes it will be unnecessary to take some
of the precautions which we thought incumbent upon us. After rinsing a
few times with distilled water the deposit may be left to soak for an hour
or so, and then after another rinsing dried over a spirit lamp. After the
lapse of another hour it may be weighed, with a risk of error not exceeding
a few tenths of a milligram.
When still stronger currents have to be dealt with, the silver volta-
meter is less convenient. Platinum bowls of large size are not usually
met with, but two or three may be combined in parallel without much
trouble. In one of our experiments the same current was passed suc-
cessively through a single voltameter, and through two arranged in
parallel. The deposit in the single bowl, thrown down in 13 minutes,
was 2*2327 grms. Those in the other bowls were 1-0114 and T2215, alto-
gether 2'2329, agreeing almost pi'ecisely. In this way with three bowls,
such as we have used, in parallel, there would be no difficulty in measuring
currents up to 5 amperes.
28. The second branch of our subject is the evaluation of the electro-
motive force of standard galvanic cells. Enough has been said as to the
means employed for measuring electric currents in absolute measure. If a
current, after passing the current-weighing apparatus, is made to traverse
a known resistance, it will generate at the extremities of that resistance
a known electromotive force. By suitably accommodating to one another
the magnitude of the resistance and the strength of the current, the
electromotive force may be made to balance that of a standard cell,
whose force is thus determined. The difficulty of the matter relates
principally to the preparation and definition of the standard cells, and in
order to test the constancy of the cells it is desirable to extend both
the absolute determinations and the comparisons of various cells over a
considerable range of time.
Before describing further the arrangements adopted for the absolute
measurements, it will be convenient to consider the comparisons of E.M.F.,
which were always made by the method of compensation, in order to
diminish as far as possible the currents actually passed through the cells
under examination. The main circuit consisted of two Leclanche cells M,
and two resistance boxes N, (joined by a short stout wire) of 10,000 ohms
each (fig. 1). Of this resistance a variable and adjustable proportion was
included between the points of derivation, and (by use of the second box)
the total was in all cases made up to 10,000. Thus, in compensating a
single Clark cell the resistance from the first box might be 4900, and
from the second 5100. By this means the constancy of the main current
is secured. The derived branch includes the cell or cells to be tested
(P), a mercury reversing key (Q), and a galvanometer (T), with which is
112] AXD ON THE ABSOLUTE ELECTROMOTIVE FORCE OF CLARK CELLS. 313
associated a resistance (S) of 10,000 ohms. The galvanometer itself was of
the Thomson pattern, and had a resistance of about 200 ohms. By the
substitution of an instrument with a longer wire and of resistance up to
10.000, a greater degree of sensitiveness might have been obtained, but with
careful reading of the galvanometer scale the arrangements were sufficient
for the purpose, and would indicate the E.M.F. to about 10 ^ 00 . In the
preliminary trials a simple contact key with platinum studs was used in
die galvanometer branch with the idea that shorter contacts would thus
suffice- But, probably from thermoelectric disturbance, the readings thus
obtained were not so consistent as with the mercury reversing key, and
the smallness of the currents actually allowed to pass rendered the longer
contacts unobjectionable. From the data already given it will be seen
that a current of 10~* amperes was sensible, and no disturbance could be
expected from currents 100 times, or more, greater than this. In order
to test whether the connexions were rightly made, the first observation
was usually taken with a still higher resistance in the galvanometer
branch, which could easily be effected by causing the current to pass
through the body of one of the observers from hand to hand. If by
accident too large a current was allowed to pass through a cell, no further
use was made of that cell until the next day*. It must be mentioned
that great care was taken, and was necessary, in respect of the insulation
of the various parts. For instance, no correct results were obtainable
when the Leclanche's stood upon the (tiled) floor, if at the same time
other parts of the combination were touched with the hand. A sheet of
paraffined paper interposed proved a remedy. In this matter we have had
several disagreeable lessons, and we cannot too strongly emphasise our advice
to take too many rather than too few precautions.
When two cells under comparison differ by a considerable fraction, they
may be compared separately with the Leclanche's, or rather expressed in
terms of the current afforded by the Leclanche' ; s through 10,000 ohms.
Thus, on Dec. 3, 1883, in order to balance Clark No. 1 (see below > 4926
were required between the points of derivation. When a standard Daniell
of Kaoult's pattern was substituted for the Clark, the number required
was 3798. In terms of No. 1 Clark the E.M.F. of the Daniell is thus
3798/4926, or '7710. At the end of a series of comparisons it is proper
to repeat the observation of the first standard cell, in order to check the
constancy of the current supplied by the Leclanche's, In our experience
there was usually no appreciable variation.
When the cells to be compared are nearly alike, it is better in the
second observation to express the difference of forces by setting the second
stringent than were really necessary.
d later (| 31) show that the precautions observed in this respect were
314 ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER, [112
cell to act against the first. Thus, the force of Clark No. 1 being expressed
as before by 4926, the corresponding resistance for the excess of the force
of Clark 1 over Clark 3 was 2 ohms. Hence, in terms of Clark 1 the
force of Clark 3 is '9996, and the result is less liable to error than if the
comparisons of each with the Leclanche's were effected separately.
29. Of the first batch of Clark's which were compared together
from November, 1883, onwards, No. 1 was set up near the beginning,
and Nos. 2, 3, 4, 5, towards the end of October. They were prepared
generally according to the directions given by Dr Alder Wright*, to
whom we have been indebted for advice and for samples of some of the
materials. The saturated solution of zinc sulphate was nearly neutral.
The metallic zinc was bought as pure from Messrs Hopkin and Williams.
The mercurous sulphate was from the same source, and the metallic
mercury was redistilled in the laboratory. We did not consider it desir-
able to take precautions against the presence of air, thinking that it was
sure to find an entrance sooner or later.
Four new cells, Nos. 6, 7, 8, 9, were set up from the same materials on
January 10, 1884. It will be seen from the table that when a fortnight
old they differed but little from the first batch.
In preparing these cells the most troublesome part of the process was
found to be the casting of the zincs. The metal, melted in a porcelain
crucible, was sucked up into a previously heated tube of hard glass, but
the operation required some address, and there was considerable waste
of zinc from oxidation and otherwise. It occurred to us to try whether
equally, or perhaps still more, satisfactory results might not be obtained
by substitution for the solid metal of an amalgam of zinc. For this
purpose a form of cell, called for brevity the ^T-cell, was contrived, and is
shown full size (fig. 5). One of the legs is charged with the amalgam
of zinc (B), the other with pure mercury ((7), covered with a layer of
mercurous sulphate (D). The whole is then filled up above the level of
the cross tube with saturated zinc sulphate (E), and a few crystals are
added. Evaporation is prevented by corks (F), closing the upper ends of
the tubes. Electrical contact with the amalgam and with the pure mercury
is made by platinum wires (A), sealed into the glass.
A preliminary experiment in which both legs of a cell were charged
with amalgam (the mercurous sulphate being dispensed with) having shown
that the E.M.F. was independent of the excess of undissolved zinc, two cells,
HI, H 2 , were set up on February 12, 1884, and submitted to various tests,
such as stirring up the amalgam with a glass rod. The amalgam was
prepared from pure mercury and the same zinc as before. Subsequently, on
March 6, six more cells were charged with a somewhat different treatment.
* Phil. Mag. July, 1883.
112] AND ON THE ARSOLUTE ELECTROMOTIVE FORCE OF CLARK CELLS. 315
The sulphate of zinc was from another sample and contained appreciable
quantities of iron. Moreover, the amalgam was differently prepared. The
mercury and zinc were shaken up together in a bottle with a little acid,
after which the acid was washed out by shaking with several changes
of water, until litmus paper was no longer reddened. Into each cell, in
addition to the fluid amalgam, there was dropped a piece of solid zinc
from the bottle. The same mercurous sulphate as before was employed,
but the washing with distilled water was dispensed with. The three re-
maining cells of this pattern H, H u , H u , were charged on March 12, 1884,
with a third sample of zinc sulphate.
r
The agreement among themselves and the constancy of the H -cells has
been all that could be wished; but some modification in preparation will
be desirable, for it has been found that the amalgam tends to harden into
compact lumps, the expansion of which is liable to burst the cells. From
this cause H 3 , H^, H 7 , succumbed at a comparatively early stage. It is
probable that the addition of solid zinc to the fluid amalgam had better
be omitted, but on this and other points we hope to make further investi-
gation. The H pattern lends itself conveniently to experiment, as it is
possible by withdrawing the corks to make any desired addition to the
contents. On more than one occasion the contents of each leg have been
vigorously stirred, without the slightest change in the E.M.F.
Since the first draft of this memoir was written two new batches of
cells of the ordinary pattern have been prepared with different materials.
316
ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER
[112
;:
I 2
3 "
X X tO X O O ~ X X' ^ X 00 1
I O 00 00 00 X X
112] AND OX THE ABSOLUTE ELECTROMOTIVE FORCE OF CLARK CELLS. 317
?
9 9
o o
!
1 3 S 3 2
iiss
999
I g s a 1
? 9 9 9
2 =1 2 2
1 lllil i s i i *
I
o o ;= is oo -* s
1 J 2 ' ' ' ' 22
I
I
I ^ - - - ~ - - - - - |
~ ^
s in s g 312 i 1 I
>S > S52" ^"^3^-^22? =
^ ^ ^^ ^--=^- |
; i s 2 1
I !S22
g
318 ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER, [112
In this case the zincs were used as supplied, without re-casting*, and the
mercurous sulphate, though distinctly acid, was not washed. The first
batch (10, 11, 12, 13) were set up on May 7, and the second batch (14,
15, 16, 17, 18, 19) on May 26.
30. Tables VI., VII., VIII. show the results of most of the com-
parisons, the value of every cell on each day being expressed in terms of
Clark No. 1. It will be seen that there are durable differences between
cells of the same batch, but that these do not much exceed y^. There
are also changes of small amount in the force of a given cell, part of
which is perhaps attributable to a difference of temperature coefficient.
Moreover the actual temperatures may possibly have differed a few tenths
of a degree in the case of various cells, many of which stood some feet
apart. Clark No. 3 does not appear in Table VII, since on January 25
it was found to be short circuited. During the later comparisons, Nos. 6
and 7 were unavailable, having been diverted to another use.
The two last batches took a longer time than usual (about three
weeks) to reach their normal values. It will be seen from Table VIII.
that when first set up these cells were too strong by as much as 1 or
2 per cent. It was thought that the process of settling down might be
quickened by closing the circuit occasionally for some minutes, through a
resistance of 1000 ohms, and the asterisk in the table indicates that on
the day previous to the comparison the cell in question had been so treated
for about ten minutes. When once the settling down is completed, further
short circuiting appear to be without effect.
31. Some observers having laid great stress upon the importance
of guarding Clark cells from the passage of sensible currents, we give a
specimen of the results of some tests to which we have subjected a few
of the cells, in order to find out how much care was really necessary in
their use to avoid polarisation. The accompanying Table IX. shows the
variations of E.M.F. of Clark No. 6 on April 28, when very rudely treated.
The other connexions remaining as usual, the poles of the cell were joined
through a resistance-box, by means of which the cell could be short cir-
cuited with any external resistance from to infinity. The numbers
entered (such as 4994) are proportional to the difference of potential
between the poles, being in fact the resistance between the points of
derivation on the Leclanche circuit. It will be seen that in the course
of a quarter of an hour the cell recovers, to within a few ten-thousandths
of its value, from the effects of being short circuited for several minutes
through such resistances as 1000 ohms. From the electromotive forces
during the short circuiting it appears that the internal resistance is high,
nearly as much as 300 ohms.
* The surface of the metal was brightened with file and sand-paper.
112] AXD OX THE ABSOLUTE ELECTROMOTIVE FORCE OF CLARK CELLS. 319
The manner in which the Clark cells have borne the tests applied to
them justifies the hope that they may be found generally available as
standards of E.M.F. But further experience is necessary as to the effect
of various modes of preparation, and it is to be hoped that this may
soon be forthcoming. As used by us. the process is so simple that no one
need be deterred from setting up cells for himself.
TABLE IX.
Time
Resistance between Poles
K.1I.F.
h. m.
3 35
X
4994
3 47
oo
4994
3 53
Changed from x to 10,000
3 56
10,000
4851
3 41
10,000
4853
4 59
Changed from 10,000 to x
5 2
00
4990
5 15
00
4991
5 47
00
4992
6 3
Changed from x to 1000
6 5
1,000
3990
6 11
1,000
3860
6 13
Changed from 1000 to x
6 19
00
4990
6 25
oo
4991
6 29
Changed from x to 500
6 34
Changed from 500 to oo
6 36
00
4985
6 37
00
4988
6 52
00
4991
32. Experiments on Daniell cells gave only a moderately good result.
Raoult's form was employed, in which the zinc and copper solutions are
placed in separate beakers, the connexion being only through a Y-tube
charged with zinc sulphate and tied over the ends with bladder. One
electrode was of pure zinc amalgamated with pure mercury, and the other
of copper freshly coated electrolytically. The zinc and copper solutions
were both of sp. gr. 1-1.
TABLE X.
November 30, December 3,
1883 1883
Decembers,
1883
' December 11, I December 12,
1883 1883
Clark No. 1
1-0000
1-0000
1-0000
1-0000
1-0000
Daniell . .
7702
7710 -7705
7698
7702
320 ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER, [112
The Daniell cell has of course to be charged freshly on each occasion,
and is thus far less convenient in use than the Clark's, which stand for
months always ready for use. The temperature of the cells at the time
of the comparisons tabulated was about 16 0.
Through the kindness of the inventor, we have had the opportunity
of comparing some De La Rue cells with the Clark's. The cells are of a
somewhat modified construction, the atmospheric oxygen being excluded
by a layer of paraffine oil. They were set up some days before the
comparisons, and short-circuited for five minutes in order to start the
chemical action.
We found
No. 1 De La Rue = 7510 Clark.
No. 2 -7512
No. 3 -7382
No. 4 = -7458
Mean = "74G5
Mr De La Rue (Phil. Trans., vol. CLXIX., Part I.) found a result decidedly
smaller, the explanation of which is to be sought in the fact that in his
experiments the cells were making a current of about yJ^ ampere, whereas
in ours the electromotive force is measured when no current passes.
It may be useful to record also a comparison between our Clark's and
a new form of Daniell, introduced by Sir W. Thomson. This cell is charged
with zinc sulphate of sp. gr. 1'02, and with saturated solution of copper
sulphate. The zinc is not amalgamated. According to Sir W. Thomson's
directions, the circuit of the cell is closed through 250 ohms, and the E.M.F.
measured is that between the poles under these conditions. After the
current had been running for about an hour and a half, the E.M.F., which
had been increasing, became fairly constant, and its value was then '743 in
terms of Clark No. 1. The comparison was made on April 8, 1884*.
33. We now pass to the description of the method adopted for the
absolute determinations. The current, after leaving the current-weighing
apparatus, is caused to traverse a wire of known resistance R, whose
stout copper terminals rest on the copper bottoms of suitable mercury
cups H, K (fig. 1). To these cups are brought also the terminals of
the derived branch, in which are included the galvanometer and the
standard cell.
On account of the strength of the currents (about ampere) the re-
sistance required to be of special construction in order to avoid too great
heating.
Two ebonite rods were held in a parallel position by a frame of wood,
and round these uncovered german silver wire was wrapped so as to be
* See notes.
112] AND ON THE ABSOLUTE ELECTROMOTIVE FORCE OF CLARK CELLS. 321
exposed to the air as much as possible. The rods are about a foot apart,
and are grooved, the better to keep the wire in its place. The resistance
is about 4 B.A., and was determined with the aid of a five and a single*.
At 17-6 its value is 4-00699 B.A.
Even this resistance-wire heats sensibly when the current of ampere
is passed through it for more than a few seconds. The increment of
resistance was determined by observations taken immediately after the
passage for some minutes of a stronger current (about 1 ampere). In
this way it was found that for the currents usually employed a correcting
factor 1-00041 must be introduced to take account of the heating, inde-
pendently of course of the correction necessary for the difference between
17 C '6 and the temperature of the atmosphere at the time of an absolute
determination.
34. In order to obtain the balance of electromotive forces two distinct
methods have been followed. In the earlier determinations there was no
electromotive force in the derived branch except that of the standard cell,
and the adjustment was effected by variation of a comparatively high
auxiliary resistance from a box, placed in multiple arc with the [4]. The
readings were taken by reversal of the galvanometer connexions at a
mercury commutator, and the small outstanding galvanometer displace-
ment was allowed for with the aid of observations of the effect of a known
change in the auxiliary resistance. In this way could be determined the
auxiliary resistance, and from it (by addition of conductivities) the effective
resistance between the points of derivation necessary for a balance with
the actual current. The value of the current at the moment in question
is deduced from the curves representing the two sets of current weighings
( 24). In the course of half-an-hour several almost independent deter-
minations of the electromotive force could be completed.
This method is the simplest, and could usually be made to work satis-
factorily. It is, however, open to the objection that if the current changes
rapidly we must either allow for a considerable galvanometer displacement
or else alter the auxiliary resistance. But the latter change reacts upon
the principal current, and renders the current weighing curves discon-
tinuous, thereby increasing the difficulty of specifying the value of the
current at the moment of observation.
35. In the second method the resistance between the points of deri-
vation is the [4] simply, and compensation is made in the galvanometer
branch by the introduction of a graduated E.M.F. (fig. 1). The arrange-
ment is in fact almost the same as in the comparison of two cells by
the method of difference ( 28), one of the cells being replaced by the
* For the methods used to ascertain the valne of ihejire the reader is referred to former
papers.
R. II. -1
322 ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER, [112
resistance [4] traversed by the main current. As the apparatus for these
comparisons was always ready for use, this method was, under the cir-
cumstances of the case, really more convenient than the other, and was
employed in the later determinations. The procedure will be best under-
stood from an example.
On March 29, 1884, determinations of silver and of electromotive force
were made simultaneously, so that the same set of current weighings
might serve for both purposes. Accordingly the main current traversed
the three voltameters, the current weighing apparatus and the resistance
[4]. In the derived branch (fig. 1) were the standard cell No. 4 Clark,
the galvanometer with its commutator, and coils from a resistance box,
through which passed the current from the two Leclanche cells ( 28).
If the compensation between the Clark and the difference of potentials at
the terminals of the [4] were incomplete the balance could be restored
by the introduction of a graduated part of the E.M.F. of the Leclanche's,
the value of which, in terms of the Clark, is found by a subsequent ex-
periment, in which the [4] is excluded. It will be understood that the
Leclanche's worked in a perfectly constant manner, the whole resistance
in circuit being always made up to 10,000 ohms (in addition to that of
the cells themselves). If E be the E.M.F. of the Clark, p the resistance
(traversed by the current of the Leclanche's) which must be used to get a
balance when the [4] is excluded, r the resistance actually required during
a set of measurements when [4] is connected, then the electromotive force
actually compensating the action of [4] is E (1 r/p).
At the beginning of the proceedings on March 29 the main current
was stronger than that required for the simple compensation of E, so
that to get a balance at the galvanometer the Leclanche's would have
had to be reversed. At 18 from the commencement the current had
fallen to the point of compensation with r = 0. At 28 m balance required
r = 20 B.A., at 34 m r = 37, and at 48 m r = 90. To take these observations,
the easiest way is to overshoot the point somewhat, and then continually
reversing the galvanometer to note the time of passing through the balance.
From the curves representing the current weighings, the double force of
attraction at the above times were found to be '9645, '956, '9495, "931,
expressed in grams. This is what has been denoted by m ( 26), and the
corresponding current is
i = -037048 Vm.
36. The resistance R between the points of derivation must be
expressed in absolute measure, if we wish E to be so expressed. But for
comparison with the results of other observers it will be convenient to
keep this question apart and, in the first instance, to express our electro-
motive forces as if the B.A. unit were correct. Any factor (such as '9867)
112] ASTD OX THE ABSOLUTE ELECTROMOTIVE FORCE OF CLARK CELLS. 323
which may be adopted to express the RJL unit in terms of the ohm will
enter also into the expression of E in true volte.
At the atmospheric temperature 13 '1 the value of the [4] is 3-9998 BJL,
whence
R = 4D0143 BJL,
correction being made for the heating effect of the current.
The formula for E is
The value of p (on the occasion in question) was 4999 RA-, and this
completes the data few the evaluation of E. The four values corresponding
to the above observations are
1-4559, 14553, 1-4553, 14566,
giving as mean
E = 1-4558 RA. volte
This result is tar No. 4 at a temperature of 13 r 'l. The value of No. * in
terms of Mo. 1 at the time in question was about -999$, so that we should
have found for Ho. 1
E = 1-4561 R A. volte
We have still to reduce to the standard temperature of 15 . The
coefficient originally given bj Latimer Clark is 1-0006 per degree ceim-
grade- Wright and Thomson* found a smaller number, viz.. 1D004I, and
with this our results were first reduced. Later, however, we found reason
to suspect that the actual change was greater than this, and accordingly
made some special observations to dear up the doubt One ceil (No. 61
was mounted in a large test tube, the gutta-percha-covered leading wires
being brought through a tightly-fitting indiarubber cork, and was kept
constantly at (T centigrade by being surrounded with ice. With this
No. 1 at the temperature of the room was compared from day to day,
with the result that its temperature coefficient is about the double <1-OOOS2)
of that given by Wright and Thomson. A similar result was found by
Helmholtz-f-, who remarks that the effect of temperature may vary according
to the preparation of the celL
Using this number to reduce the result of March *9, we have to subtract
-0022, thus obtaining
# = 1-4539 BJL volts
as the electromotive of Xo. 1 Clark at 15".
- PWL Jfaf . Jrfy, 1SSB, PL 36.
t Sitxmmy&r. d. KSm. Atmd. 4. llu*. s BerioL, Fcfew?, 18S.
213
324
ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER,
[112
37. This determination and twelve others, made at intervals from
Oct., 1883, to April, 1884, are exhibited in Table XL* They are all
TABLE XI.
I.
II.
III.
IV.
V.
VI.
VII.
VIII.
E.M.F.
Date,
1883 and 1884
Cell used
Tempera-
ture
E.M.F.
in B.A.
volts
E.M.F.
relative
to No. 1
E.M.F.
ofNo.l
Correction
to 15
in B.A. volts
corrected to
15
October 23 .
Clark No. 1
15-9
1-4542
1-0000
1-4542
+ -0010
1-4552
November 20
No. 2
15-3
1-4549
1-0006
1-4540
+ -0004
1-4544
,, 21
No. 1
14-9
1-4543
1-0000
1 -4543
- -0002
1-4541
22
No. 1
14-9
1-4533
1-0000
1-4533
- -0002
1-4531
December 4
No. 1
15-8
1-4524
1-0000
1-4524
+ -0010
1-4534
11
No. 1
17-2
1-4524
1-0000
1-4524
+ -0026
1-4550
12
No. 2
15-8
1-4549
1-0008
1-4537
+ -0010
1-4547
January 28 .
No. 2
15-0
1-4541
1-0000
1-4541
+ -oooo
1-4541
March 20 . .
No. 4
15-8
1-4533
9998
1-4536
+ -0010
1-4546
ii 25 -
No. 1
13-5
1-4560
1-0000
1-4560
- -0018
1-4542
29. .
No. 4
13-1
1-4558
9998
1-4561
- -0022
1-4539
April 2 . .
No. 1
16-1
1-4524
1-0000
1-4524
+ 0014
1-4538
7 . .
No. 1
15-5
1-4535
1-0000
1-4535
+ -0006
1-4541
Mean . . .
15-3
1-4542
deduced from observations with the current-weighing apparatus. It will
be seen that there is little or no evidence of any progressive change. The
casual fluctuations are of course partly due to errors of observation, but it
would seem are principally to be attributed to real variations of electro-
motive force of the same kind as appear in the Tables VI., VII., VIII.,
showing the relative values of the various cells. The mean temperature
at the times of the determinations differs so little from 15, that the final
number for that temperature is almost independent of the temperature
coefficient.
We may take as applicable with but little error to all the cells of this
type that have been experimented upon
# = 1-454 B.A. volts at 15.
The value for the #-cells would be a little higher. (See Tables.)
The corresponding number found by Mr Latimer Clark was
# = 1-457 B.A. volts,
so that the difference between us is small, and perhaps even dependent
upon variations in the materials or construction of the cells.
To express our results in true volts we have only to introduce the factor
expressive of the B.A. unit in terms of the ohm. If in accordance with our
* For continuation of Table XI. see notes.
112] AXD OX THE ABSOLUTE ELECTROMOTIVE FORCE OF CLARK CELLS. 325
own determinations we take
1 RA. unit = -9867 ohm,
we shall have as the value of a Clark cell at 15
'=1-435 volt
38. It has been mentioned that on March 29 silver deposits were
made at the same time as the observations of E.1LF. One object of this
was to exemplify the procedure which will probably be in future the most
convenient for the determination of E.M F. when the very highest accuracy
is not required. It is evident that if we assume a knowledge of the
electro-chemical equivalent of silver, the weights obtained in a given time
on March 29 will lead to a determination of E.M.F., independently of tie
current neighing*. We propose to exhibit the method of calculation,
ignoring altogether the use of the current-weighing apparatus, whose only
effect will be that of a resistance of about 40 ohms. If IT be the weight
of silver deposited in the time f, ic the electro-chemical equivalent, we
have as the relation between IT and E,
On this occasion IT = 1-4531 grnis., f = 3599 seconds. ^ = 40014 B.A..
p = 4099 R.A.. as before. If IT be assumed, the only other element required
for the evaluation of E is
rdt
viz., the mean value of r necessary for a balance of E.ILF. during the time
that the current ran through the voltameters. To get this the actual
observations of r are plotted, the times being taken as abscissae, and a
curve constructed representing the value of r throughout the coarse of
the experiment*. From this curve the ordinates are measured, which
correspond to the middle of every five minutes' period. The values of r
thus obtained are
TABLE XH.
+ 32
7! -16
37! ~ *^
13| -W
42| +
17J - 2
47! + *
--r
-
52| +112
- 7:
+18
57! + 140
= +38-3.
of the core use was made of observations in which the galvanometer
the nloe of the sale dhiaoM bag afpretiMlfiy
326 ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER, [112
The rapid falling-off of the current towards the end of the hour is
believed to be due to the formation of crystals upon the anodes of the
cells charged with silver chlorate. The value of
_ [ rdt
1 t
being thus found to be 4960'7, the calculation of E may be completed.
Taking w= 1-1180 x 10~ 2 , we get
E= 1-4562 B.A. volts,
as the electromotive force of No. 4 Clark at 13'l.
On April 2 an equally satisfactory result was found from the silver
deposits without use of the current weighings. It will be seen that in
this way anyone may determine the E.M.F. of his standard battery with
a very moderate expenditure of trouble and without the need of any special
apparatus. So large a resistance in the main circuit as in the above
example, due to the idle coils of the current-measuring apparatus, is not
necessary, but some resistance in addition to R and that of the battery
and voltameters would probably be advisable. Otherwise the magnitude
of the current would be too sensitive to the resistance of the volta-
meters, which cannot be included in the circuit until the experiment
actually begins. In the preliminary adjustments the resistance of the
voltameters should be represented by an estimated equivalent of wire
resistance, and this should not be too large a fraction of the whole. In
our case the resistance of the three voltameters charged with nitrate
solution of 15 per cent, was a little under two ohms, and the condi-
tions under which we worked would be sufficiently imitated by a circuit
containing, besides the [4] and the voltameters, an extra resistance of
10 ohms. A battery of three or four Grove cells would then be sufficient
for the generation of the current.
112] AND OX THE ABSOLUTE ELECTROMOTIVE FORCE OF CLARK CELLS. 327
APPENDIX (see 15).
TABLE of the [logarithm] of sin 7 [2F y -(l +sec*<y)E y } from 7 = 00 to 7 = 70
' ~ '
t
55
1-9198899 60
1786408
65
4433405
55 6
1-9250674 60 6 -1838431
65 6
4487720
55 12
1^302440 60 12 -1890478
65 12
4542107
55 18
1-9354198 60 18 -1942546
65 18
4596565
55 24
1-9405945 60 24 -1994636
65 24
4651097
55 30
1-9457677 60 30 -2046748
65 30
4705707
55 36
1-9509400 60 36 -2098887
65 36
4760395
55 42
1-9561123 60 42 -2151058
65 42
4815165
55 48
1-9612837 60 48
2203260
65 48
4870015
55 54
1-9664536 60 54
2255491
65 54
4924944
56
1-9716227 61
2307753
66
4979956
56 6
1-9767918 61 6
-2360045
66 6
5035052
56 12
1-9819605 61 12
2412367
66 12
5090234
56 18
1-9871288 61 18
2464720
66 18
5145504
56 24
1-9922966
61 24
-2517106
66 24
-5200861
56 30 1-9974637
61 30
2569525
66 30
5256304
56 36 -0026304 61 36
2621981
66 36
531183-8
56 42 -0077970 61 42
2674478
66 42
5367469
56 48 -0129635
61 48
2727014
66 48
54-23105
56 54 -0181298
61 54 2779545
66 54
5479017
57
0232962
62
2832194
67
5534935
57 6
-0284628
62 6
2884843
67 6
5590948
57 12
-0336297
62 12
-2937533
67 12
56470*50
57 18
1X387966
62 18
2990263
67 18
5703-278
57 24 O439638
62 24
-3043035
67 24
5759599
57 30 iH91317
62 30
3095854
67 30
-58160-2-2
57 36 -0542999
62 36
3148717
67 36
58725-543
57 42
0594684
62 42
3201621
67 42
59-2918.8
-- ^
0646364
r;0 -.
3254571
67 48
5985936
57 54
-0698062
62 54
3307575
67 54
CO42795
58
0749769
63
3360628
68
6099767
58 6
0801480
63 6
3413729
68 6
6156351
58 12
0853198
63 12
3466879
68 12
6214051
66 1-
0904926 63 18
3520081
68 18
-6271370
58 24
-0956665
63 24
3573335
68 24
6328810
58 30
1008414
63 30
3626642
68 30
6386371
58 36 -1060175
63 36
3680004
68 36
6444054
58 4. -1111950
63 42
3733422
68 42
6501859
58 48 -1163737
63 48
3786896
68 48
6559791
58 54
1215535
63 51
3840425
68 54
6617852
59
1267346
64
3894014
69
667C045
59 6
1319170
64 6
3947666
69 6
6734371
59 12
1371009 64 12
4001380
69 12
6792833
59 18
1422865
64 18
4055155
69 18
6851433
59 24
1474739
64 24
4108993
69 24
6910170
59 30
1526636
64 30
4162893
69 30
6969043
59 36
1578552
64 36
4216858
69 36
7028058
59 42
1630186
64 42
4270894
69 42
7087220
H 4- -1682439
64 48
4324998
69 48
7146529
59 54 -1734412
64 54
4379166
69 54
7205985
328 ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER, [112
EXPLANATION OF FIGURES.
Fig. 1. A. Principal battery of Grove's or storage cells.
B. Resistance for adjustment of current.
C. Voltameters.
D. Rough tangent galvanometer.
E. Reversing key of current-weighing apparatus.
F. Fixed coils.
G. Suspended coil.
H, K. Mercury cups, into which dip the terminals of resistance R.
L. Earth connexion.
M. Leclanche's of E.M.F. compensator.
N, 0. Resistance-boxes of same.
P. Standard galvanic cell.
Q. Galvanometer commutator.
8. Associated resistance of 10,000 ohms.
T, Galvanometer.
Fig. 2. Section of ebonite ring (full size).
Fig. 3, 14. Connexions for comparison of galvanometer constants.
A. Daniell cell.
B. Mercury reversing key.
C. Point where current divides.
D. Coil of electro-dynamometer.
E. Ebonite coil.
F. H, L, M. Mercury cups.
G. Bridge galvanometer.
K. Resistance-box in multiple arc with [10].
P. Short circuiting piece to connect F and H.
N. Resistance added to E.
Fig. 4, 24. Curves of current weighings. In the original drawing two
divisions along the line of abscissa represent one minute,
and two divisions along the line of ordinates represent
one milligram. Of these divisions every tenth only is
shown in the Figure.
Fig. 5, 29. ^-pattern of Clark cell.
A. Platinum wires sealed through glass.
B. Amalgam of zinc.
C. Pure mercury.
D. Mercurous sulphate.
E. Saturated solution of zinc-sulphate.
F. Corks.
112] AXD OX THE ABSOLUTE ELECTROMOTIVE FORCE OF CLARK CELLS. 3V
NOTES.
(Added December, 18*4.)
Note to 25.
In order to investigate the effect (if any) of temperature upon the
amount of silver deposits, we have made experiments in which volta-
meters maintained at different temperatures were exposed to the same
current. The results, exhibited in the accompanying table, show a small
but apparently real increase in the weight of the deposit as the tempera-
ture rises. Had the effect been in the other direction, we should have
been disposed to attribute it to imperfections of manipulation, for the
deposits from the warm solutions were always coarser and looser in
texture than the corresponding deposits (upon the same area) from the
cold solutions.
\ After ami washing and drying at 16flF After fc*^"^g to icqge of redness ;
__ :-;-....
1*34 .. : - , :
ea -w- --&--! ~ u
May 27
..
2-3915
2 -3305
0010
Jane 4
2-0230
2-0220
2-0229
2-0221
twos
;.;j ;:
1-9050
1-9043
[ 1-S049
1-3O43
-0006
July 31
1-9438
1-9433 1-9430
! 1-9440 1-W32
1^431
-0009
The solution was a 15 per cent, solution of pure nitrate of silver, and
the anodes were of pure metal. The current was about | ampere, and
passed for rather more than an hour.
The results here disclosed diminish, of course, the chemical significance
of the number given as representing the electro-chemical equivalent of
silver, but the variation is so small at ordinary laboratory temperatures
that the use of the silver voltameter as a means of defining electric
quantity is not practically interfered with.
Note to 26.
M. Mascart (Journal de Physique, t. iii; Juillet, 1884) has recently
revised the calculation of the constant of his apparatus, by which revision
the final number is altered from "01124 to -011156.
330
ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER,
[112
Note to 27.
Although there can be no doubt that silver is greatly preferable to
copper for the electrolytic measurement of currents, we have thought that
it might be useful to make a few comparisons of the two metals, so as to
allow copper to be referred to on an emergency with as much success as
the nature of the case admits. The copper deposits were taken in the
same way as the silver upon platinum bowls, the anodes being wrapped
in filter-paper and suspended at the top of the liquid. On account of
the tendency to oxidation it is not advisable to allow the copper deposits
to soak for a long time. They were washed in boiling water for about
half-an-hour, and then dried off in the hot closet at 150. The solutions
were made from sulphate, bought as pure, no acid being added. Of the
four bowls I., II. are large and somewhat deep, III., IV. are shallow saucers
about 3 inches in diameter. In the large bowls the area of deposit was
about 32 sq. centims., in the smaller about 25 sq. centims. The strength
of current on the first two occasions was about ^ ampere, on the last about
f ampere, thus representing the circumstances for the measurement of the
current through an incandescent lamp.
Weight
Ratio of
Equivalent
Date, 1884
Bowl
Solution
of
Mean
copper to
of copper
deposits
silver
(silver =108)
Nov. 20 . .
I.
III.
Silver nitrate 15 per cent.
1-3874]
1-3870J
1-3872
2937
31-72
"
II.
IV.
Copper sulph. sp. gr. 1-174
40651
4082J
4074
Nov. 27 . .
II.
IV.
Silver nitrate 15 per cent.
1-0523|
1-0522J
1-0522
2934
31-69
. .
III.
Copper sulph. sp. gr. 1-115
3094)
.OAQ7
I.
3081/
oUof
Dec. 11 . .
II.
Silver nitrate 15 per cent.
3-04891
3-0488
,,
IV.
3-0487J
2938
31-74
"
III.
I.
Copper sulph. sp. gr. 1-115
89561
8962J
8959
Mean . .
2936
31-72
Multiplying '2936 by 4'0246 we get 1182 grms. as the amount of copper
deposited per ampere per hour.
112] AND ON THE ABSOLUTE ELECTROMOTIVE FOBCE OF CLARK CELLS. 331
Note to 30.
Observations made at intervals since this paper was read may here be
given in continuation of Tables VII. and VIII.
Jane 26
July 14
July 21, 22
Aug. 6
Oct. 8
Oct. 28
NOT. 14
Dec. 5
Clark 1. .
1-0000
1-0000
1-0000
1-0000
1-0000
1-0000
1-0000
1-0000
4. .
9998
1-0000
1-0000
1-0004
99%
9997
o-oooo
o-oooo
5 . .
9997
1-0007
9998
1-0006
9990
9994
9993
9997
8. .
9997
9998
9998
9996
9997
9998
1-0000
99%
9. .
1-0000
1-0000
1-0002
9998
1-0000
1-0002
1-0002
9999
10. .
1-0003
1-0003
1-0003
1-0003
1-0003
1-0003
1-0003
1-0003
11 . .
1-0003
1-0004
1-0003
1-0007
1-0003
1-0003
1-0003
1-0004
,. 12 ..
1-0004
1-0004
1-0004
1-0007
1-0003
1-0003
1-0003
1-0003
13 ..
1-000-2
1-0002
1-0002
1-0000
1-0002
1-0002
1-0003
1-0001
14 ..
1-0026
1-0003
1-0003
1-0002
1-0003
1-0003
1-0003
1-0003
16 ..
1-0004
1-0003
1-0004
1-0002
1-0002
1-0003
1-0003
1-0003
18. .
9974
9920
9900
9860
9800
9810
9760
ansteady
19 ..
1-0004
1-0000
1-0000
9997
1-0000
1-0002
1-0001
9999
H, .
1-0005
1-0004
1-0005
1-0005
1 0003
1-0003
1-0004
10000
H f .
1-0007
1-0004
1-0005
1-0005
1-0004
1-0002
1-0003
U M
1-0004
1-0005
1-0005
1-0003
1-0003
1-0003
1 -00-34
1-0004
H u
1-0004
1-0005
1-0005
1-0003
1-0002
1-0003
1-0003
1-0003
jf
1-0030
1-0004
1-0003
1-0003
1-0003
H 13
1-0009
1-0003
1-0002
1-0003
1-0003
1-0001
Some ZT-cells have been set up by Mr Threlfall, with amalgams of
known composition, varying from ^ zinc to ^ zinc by weight. The dura-
tion of the test has as yet been scarcely adequate, but it appears that the
smaller quantity of zinc is sufficient.
Note to 32.
Comparisons of standard Daniell cells of the Post Office pattern sent
me by Mr Preece have been made on several days, but did not give satis-
factory results. The E.M.F. rises about 1 per cent, during the half-hour
following the placing of the zincs and porous cells in the working compart-
ment, and the two specimens differed about 2^ per cent. The mean values
were about 1-081 and 1-056 true volts.
Note 1 to 37.
An examination of the recent comparisons of cells of different ages will
probably lead to the conclusion that no important absolute change of E.M.F.
332
ON THE ELECTRO-CHEMICAL EQUIVALENT OF SILVER.
[112
can have occurred during the thirteen months; but since the cells have
been employed as standards for the determination of electric currents in
various experiments, e.g., for the determination of the constant of magnetic
rotation (Proc., June, 1884), it seemed desirable to supplement Table XI.
with observations of later date. Two further absolute determinations have
accordingly been made on November 21 and November 27, 1884, by the
method of 38, with the following results :
TABLE XL (continued).
Date
Cell used
Temperature
K.M.F. in B.A.
volts
Correction
to 15
E.M.F. in B.A.
volts corrected
to 15
November 21
Clark No. 1 .
13-7
1-4548
- -0016
1-4532
27
No. 1 .
13-4
1-4555
-0019
1-4536
Mean . .
1-4534
The difference between 14534 and the mean of Table XL, viz., 1'4542,
would indicate a fall of about ^W- but the determinations are hardly precise
enough to warrant us in regarding this fall as an established fact.
Note 2 to 37.
Two further determinations of the E.M.F. of Clark cells have been
published since this paper was communicated to the Royal Society. They
both depend upon the evaluation of currents by means of silver, as in
38.
A. v. Ettingshausen (Zeitschrift fur Elektrotechnik, 1884, xvi. Heft)
finds at 15 0- 5 the value 1*433 volt, using Kohlrausch's (second) value of
the electro-chemical equivalent.
Again (Amer. Journ. Sci., Nov., 1884) Mr Carhart obtains T434 volt.
This appears to correspond to a temperature of 18.
These results are satisfactory as tending to show that Clark cells may
be set up in different places and by different hands so as to give nearly
identical E.M.F.
113.
PRESIDENTIAL ADDRESS.
[British Associatio* Report, pp. 123. Momtrral, 1&S4.]
LADIES AND GEXTLEMEX
IT is no ordinary meeting of the British Association which
I have now the honour of addressing. For more than fifty years the Asso-
ciation has held its autumn gathering in various towns of the Unlnei
Kingdom, and within those limits there is, I suppose, no place of im-
portance which we hare not visited. And now r not satisfied with past
successes, we are seeking new worlds to conquer. When it was fet
proposed to visit Canada, there were some who viewed the project wl-h
hesitation. For my own part, I never quite understood the gn:>un<-ls of
their apprehension. Perhaps they feared the thin edge of the we*ige.
When once the principle was admitted, there was no knowing to* what
it might lead. So rapid is the development of the British Empire, tfeaS
the time might come when a visit to such out-of-the-way places as
London or Manchester could no longer be claimed as a right, but only
asked for as a concession to the susceptibilities of the English. Bat
seriously, whatever objections may have at first been felt soon were out-
weighed by the consideration of the magnificent opportunities which your
hospitality affords of extending the sphere of our influence and of be-
coming acquainted with a part of the Queen's dominion which, associated
with splendid memories of the past, is advancing daily by leaps and
bounds to a position of importance such as not long ago was scarcely
dreamed of. For myself. I am not a stranger to your shores. I re-
member well the impression made upon me, seventeen years ago, by
the wild rapids of the St Lawrence, and the gloomy grandeur of the
Saguenay. If anything impressed me more, it was the kindness with
which I was received by yourselves, and which I doubt not will be again
334 PRESIDENTIAL ADDRESS. [113
extended not merely to myself, but to all the English members of the
Association. I am confident that those who have made up their minds
to cross the ocean will not repent their decision, and that, apart alto-
gether from scientific interests, great advantage may be expected from
this visit. We Englishmen ought to know more than we do of matters
relating to the Colonies, and anything which tends to bring the various
parts of the Empire into closer contact can hardly be overvalued. It is
pleasant to think that this Association is the means of furthering an
object which should be dear to the hearts of all of us; and I venture
to say that a large proportion of the visitors to this country will be
astonished by what they see, and will carry home an impression which
time will not readily efface.
To be connected with this meeting is, to me, a great honour, but
also a great responsibility. In one respect, especially, I feel that the
Association might have done well to choose another President. My own
tastes have led me to study mathematics and physics rather than geology
and biology, to which naturally more attention turns in a new country,
presenting as it does a fresh field for investigation. A chronicle of achieve-
ments in these departments by workers from among yourselves would have
been suitable to the occasion, but could not come from me. If you would
have preferred a different subject for this address, I hope, at least, that you
will not hold me entirely responsible.
At annual gatherings like ours the pleasure with which friends meet
friends again is sadly marred by the absence of those who can never more
take their part in our proceedings. Last year my predecessor in this
office had to lament the untimely loss of Spottiswoode and Henry Smith,
dear friends of many of us, and prominent members of our Associa-
tion. And now, again, a well-known form is missing. For many years
Sir W. Siemens has been a regular attendant at our meetings, and to few
indeed have they been more indebted for success. Whatever the occasion,
in his Presidential Address of two years ago, or in communications to
the Physical and Mechanical Sections, he had always new and interesting
ideas, put forward in language which a child could understand, so great
a master was he of the art of lucid statement in his adopted tongue.
Practice with Science was his motto. Deeply engaged in industry, and
conversant, all his life, with engineering operations, his opinion was never
that of a mere theorist. On the other hand, he abhorred rule of thumb,
striving always to master the scientific principles which underlie rational
design and invention.
It is not necessary that I should review in detail the work of Siemens.
The part which he took, during recent years, in the development of the
dynamo machine must be known to many of you. We owe to him the
113]
practical adoption of the method, first
into a shunt the eoik of the field magnets, by which a greatly
steadiness of action is obtained The same characteristics are observable
throughout a definite object in view and a well-directed perseverance in
overcoming the difficulties by which the path is usually obstructed.
These are, indeed, the conditions of successful invention, The world
knows little of such things, and regards the new machine or the new
method as the immediate outcome of a happy idea. Probably, if the truth
were known, we should see that, in nine cases out of ten,
as much upon good judgment and perseverance as upon fertility of
nation, The labours of our great inventors are not unappreciated^ but
I doubt whether we adequately realise the enormous obligations under
which we fie. It is no exaggeration to say that the fife of such a man
as Siemens is spent in the public service: the advantages which he reaps
for himself being as nothing in comparison with those which be
upon the community at large.
As an example of tins it will be sufficient to mention one of the
valuable achievements of his active fife his mtaodnction, in
with his brother, of the Regenerative Gas Furnace, by which am
economy of fuel (estimated at millions of tons annually ) has been
in the manufacture of steel and glass. The nature of this eooawmiy
easily explained. Whatever may be the work to be done by the burni
of fuel, a certain temperettmrtt is necessary. For example, n> a-;:.
heat in the form of boiling water, would be of any avail for the
of steel When the products of combustion are cooled down to I
the heat which they still contain is msefess as regards the
view. The importance of this consideration depends enanijne IT
upon the working temperature. If the object be the evajwicad^ffii of water
or the warming of a house, almost all the heat may be extracted from
the fuel without special arrangements. Bat it is otherwise when the
temperature required is not much below that of combustion itself; for
then the escaping gases carry away with them the larger part of the
whole heat developed. It was to meet this difficulty that the regene-
rative furnace was (devised. The products of combustion, before iania|
into the chimney, are caused to pass through piles of taoady stacked
fire-brick, to which they give up their heal After a time the fire-brick,
upon which the gases first impinge, becomes nearly as hot as the fnruace
By suitable valves the burnt gases are then diverted through
stack of brickwork, which they heat up in fike manner, while
the heat stored up in the first stack' is utilised to warm the unburnt
gas and air on their way to the furnace. In this way almost all the
heat developed at a high temperature during the combustion
available for the work in
336 PRESIDENTIAL ADDRESS. [113
As it is now several years since your presidential chair has been occu-
pied by a professed physicist, it may naturally be expected that I should
attempt some record of recent progress in that branch of science, if indeed
such a term be applicable. For it is one of the difficulties of the task
that subjects as distinct as Mechanics, Electricity, Heat, Optics and
Acoustics, to say nothing of Astronomy and Meteorology, are included
under Physics. Any one of these may well occupy the life-long attention
of a man of science, and to be thoroughly conversant with all of them
is more than can be expected of any one individual, and is probably
incompatible with the devotion of much time and energy to the actual
advancement of knowledge. Not that I would complain of the association
sanctioned by common parlance. A sound knowledge of at least the prin-
ciples of general physics is necessary to the cultivation of any department.
The predominance of the sense of sight as the medium of communication
with the outer world, brings with it dependence upon the science of optics ;
and there is hardly a branch of science in which the effects of temperature
have not (often without much success) to be reckoned with. Besides, the
neglected borderland between two branches of knowledge is often that
which best repays cultivation, or, to use a metaphor of Maxwell's, the
greatest benefits may be derived from a cross fertilisation of the sciences.
The wealth of material is an evil only from the point of view of one of
whom too much may be expected. Another difficulty incident to the task,
which must be faced but cannot be overcome, is that of estimating rightly
the value, and even the correctness, of recent work. It is not always that
which seems at first the most important that proves in the end to be so.
The history of science teems with examples of discoveries which attracted
little notice at the time, but afterwards have taken root downwards and
borne much fruit upwards.
One of the most striking advances of recent years is in the produc-
tion and application of electricity upon a large scale a subject to which
I have already had occasion to allude in connection with the work of
Sir W. Siemens. The dynamo machine is indeed founded upon discoveries
of Faraday now more than half a century old ; but it has required the
protracted labours of many inventors to bring it to its present high degree
of efficiency. Looking back at the matter, it seems strange that progress
should have been so slow. I do not refer to details of design, the elabo-
ration of which must always, I suppose, require the experience of actual
work to indicate what parts are structurally weaker than they should be,
or are exposed to undue wear and tear. But with regard to the main
features of the problem, it would almost seem as if the difficulty lay in
want of faith. Long ago it was recognised that electricity derived from
chemical action is (on a large scale) too expensive a source of mechanical
power, notwithstanding the fact that (as proved by Joule in 1846) the
113] PBESIDEXTIAL ADDRESS. ;. 7
conversion of electrical into mechanical work can be effected with great
economy. From this it is an evident consequence that electrimv mav
advantageously be obtained from mechanical power; and one cannot help
thinking that if the fact had been borne steadily in mind, the development
of the dynamo might have been much mote rapid. Bat discoveries and
inventions are apt to appear obvious when regarded from the fl*iw?|HHFrt-
of accomplished fact; and I draw attention to the matter only to point
the moral that we do well to posh the attack persistently when we can
be sure beforehand that the obstacles to be overcome are only difficulties
of contrivance, and that we are not vainly fighting unawares against a law
: y - :
The present development of electricity on a large scale depends, how-
ever, almost as much upon the incandescent lamp as upon the dynamo.
The success of these lamps demands a very perfect vacuum not more
than about one-millionth of the normal quantity of air should remain.
and it is interesting to recall that, twenty years ago, such vacua were
rare even in the laboratory of the physicist. It is pretty safe to sav rLi:
these wonderful results would never have been accomplished had practical
applications alone been in view. The way was prepared by an army of
scientific men whose main object was the 'advancement of knowledge. ac*i
who could scarcely have imagined that the processes which they elaborated
would soon be in use on a commercial scale and entrusted to the hands
of ordinary workmen.
When I speak in hopeful language of practical electricity. I '!:- no:
forget the disappointment within the last year or two of many ->ver-
sanguine expectations. The enthusiasm of the inventor and promoter are
ni!wmj to progress, and it seems to be almost a law of nature that it
should overpass the bounds marked out by reason and experience. What
is most to be regretted is the advantage taken by speculators of the
often uninstructed interest felt by the public in novel schemes by which
its imagination is fired. But looking forward to the future of electric
lighting, we have good ground for encouragement. Already the lighting; of
large passenger ships is an assured success, and one which will be highly
appreciated by those travellers who have experienced the tedium of long
winter evenings unrelieved by adequate illumination. Here, no doubt* the
conditions are in many respects especially favourable. As regards space,
life on board ship is highly concentrated : while unity of management and
the presence on the spot of skilled engineers obviate some of the difficul-
ties that are met with under other circumstances. At present we have
no experience of a house-to-house system of illumination on a great scale
and in competition with cheap gas: but preparations are already far
advanced for trial on an adequate scale in London. In large institutions,
338 PRESIDENTIAL ADDRESS. [113
such as theatres and factories, we all know that electricity is in successful
and daily extending operation.
When the necessary power can be obtained from the fall of water,
instead of from the combustion of coal, the conditions of the problem are
far more favourable. Possibly the severity of your winters may prove an
obstacle, but it is impossible to regard your splendid river without the
thought arising that the day may come when the vast powers now running
to waste shall be bent into your service. Such a project demands of course
the most careful consideration, but it is one worthy of an intelligent and
enterprising community.
The requirements of practice react in the most healthy manner upon
scientific electricity. Just as in former days the science received a stimu-
lus from the application to telegraphy, under which everything relating to
measurement on a small scale acquired an importance and development
for which we might otherwise have had long to wait, so now the require-
ments of electric lighting are giving rise to a new development of the
art of measurement upon a large scale, which cannot fail to prove of
scientific as well as practical importance. Mere change of scale may not
at first appear a very important matter, but it is surprising how much
modification it entails in the instruments, and in the processes of measure-
ment. For instance, the resistance coils on which the electrician relies in
dealing with currents whose maximum is a fraction of an ampere, fail
altogether when it becomes a question of hundreds, not to say thousands,
of amperes.
The powerful currents, which are now at command, constitute almost
a new weapon in the hands of the physicist. Effects, which in old days
were rare and difficult of observation, may now be produced at will on the
most conspicuous scale. Consider for a moment Faraday's great discovery
of the ' Magnetisation of Light,' which Tyndall likens to the Weisshorn
among mountains, as high, beautiful, and alone. This judgment (in which
I fully concur) relates to the scientific aspect of the discovery, for to the
eye of sense nothing could have been more insignificant. It is even
possible that it might have eluded altogether the penetration of Faraday,
had he not been provided with a special quality of very heavy glass. At
the present day these effects may be produced upon a scale that would
have delighted their discoverer, a rotation of the plane of polarization
through 180 being perfectly feasible. With the aid of modern appli-
ances, Kundt and Rontgen in Germany, and H. Becquerel in France,
have detected the rotation in gases and vapours, where, on account of its
extreme smallness, it had previously escaped notice.
Again, the question of the magnetic saturation of iron has now an
importance entirely beyond what it possessed at the time of Joule's early
113] PRESIDENTIAL ADDRESS. 339
observations. Then it required special arrangements purposely contrived
to bring it into prominence. Now in every dynamo machine, the iron of
the field-magnets approaches a state of saturation, and the very elements
of an explanation of the action require us to take the fact into account.
It is indeed probable that a better knowledge of this subject might lead
to improvements in the design of these machines.
Notwithstanding the important work of Rowland and Stoletow, the whole
theory of the behaviour of soft iron under varying magnetic conditions is
still somewhat obscure. Much may be hoped from the induction balance
of Hughes, by which the marvellous powers of the telephone are applied
to the discrimination of the properties of metals, as regards magnetism
and electric conductivity.
The introduction of powerful alternate-current in machines by Siemens,
Gordon, Ferranti, and others, is likely also to have a salutary effect in
educating those so-called practical electricians whose ideas do not easily
rise above ohms and volts. It has long been known that when the changes
are sufficiently rapid, the phenomena are governed much more by induc-
tion, or electric inertia, than by mere resistance. On this principle much
may be explained that would otherwise seem paradoxical. To take a
comparatively simple case, conceive an electro-magnet wound with two
contiguous wires, upon which acts a given rapidly periodic electro-motive
force. If one wire only be used, a certain amount of heat is developed in
the circuit. Suppose now that the second wire is brought into operation
in parallel a proceeding equivalent to doubling the section of the original
wire. An electrician accustomed only to constant currents would be sure
to think that the heating effect would be doubled by the change, as much
heat being developed in each wire separately as was at first in the single
wire. But such a conclusion would be entirely erroneous. The total
current, being governed practically by the self-induction of the circuit,
would not be augmented by the accession of the second wire, and the total
heating effect, so far from being doubled, would, in virtue of the superior
conductivity, be halved.
During the last few years much interest has been felt in the reduction
to an absolute standard of measurements of electro-motive force, current,
resistance, etc., and to this end many laborious investigations have been
undertaken. The subject is one that has engaged a good deal of my own
attention, and I should naturally have felt inclined to dilate upon it, but
that I feel it to be too abstruse and special to be dealt with in detail
upon an occasion like the present. As regards resistance, I will merely
remind you that the recent determinations have shown a so greatly im-
proved agreement, that the Conference of Electricians assembled at Paris,
in May, have felt themselves justified in defining the ohm for practical
222
340 PRESIDENTIAL ADDRESS. [113
use as the resistance of a column of mercury of C., one square milli-
metre in section, and 106 centimetres in length a definition differing by
a little more than one per cent, from that arrived at twenty years ago
by a committee of this Association.
A standard of resistance once determined upon can be embodied in a
'resistance coil/ and copied without much trouble, and with great accu-
racy. But in order to complete the electrical system, a second standard of
some kind is necessary, and this is not so easily embodied in a permanent
form. It might conveniently consist of a standard galvanic cell, capable of
being prepared in a definite manner, whose electro-motive force is once for
all determined. Unfortunately, most of the batteries in ordinary use are for
one reason or another unsuitable for this purpose, but the cell introduced
by Mr Latimer Clark, in which the metals are zinc in contact with satu-
rated zinc sulphate and pure mercury in contact with mercurous sulphate,
appears to give satisfactory results. According to my measurements, the
electro- motive force of this cell is T435 theoretical volts.
We may also conveniently express the second absolute electrical mea-
surement necessary to the completion of the system by taking advantage
of Faraday's law, that the quantity of metal decomposed in an electro-
lytic cell is proportional to the whole quantity of electricity that passes.
The best metal for the purpose is silver, deposited from a solution of the
nitrate or of the chlorate. The results recently obtained by Professor
Kohlrausch and by myself are in very good agreement, and the conclu-
sion that one ampere flowing for one hour decomposes 4'025 grains of
silver, can hardly be in error by more than a thousandth part. This
number being known, the silver voltameter gives a ready and very accu-
rate method of measuring currents of intensity, varying from -^ ampere
to four or five amperes.
The beautiful and mysterious phenomena attending the discharge of
electricity in nearly vacuous spaces have been investigated and in some
degree explained by De La Rue, Crookes, Schuster, Moulton, and the
lamented Spottiswoode, as well as by various able foreign experimenters.
In a recent research Crookes has sought the origin of a bright citron-
coloured band in the phosphorescent spectrum of certain earths, and after
encountering difficulties and anomalies of a most bewildering kind, has
succeeded in proving that it is due to yttrium, an element much more
widely distributed than had been supposed. A conclusion like this is
stated in a few words, but those only who have undergone similar expe-
rience are likely to appreciate the skill and perseverance of which it is
the final reward.
A remarkable observation by Hall of Baltimore, from which it appeared
that the flow of electricity in a conducting sheet was disturbed by mag-
netic force, has been the subject of much iftii Mr Shelford Bklweil
has brought forward experiments tending to prove that the effect is of
a secondary character, due in the first instance to the
operating upon the conductor of an electric current when
powerful magnetic field. Mr Bidwell s view agrees in the
Mr Hall's division of die metals into two groups according to the direc-
tion of the effect.
Without doubt the most important achievement of the older genera-
tion of scientific men has been the ffatahfehment and application of the
great laws of Thermo-djnamics, or, as it is often called, the M-fcmiMl
Theory of Heat. The first law, which asserts that heat and mechanical
work can be transformed one into the other at a certain fixed rate, is now
well understood by every student of physics, and the number expressing
the mechanical equivalent of heat resulting from the experiments or' Jo-ile.
has been confirmed by the researches of others, and especially of Rowland.
But the second law, which practically is even more important than the
first, is only now beginning to receive the rail appreciation due to it.
One reason of this may be found in a not unnatural coefesioc. of :<ie:is.
Words do not always lend themselves readily to the demands that are
made upon them by a gloving science., and I think that she almoisi
unavoidable use of the word equivalent in the statement of :he dr>- law
is partly responsible for the little attention that is given to the se:-i:nL
For the second law so w contradicts the usual statement of the tirs~ r
as to assert that equivalents of heat and work are not of equal val-e.
While work can always be converted into heat,, heat can only be dm versed
into work under certain limitations. For every practical pirpoee the work
is worth the most, and when we speak of equivalents, we use the word
in the same sort of "in*"?* 1 sense as that in which chemists speak of
equivalents of gold and iron. The second law teaches us that the real
value of heat,, as a source of mechanical power,, depends upon the tempera-
ture of the body in which it resides: the hotter the body in relation to
its surroundings, the more available the heat.
In order to see the relations which obtain between the first and the
second law of Tbermo-dynamks, it is only necessary for us to glance at
the theory of the steam-engine. Not many years ago calculations woe
plentiful, demonstrating the inefficiency of the steam-engine on the basis
of a comparison of the work actually got out of the engine with the
equivalent of the heat supplied to the boiler. Such caksJa-
only the first law of Tnermo-dynamics, which deals
with the equivalents of heat and work, and have very hide bearing upon
the practical question of efficiency, which requires us to have regard abo
to the second law. According to that law the fraction of the total energy
342 PRESIDENTIAL ADDRESS. [113
which can be converted into work depends upon the relative temperatures
of the boiler and condenser; and it is, therefore, manifest that, as the
temperature of the boiler cannot be raised indefinitely, it is impossible to
utilise all the energy which, according to the first law of Thermo-dynamics,
is resident in the coal.
On a sounder view of the matter, the efficiency of the steam-engine is
found to be so high, that there is no great margin remaining for improve-
ment. The higher initial temperature possible in the gas-engine opens out
much wider possibilities, and many good judges look forward to a time
when the steam-engine will have to give way to its younger rival.
To return to the theoretical question, we may say with Sir W. Thomson,
that though energy cannot be destroyed, it ever tends to be dissipated,
or to pass from more available to less available forms. No one who has
grasped this principle can fail to recognise its immense importance in the
system of the Universe. Every change chemical, thermal, or mechanical
which takes place, or can take place, in Nature does so, at the cost of
a certain amount of available energy. If, therefore, we wish to inquire
whether or not a proposed transformation can take place, the question
to be considered is whether its occurrence would involve dissipation of
energy. If not, the transformation is (under the circumstances of the
case) absolutely excluded. Some years ago, in a lecture at the Royal
Institution*, I endeavoured to draw the attention of chemists to the im-
portance of the principle of dissipation in relation to their science, pointing
out the error of the usual assumption that a general criterion is to be
found in respect of the development of heat. For example, the solution
of a salt in water is, if I may be allowed the phrase, a downhill trans-
formation. It involves dissipation of energy, and can therefore go forward ;
but in many cases it is associated with the absorption rather than with
the development of heat. I am glad to take advantage of the present
opportunity in order to repeat my recommendation, with an emphasis
justified by actual achievement. The foundations laid by Thomson now
bear an edifice of no mean proportions, thanks to the labours of several
physicists, among whom must be especially mentioned Willard Gibbs and
Helmholtz. The former has elaborated a theory of the equilibrium of
heterogeneous substances, wide in its principles, and we cannot doubt far-
reaching in its consequences. In a series of masterly papers Helmholtz has
developed the conception of free energy with very important applications
to the theory of the galvanic cell. He points out that the mere tendency
to solution bears in some cases no small proportion to the affinities more
usually reckoned chemical, and contributes largely to the total electro-
motive force. Also in our own country Dr Alder Wright has published
some valuable experiments relating to the subject.
* Vol. i. p. 238.
113] PRESIDEXTIJLL ADDRESS. 343
From the further study of electrolysis we may expect to gain improved
views as to the nature of the chemical reactions, and of the forces con-
cerned in bringing them about. I am not qualified I wish I were to
speak to you on recent progress in general chemistry. Perhaps my feelings
towards a first love may blind me, but I cannot help thinking that the
next great advance, of which we have already some foreshadowing, will
come on this side. And if I might without presumption venture a word
of recommendation, it would be in favour of a more minute study of the
simpler chemical phenomena.
Under the head of scientific mechanics it is principally in relation
to fluid motion that advances may be looked for. In speaking upon
this subject I must limit myself almost entirely to experimental work.
Theoretical hydro-dynamics, however important and interesting to the
mathematician, are eminently nnsuited to oral exposition. All I can do
to attenuate an injustice, to which theorists are pretty well accustomed,
is to refer you to the admirable reports of Mr Hicks, published under
the auspices of this Association.
The important and highly practical work of the late Mr Froude in
relation to the propulsion of ships is doubtless known to most of you.
Recognising the fallacy of views then widely held as to the nature of the
resistance to be overcome, he showed to demonstration that, in the case
of fair-shaped bodies, we have to deal almost entirely with resist an ce
dependent upon skin friction, and at high speeds upon the generation
of surface waves by which energy is carried off. At speeds which are
moderate in relation to the size of the ship, the resistance is practically
dependent upon skin friction only. Although Professor Stokes and other
mathematicians had previously published calculations pointing to the same
conclusion, there can be no doubt that the view generally entertained was
very different. At the first meeting of the Association which I ever
attended, as an intelligent listener, at Bath in 1864, I well remember
the surprise which greeted a statement by Rankine that he regarded
skin friction as the only legitimate resistance to the progress of a well-
designed ship. Mr Froude's experiments have set the question at rest
in a manner satisfactory to those who had little confidence in theoretical
prevision.
In speaking of an explanation as satisfactory in which skin friction is
accepted as the cause of resistance, I must guard myself against being
supposed to mean that the nature of skin friction is itself well under-
stood. Although its magnitude varies with the smoothness of the surface,
we have no reason to think that it would disappear at any degree of
smoothness consistent with an ultimate molecular structure. That it is
344 PRESIDENTIAL ADDRESS. [113
connected with fluid viscosity is evident enough, but the modus operandi
is still obscure.
Some important work bearing upon the subject has recently been
published by Professor O. Reynolds, who has investigated the flow of
water in tubes as dependent upon the velocity of motion and upon the
size of the bore. The laws of motion in capillary tubes, discovered
experimentally by Poiseuille, are in complete harmony with theory. The
resistance varies as the velocity, and depends in a direct manner upon
the constant of viscosity. But when we come to the larger pipes and
higher velocities with which engineers usually have to deal, the theory
which presupposes a regularly stratified motion evidently ceases to be
applicable, and the problem becomes essentially identical with that of skin
friction in relation to ship propulsion. Professor Reynolds has traced with
much success the passage from the one state of things to the other,
and has proved the applicability under these complicated conditions of
the general laws of dynamical similarity as adapted to viscous fluids by
Professor Stokes. In spite of the difficulties which beset both the theo-
retical and experimental treatment, we may hope to attain before long to
a better understanding of a subject which is certainly second to none in
scientific as well as practical interest.
As also closely connected with the mechanics of viscous fluids, I must
not forget to mention an important series of experiments upon the friction
of oiled surfaces, recently executed by Mr Tower for the Institution of
Mechanical Engineers. The results go far towards upsetting some ideas
hitherto widely admitted. When the lubrication is adequate, the friction
is found to be nearly independent of the load, and much smaller than is
usually supposed, giving a coefficient as low as j^Vs- When the layer of
oil is well formed, the pressure between the solid surfaces is really borne
by the fluid, and the work lost is spent in shearing, that is, in causing
one stratum of the oil to glide over another.
In order to maintain its position, the fluid must possess a certain degree
of viscosity, proportionate to the pressure ; and even when this condition is
satisfied, it would appear to be necessary that the layer should be thicker
on the ingoing than on the outgoing side. We may, I believe, expect from
Professor Stokes a further elucidation of the processes involved. In the
meantime, it is obvious that the results already obtained are of the utmost
value, and fully justify the action of the Institution in devoting a part of
its resources to experimental work. We may hope indeed that the example
thus wisely set may be followed by other public bodies associated with
various departments of industry.
I can do little more than refer to the interesting observations of
Prof. Darwin, Mr Hunt, and M. Forel on Ripplemark. The processes
113] PRESIDENTIAL ADDRESS. 345
concerned would seem to be of a rather intricate character, and largely
dependent upon fluid viscosity. It may be noted indeed that most of the
still obscure phenomena of hydro-dynamics require for their elucidation a
better comprehension of the laws of viscous motion. The subject is one
which offers peculiar difficulties. In some problems in which I have lately
been interested, a circulating motion presents itself of the kind which the
mathematician excludes from the first when he is treating of fluids desti-
tute altogether of viscosity. The intensity of this motion proves, however,
to be independent of the coefficient of viscosity, so that it cannot be cor-
rectly dismissed from consideration as a consequence of a supposition that
the viscosity is infinitely small. The apparent breach of continuity can
be explained, but it, shows how much care is needful in dealing with the
subject, and how easy it is to fall into error.
The nature of gaseous viscosity, as due to the diffusion of momentum,
has been made clear by the theoretical and experimental researches of
Maxwell A flat disc moving in its own plane between two parallel solid
surfaces is impeded by the necessity of shearing the intervening layers of
gas, and the magnitude of the hindrance is proportional to the velocity
of the motion and to the viscosity of the gas, so that under similar cir-
cumstances this effect may be taken as a measure, or rather definition, of
the viscosity. From the dynamical theory of gases, to the development
of which he contributed so much, Maxwell drew the startling conclusion
that the viscosity of a gas should be independent of its density, that
within wide limits the resistance to the moving disc should be scarcely
diminished by pumping out the gas, so as to form a partial vacuum.
Experiment fully confirmed this theoretical anticipation, one of the most
remarkable to be found in the whole history of science, and proved that
the swinging disc was retarded by the gas, as much when the barometer
stood at half an inch as when it stood at thirty inches. It was obvious,
of course, that the law must have a limit, that at a certain point of
exhaustion the gas must begin to lose its power; and I remember dis-
cussing with Maxwell, soon after the publication of his experiments, the
whereabouts of the point at which the gas would cease to produce its
ordinary effect. His apparatus, however, was quite unsuited for high
degrees of exhaustion, and the failure of the law was first observed
by Kundt and Warburg, as pressures below 1 mm. of mercury. Subse-
quently the matter has been thoroughly examined by Crookes, who ex-
tended his observations to the highest degrees of exhaustion as measured
by MacLeod's gauge. Perhaps the most remarkable results relate to
hydrogen. From the atmospheric pressure of 760 mm. down to about
^ mm. of mercury the viscosity is sensibly constant. From this point
to the highest vacua, in which less than one-millionth of the original
gas remains, the coefficient of viscosity drops down gradually to a small
346 PRESIDENTIAL ADDRESS. [113
fraction of its original value. In these vacua Mr Crookes regards the
gas as having assumed a different, ultra-gaseous, condition ; but we must
remember that the phenomena have relation to the other circumstances
of the case, especially the dimensions of the vessel, as well as to the
condition of the gas.
Such an achievement as the prediction of Maxwell's law of viscosity has,
of course, drawn increased attention to the dynamical theory of gases. The
success which has attended the theory in the hands of Clausius, Maxwell,
Boltzmann and other mathematicians, not only in relation to viscosity, but
over a large part of the entire field of our knowledge of gases, proves that
some of its fundamental postulates are in harmony with the reality of
Nature. At the same time, it presents serious difficulties; and we cannot
but feel that while the electrical and optical properties of gases remain
out of relation to the theory, no final judgment is possible. The growth
of experimental knowledge may be trusted to clear up many doubtful
points, and a younger generation of theorists will bring to bear improved
mathematical weapons. In the meantime we may fairly congratulate our-
selves on the possession of a guide which has already conducted us to a
position which could hardly otherwise have been attained.
In Optics attention has naturally centred upon the spectrum. The
mystery attaching to the invisible rays lying beyond the red has been
fathomed to an extent that, a few years ago, would have seemed almost
impossible. By the use of special photographic methods Abuey has mapped
out the peculiarities of this region with such success that our knowledge
of it begins to be comparable with that of the parts visible to the eye.
Equally important work has been done by Langley, using a refined inven-
tion of his own based upon the principle of Siemens' pyrometer. This
instrument measures the actual energy of the radiation, and thus ex-
presses the effects of various parts of the spectrum upon a common scale,
independent of the properties of the eye and of sensitive photographic
preparations. Interesting results have also been obtained by Becquerel,
whose method is founded upon a curious action of the ultra-red rays in
enfeebling the light emitted by phosphorescent substances. One of the
most startling of Langley 's conclusions relates to the influence of the
atmosphere in modifying the quality of solar light. By the comparison
of observations made through varying thicknesses of air, he shows that the
atmospheric absorption tells most upon the light of high refrangibility ; so
that, to an eye situated outside the atmosphere, the sun would present a
decidedly bluish tint. It would be interesting to compare the experimental
numbers with the law of scattering of light by small particles given some
years ago as the result of theory*. The demonstration by Langley of the
* Conf. vol. i. p. 95.
113] PRESIDENTIAL ADDRESS. 347
inadequacy of Cauchy's law of dispersion to represent the relation between
refraugibility and wave-length in the lower part of the spectrum must have
an important bearing upon optical theory.
The investigation of- the relation of the visible and ultra-violet spectrum
to various forms of matter has occupied the attention of a host of able
workers, among whom none have been more successful than my colleagues
at Cambridge, Professors Liveing and Dewar. The subject is too large
both for the occasion and for the individual, and I must pass it by. But,
as more closely related to Optics proper, I cannot resist recalling to your
notice a beautiful application of the idea of Doppler to the discrimina-
tion of the origin of certain lines observed in the solar spectrum. If a
vibrating body have a general motion of approach or recession, the waves
emitted from it reach the observer with a frequency which in the first
case exceeds, and in the second case falls short of, the real frequency of
the vibrations themselves. The consequence is that, if a glowing gas be
in motion in the line of sight, the spectral lines are thereby displaced
from the position that they would occupy were the gas at rest u principle
which, in the hands of Huggins and others, has led to a determination of
the motion of certain fixed stars relatively to the solar system. But the
sun is itself in rotation, and thus the position of a solar spectral line is
slightly different according as the light comes from the advancing or from
the retreating limb. This displacement was, I believe, first observed by
Thollon; but what I desire now to draw attention to is the application
of it by Cornu to determine whether a line is of solar or atmospheric
origin. For this purpose a small image of the sun is thrown upon the slit
of the spectroscope, and caused to vibrate two or three times a second, in
such a manner that the light entering the instrument comes alternately
from the advancing and retreating limbs. Under these circumstances a
line due to absorption within the sun appears to tremble, as the result of
slight alternately opposite displacements. But if the seat of the absorp-
tion be in the atmosphere, it is a matter of indifference from what part of
the sun the light originally proceeds, and the line maintains its position
in spite of the oscillation of the image upon the slit of the spectroscope.
In this way Cornu was able to make a discrimination which can only
otherwise be effected by a difficult comparison of appearances under
various solar altitudes.
The instrumental weapon of investigation, the spectroscope itself, has
made important advances. On the theoretical side, we have for our guid-
ance the law that the optical power in gratings is proportional to the total
number of lines accurately ruled, without regard to the degree of closeness,
and in prisms that it is proportional to the thickness of glass traversed.
The magnificent gratings of Rowland are a new power in the hands of the
spectroscopist, and as triumphs of mechanical art seem to be little short
348 PRESIDENTIAL ADDRESS. [113
of perfection. In our own report for 1882, Mr Mallock has described a
machine, constructed by him, for ruling large diffraction gratings, similar
in some respects to that of Rowland.
The great optical constant, the velocity of light, has been the subject
of three distinct investigations by Cornu, Michelson, and Forbes. As may
be supposed, the matter is of no ordinary difficulty, and it is therefore not
surprising that the agreement should be less decided than could be wished.
From their observations, which were made by a modification of Fizeau's
method of the toothed wheel, Young and Forbes drew the conclusion that
the velocity of light in vacuo varies from colour to colour, to such an extent
that the velocity of blue light is nearly two per cent, greater than that of
red light. Such a variation is quite opposed to existing theoretical notions,
and could only be accepted on the strongest evidence. Mr Michelson, whose
method (that of Foucault) is well suited to bring into prominence a varia-
tion of velocity with wave-length, informs me that he has recently repeated
his experiments with special reference to the point in question, and has
arrived at the conclusion that no variation exists comparable with that
asserted by Young and Forbes. The actual velocity differs little from
that found from his first series of experiments, and may be taken to be
299,800 kilometres per second.
It is remarkable how many of the playthings of our childhood give
rise to questions of the deepest scientific interest. The top is, or may be
understood, but a complete comprehension of the kite and of the soap-
bubble would carry us far beyond our present stage of knowledge. In
spite of the admirable investigations of Plateau, it still remains a mystery
why soapy water stands almost alone among fluids as a material for
bubbles. The beautiful development of colour was long ago ascribed to
the interference of light, called into play by the gradual thinning of the
film. In accordance with this view the tint is determined solely by the
thickness of the film, and the refractive index of the fluid. Some of the
phenomena are however so curious, as to have led excellent observers like
Brewster to reject the theory of thin plates, and to assume the secretion
of various kinds of colouring matter. If the rim of a wine-glass be dipped
in soapy water, and then held in a vertical position, horizontal bands soon
begin to show at the top of the film, and extend themselves gradually,
downwards. According to Brewster these bands are not formed by the
'subsidence and gradual thinning of the film,' because they maintain
their horizontal position when the glass is turned round its axis. The
experiment is both easy and interesting; but the conclusion drawn from
it cannot be accepted. The fact is that the various parts of the film
cannot quickly alter their thickness, and hence when the glass is rotated
they re-arrange themselves in order of superficial density, the thinner
113] PRESIDENTIAL ADDRESS. 349
parts floating up over, or through, the thicker parts. Only thus can the
tendency be satisfied for the centre of gravity to assume the lowest
possible position.
When the thickness of a film falls below a small fraction of the
length of a wave of light, the colour disappears and is replaced by an
intense blackness. Professors Remold and Riicker have recently made
the remarkable observation that the whole of the black region, soon after
its formation, is of uniform thickness, the passage from the black to the
coloured portions being exceedingly abrupt. By two independent methods
they have determined the thickness of the black film to lie between seven
and fourteen millionths of a millimetre; so that the thinnest films corre-
spond to about one-seventieth of a wave-length of light. The importance
of these results in regard to molecular theory is too obvious to be insisted
upon.
The beautiful inventions of the telephone and the phonograph, although
in the main dependent upon principles long since established, have imparted
a new interest to the study of Acoustics. The former, apart from its uses
in every-day life, has become in the hands of its inventor, Graham Bell,
and of Hughes, an instrument of first-class scientific importance. The
theory of its action is still in some respects obscure, as is shown by the
comparative failure of the many attempts to improve it. In connection
with some explanations that have been offered, we do well to remember
that molecular changes in solid masses are inaudible in themselves, and
can only be manifested to our ears by the generation of a to and fro
motion of the external surface extending over a sensible area. If the
surface of a solid remains undisturbed, our ears can tell us nothing of
what goes on in the interior.
In theoretical acoustics progress has been steadily maintained, and
many phenomena, which were obscure twenty or thirty years ago, have
since received adequate explanation. If some important practical ques-
tions remain unsolved, one reason is that they have not yet been definitely
stated. Almost everything in connection with the ordinary use of our
senses presents peculiar difficulties to scientific investigation. Some kinds
of information with regard to their surroundings are of such paramount
importance to successive generations of living beings, that they have
learned to interpret indications which, from a physical point of view, are
of the slenderest character. Every day we are in the habit of recog-
nising, without much difficulty, the quarter from which a sound proceeds,
but by what steps we attain that end has not yet been satisfactorily
explained. It has been proved* that when proper precautions are taken
we are unable to distinguish whether a pure tone (as from a vibrating
* Cont Tol. i. pp. 277, 314.
350 PRESIDENTIAL ADDRESS. [113
tuning-fork held over a suitable resonator) comes to us from in front or
from behind. This is what might have been expected from an a priori
point of view ; but what would not have been expected is that with
almost any other sort of sound, from a clap of the hands to the clearest
vowel sound, the discrimination is not only possible but easy and in-
stinctive. In these cases it does not appear how the possession of two
ears helps us, though there is some evidence that it does; and even
when sounds come to us from the right or left, the explanation of the
ready discrimination which is then possible with pure tones, is not so
easy as might at first appear. We should be inclined to think that the
sound was heard much more loudly with the ear that is turned towards
than with the ear that is turned from it, and that in this way the direc-
tion was recognised. But if we try the experiment, we find that, at any
rate with notes near the middle of the musical scale, the difference of
loudness is by no means so very great. The wave-lengths of such notes
are long enough in relation to the dimensions of the head to forbid the
formation of anything like a sound shadow in which the averted ear might
be sheltered.
In concluding this imperfect survey of recent progress in physics,
I must warn you emphatically that much of great importance has been
passed over altogether. I should have liked to speak to you of those far-
reaching speculations, especially associated with the name of Maxwell, in
which light is regarded as a disturbance in an electro-magnetic medium.
Indeed, at one time, I had thought of taking the scientific work of Maxwell
as the principal theme of this address. But, like most men of genius,
Maxwell delighted in questions too obscure and difficult for hasty treat-
ment, and thus much of his work could hardly be considered upon such
an occasion as the present. His biography has recently been published,
and should be read by all who are interested in science and in scientific
men. His many-sided character, the quaintness of his humour, the pene-
tration of his intellect, his simple but deep religious feeling, the affection
between son and father, the devotion of husband to wife, all combine to
form a rare and fascinating picture. To estimate rightly his influence
upon the present state of science, we must regard not only the work
that he executed himself, important as that was, but also the ideas and
the spirit which he communicated to others. Speaking for myself as one
who in a special sense entered into his labours, I should find it difficult to
express adequately my feeling of obligation. The impress of his thoughts
may be recognised in much of the best work of the present time. As a
teacher and examiner he was well acquainted with the almost universal
tendency of uninstructed minds to elevate phrases above things : to refer,
for example, to the principle of the conservation of energy for an explaua-
113] PRESIDENTIAL ADDRESS. 351
tion of the persistent rotation of a fly-wheel, almost in the style of the
doctor in Le Malade Imaginaire, who explains the fact that opium sends
you to sleep by its soporific virtue. Maxwell's endeavour was always to
keep the facts in the foreground, and to his influence, in conjunction
with that of Thomson and Helmholtz, is largely due that elimination of
unnecessary hypothesis which is one of the distinguishing characteristics
of the science of the present day.
In speaking unfavourably of superfluous hypothesis, let me not be
misunderstood. Science is nothing without generalisations. Detached and
ill-assorted facts are only raw material, and in the absence of a theo-
retical solvent, have but little nutritive value. At the present time and
in some departments, the accumulation of material is so rapid that there
is danger of indigestion. By a fiction as remarkable as any to be found
in law, what has once been published, even though it be in the Russian
language, is usually spoken of as 'known,' and it is often forgotten that
the rediscovery in the library may be a more difficult and uncertain
process than the first discovery in the laboratory. In this matter we
are greatly dependent upon annual reports and abstracts, issued prin-
cipally in Germany, without which the search for the discoveries of a
little-known author would be well-nigh hopeless. Much useful work has
been done in this direction in connection with our Association. Such
critical reports as those upon Hydro-dynamics, upon Tides, and upon
Spectroscopy, guide the investigator to the points most requiring atten-
tion, and in discussing past achievements contribute in no small degree
to future progress. But though good work has been done, much yet
to do.
If, as is sometimes supposed, science consisted in nothing but the
laborious accumulation of facts, it would soon come to a stand-still,
crushed, as it were, under its own weight. The suggestion of a new
idea, or the detection of a law, supersedes much that had previously been
a burden upon the memory, and by introducing order and coherence facili-
tates the retention of the remainder in an available form. Those who are
acquainted with the writings of the older electricians will understand my
meaning when I instance the discovery of Ohm's law as a step by which
the science was rendered easier to understand and to remember. Two
processes are thus at work side by side, the reception of new material and
the digestion and assimilation of the old ; and as both are essential, we may
spare ourselves the discussion of their relative importance. One remark,
however, should be made. The work which deserves, but I am afraid
does not always receive, the most credit is that in which discovery and
explanation go hand in hand, in which not only are new facts presented,
but their relation to old ones is pointed out.
352 PRESIDENTIAL ADDRESS. [113
In making oneself acquainted with what has been done in any subject,
it is good policy to consult first the writers of highest general reputation.
Although in scientific matters we should aim at independent judgment,
and not rely too much upon authority, it remains true that a good deal
must often be taken upon trust. Occasionally an observation is so simple
and easily repeated, that it scarcely matters from whom it proceeds ; but
as a rule it can hardly carry full weight when put forward by a novice
whose care and judgment there has been no opportunity of testing, and
whose irresponsibility may tempt him to ' take shots,' as it is called.
Those who have had experience in accurate work know how easy it would
be to save time and trouble by omitting precautions and passing over
discrepancies, and yet, even without dishonest intention, to convey the im-
pression of conscientious attention to details. Although the most careful
and experienced cannot hope to escape occasional mistakes, the effective
value of this kind of work depends much upon the reputation of the
individual responsible for it.
In estimating the present position and prospects of experimental science,
there is good ground for encouragement. The multiplication of laboratories
gives to the younger generation opportunities such as have never existed
before, and which excite the envy of those who have had to learn in middle
life much that now forms part of an undergraduate course. As to the
management of such institutions there is room for a healthy difference of
opinion. For many kinds of original work, especially in connection with
accurate measurement, there is need of expensive apparatus ; and it is
often difficult to persuade a student to do his best with imperfect ap-
pliances when he knows that by other means a better result could be
attained with greater facility. Nevertheless it seems to me important to
discourage too great reliance upon the instrument maker. Much of the
best original work has been done with the homeliest appliances ; and the
endeavour to turn to the best account the means that may be at hand
develops ingenuity and resource more than the most elaborate determina-
tions with ready-made instruments. There is danger otherwise that the
experimental education of a plodding student should be too mechanical
and artificial, so that he is puzzled by small changes of apparatus much
as many school-boys are puzzled by a transposition of the letters in a
diagram of Euclid.
From the general spread of a more scientific education, we are war-
ranted in expecting important results. Just as there are some brilliant
literary men with an inability, or at least a distaste practically amounting
to inability, for scientific ideas, so there are a few with scientific tastes
whose imaginations are never touched by merely literary studies. To
save these from intellectual stagnation during several important years
113] PRESIDENTIAL ADDRESS. 353
of their lives is something gained: bat the thorough-going advocates of
scientific education aim at much more. To them it appears strange, and
almost monstrous, that the dead languages should hold the place thev do
in general education ; and it can hardly be denied that their supremacy is
the result of routine rather than of argument. I do not, mvself, take up
the extreme position. I doubt whether an exclusively scientific training
would be satisfactory: and where there is plenty of time and a literarv
aptitude I can believe that Latin and Greek mav make a good founda-
tion. But it is useless to discuss the question upon the supposition that
the majority of boys attain either to a knowledge of the languages or to
an appreciation of the writings of the ancient authors. The contrary is
notoriously the truth: and the defenders of the existing system usually
take their stand upon the excellence of its discipline. From this point
of view there is something to be said The laziest boy must exert him-
self a little in puzzling out a sentence with grammar and dictionary,
while instruction and supervision are easy to organise and not too costlv.
But when the case is stated plainly, few will agree that we can afford
so entirely to disregard results. In after life the intellectual energies
are usually engrossed with business, and no further opportunity is found
for attacking the difficulties which block the gateways of knowledge.
Mathematics, especially, if not learned young, are likely to remain un-
learned. I will not further insist upon the educational importance of
mathematics and science, because with respect to them I shall probably
be supposed to be prejudiced. But of modern languages I am ignorant
enough to give value to my advocacy. I believe that French and German,
if properly taught, which I admit they rarely are at present, would go
far to replace Latin and Greek from a disciplinary point of view, while
the actual value of the acquisition would, in the majority of cases, be
incomparably greater. In half the time usually devoted, without success,
to the classical languages, most boys could acquire a really serviceable
knowledge of French and German. History and the serious study of
English literature, now shamefully neglected, would also find a place in
such a scheme.
There is one objection often felt to a modernised education, as to
which a word may not be without use. Many excellent people are afraid
of science as tending towards materialism. That such apprehension should
exist is not surprising, for unfortunately there are writers, speaking in the
name of science, who have set themselves to foster it. It is true that
among scientific men. as in other classes, crude views are to be met
with as to the deeper things of Nature; but that the life-long beliefs of
Newton, of Faraday, and of Maxwell, are inconsistent with the scientific
habit of mind, is surely a proposition which I need not pause to refute.
It would be easy, however, to lay too much stress upon the opinions of
. n. 23
354 PRESIDENTIAL ADDRESS. [113
even such distinguished workers as these. Men, who devote their lives to
investigation, cultivate a love of truth for its own sake, and endeavour
instinctively to clear up, and not, as is too often the object in business
and politics, to obscure a difficult question. So far the opinion of a scien-
tific worker may have a special value ; but I do not think that he has a
claim, superior to that of other educated men, to assume the attitude of
a prophet. In his heart he knows that underneath the theories that he
constructs there lie contradictions which he cannot reconcile. The higher
mysteries of being, if penetrable at all by human intellect, require other
weapons than those of calculation and experiment.
Without encroaching upon grounds appertaining to the theologian and
the philosopher, the domain of natural science is surely broad enough to
satisfy the wildest ambition of its devotees. In other departments of
human life and interest, true progress is rather an article of faith than
a rational belief; but in science a retrograde movement is, from the
nature of the case, almost impossible. Increasing knowledge brings with
it increasing power, and great as are the triumphs of the present century,
we may well believe that they are but a foretaste of what discovery and
invention have yet in store for mankind. Encouraged by the thought that
our labours cannot be thrown away, let us redouble our efforts in the
noble struggle. In the Old World and in the New, recruits must be
enlisted to fill the place of those whose work is done. Happy should
I be if, through this visit of the Association, or by any words of mine, a
larger measure of the youthful activity of the West could be drawn into
this service. The work may be hard, and the discipline severe ; but the
interest never fails, and great is the privilege of achievement.
114.
A LECTURE EXPERIMENT ON INDUCTION.
1SS4.]
IT i* well known that an efeetro-magndtb nmttiorpoised iim Ae rarejaitt of an
alternate content MAim^ dmifmhga; tike idfifort fair MSKffv than in -a 'degree
ocine5|MMdiing to ttfc inesisaaDMae' dF ttfe adfditoofflmll wire. TM? ferfuaTio-iar f an
dedio-flagnel maj be eadnhifted to an andinnrr in an insnrumve inajm-r.
by one of a bdix wound with two eontignorats wires iswdh ;as .-sure ocmiD ;.IL]T
used iv luge in$ttramaBltK^ one of widen is imdfaBdadl m tie circ-n: of a
De M<p*i*^it **fc:g> ana] a fe, w ineandeaeenlt lamps. If tite eir-c-ai-i o>f i-be
aeoand wire be open, the mftiwlaetion of a few flbonat Jupom -wire? im,o iLe heiis
euoBB a ~vay mif%iri Cdfing off in ttae in
devdope th^nseiTes in lit of sssA a Mmd
fke sdfeudiidtian, and tne Hgiilte veeoror thair briElaaaacT
inn, the effect of donng the second circmil is
degree of incandenence be sniinble.
An my iMiil wirifalrlp' fa* ilhuiiaJiing the saooe
current* of MI ill inftenaty was described in Jf'aftune for Mar 23.
[Ait. 20, voL i. p. 167.]
115.
ON TELEPHONING THROUGH A CABLE.
[British Association Report, pp. 632, 633, 1884.]
THE principles of this subject were laid down thirty years since by
Thomson, but the author had not met with an application to the circum-
stances of the telephone.
A periodic variation of potential, imposed at one end, is propagated along
the line in accordance with the law
y = e V2*'* cos i nt _
in which n/2?r is the frequency of the electrical vibration. For Atlantic
cables the constant k, depending upon the resistance and the capacity, has in
C.G.S. measure such a value as 2 x 10 16 . The distance, in traversing which
the amplitude is reduced in the ratio e : 1, is given by
2k 2 x 10 8
= - centimetres.
If we take a pitch rather more than an octave above that of middle c, we
have n = 3,600, >Jn = 60, so that
x = 3 x 10 6 centimetres = 20 miles approximately.
A distance of twenty miles would thus reduce the intensity of sound to
almost a tenth, an operation which could not be often repeated without
rendering it inaudible. With such a cable the practical limit would not be
likely to exceed fifty miles, more especially as the easy intelligibility of
speech requires the presence of notes still higher than is supposed in the
above numerical example.
116.
ON A GALVANOMETER WITH TWENTY WffiESL
\BritiA Aaoaatiam Sepmt, p. 633, 18St]
GALVAXOMETOB suitable for currents of an ampare <or ttw* are-
accurately standardised bj means of the solver vclttaraneter; Iran nfcjs is>r ~L:ii
ceases, to lie convenient when the current to be. dealt wilt in ribfs afoio^e ST-
amperes. The present instrument k a kind of dnTeiential galvanometer.
provided with two eJeetrieaDj distinct coGs, whose constants are ii riT: : :>!'
ten to one. A eorrent of one ampere through one mil ttBnm~ Tbaflaorties a
current of ten amperes through the other. If the first be measimred inn tteirmi-
of sfrner, the second serves to standardise an imstrmEBDeM smit^Mc for ttbxe
lugvt cuiieuL
The norehj consists in the manner in which the ttemi n.> :-) ?as: : :f
seeoredL Twenty pieces of Not 17 eotton-eovered wirr r briim^ cmn no> -..-^
lengths of about eight fee** were twisted dbedk H'Og^fe ITW..J. arnrd nw> r so as
to form ten pairs r which ten pairs were a^dn in nfcie-ir tKounm tur^raJ fHi^iuttEj
together so as to form a rape. In each of the t wx> eircmiite ttiuare aurt itaa
wires. In one, that intended for the large-r cwnnrmitL ttlue^e wires. &JM- m
parallel; in the other orcnit the ten wirts aune- im $ri<es.. Off t&adh <ol' liM*
original twists one wire befcogs to the paraOel and vw? n> tine- seiies gmoKaip.
Now the two wires forming an original twist are e^m-aiihr dffi ^ctorte- upon a
needle suspended in any reasonable situation with nesprM5 tt"> nfiBenm. and thus
if the ten wires in parallel hare the same resistance, the cipnain tfofumeidl by
the ten wires in series will be precisely ten times as efleetiire as ttlne dremiit
formed by the ten wires in parallel flra re inlepemdlent of the dibpoeitaon
of the ten original pairsy but by winding them tandy intte* a rxxpe we gain an
additional security in case the ten parallel wires, though of the same length
and cut from the same hank, should hare slightly different res^tance% If
all the twenty wires could be assumed to hare equal efficiency
358 ON A GALVANOMETER WITH TWENTY WIRES. [116
the needle, the equality of resistances of the wires in parallel would be of no
moment.
The rope is bent into a single circle of about a foot diameter with leads
two feet long. At this distance the necessary junctions can be effected
without fear of disturbance. The electrodes for the heavy currents are
formed of parallel copper strips, separated by an insulating layer, and the
current is brought up through twisted leads as in Sir W. Thomson's graded
galvanometers. In the case of the smaller current, which embraces the
needle ten times, so much precaution is not required.
After the wires in parallel had been soldered up, but while those destined
to be joined in series were still disconnected, insulation tests were made
between each wire of the series group and the other wires of that group as
well as the group in parallel. The resistance between each series wire and
the parallel group was about 2| megohms, and (as might be expected) about
twice as much between any pair of wires of the series group.
It will be seen that when, in the use of the instruments, two currents are
balancing one another, every one of the twenty wires carries the same current.
In the actual instrument this current might amount, without undue heating,
to four amperes, so that the heavy current would be 40 amperes. If it be
not thought necessary to deal with currents heavier than 10 amperes, the
gauge of wire might be reduced, a change which would facilitate the winding
of the rope.
The magnet and mirror should be of the kind used in reflecting galvano-
meters, and may be hung at the centre of the circle.
117.
ON CLARK'S STANDARD CELLS.
[British Association Report, pp. 651, 652, 1884.]
IN the hope of finding a clue as to the origin of some of the minor
anomalies of Clark's cells, I have made experiments upon the E.M.F. of com-
binations, in which two different strengths of zinc amalgam take the place of
the zinc and pure mercury of the Clark cell. No mercurous sulphate is
employed, the liquid being simply a saturated solution of zinc sulphate.
If the same kind of amalgam be used for both poles, the symmetry is
complete, and there should be no E.M.F. But if we take for one pole a strong,
but fluid, amalgam, and for the other the same amalgam diluted with an
equal volume of pure mercury, we find a very sensible E.M.F., the strong
amalgam corresponding to the zinc of the ordinary Clark. In my experiment
the E.M.F. was 004 Clark, and remained pretty constant from day to day.
In another cell the same strong amalgam was used for one pole, and for the
other pole was diluted with three times its volume of pure mercury. In this
case the E.M.F. was O0.9 Clark.
If we replace the diluted amalgam with pure mercury, we obtain (without
mercurous sulphate) nearly the full E.M.F. of the Clark cell, but, as might be
expected, the force is very unsteady. From this it would seem that the func-
tion of the mercurous sulphate in the usual form of cell is to retain the
purity of the mercury, and that the E.M.F. is largely due to the affinity of
mercury for zinc.
118.
ON THE CONSTANT OF MAGNETIC ROTATION OF LIGHT
IN BISULPHIDE OF CARBON.
[Philosophical Transactions, 176, pp. 343366, 1885.]
1. THE phenomenon, to which the present investigation relates, is
Faraday's discovery of the " Magnetisation of Light," or in more usual
language the rotation of the plane of polarisation of light in traversing
certain media exposed to powerful magnetic force. One of the character-
istics of this rotation is that it takes place in the same absolute direction
whichever way the light may be travelling, differing in this respect from
the rotation which occurs without the operation of magnetic force in quartz
and many organic liquids. Advantage of this property has been taken by
Faraday and others in order to magnify the effect. By reflecting the light
backwards and forwards it is possible to make it traverse several times a
field of force whose length is limited.
A consequence remarkable from the theoretical point of view is the
possibility of an arrangement in which the otherwise general optical law
of reciprocity shall be violated. Consider, for example, a column of dia-
magnetic medium exposed to such a force that the rotation is 45, and
situated between two Nicols whose principal planes are inclined to one
another at 45. Under these circumstances light passing one way is
completely stopped by the second Nicol, but light passing the other
way is completely transmitted. A source of light at one point A would
thus be visible at a second point B, when a source at B would be in-
visible at A ; a state of things at first sight inconsistent with the second
law of thermodynamics.
2. It is known that the rotation may be considered to be due to the
propagation at slightly different velocities of the two circularly polarised
118] ON THE CONSTANT OF MAGNETIC ROTATION OF LIGHT, ETC. 361
components into which plane polarised light may be resolved; and it is
interesting to consider what difference of velocity our instrumental ap-
pliances enable us to detect. A retardation, amounting to one wave-
length (X), of one circularly polarised component relatively to the other
would correspond to a rotation of the plane of polarisation through 180-
If we can observe a rotation of one minute, we are in a position to detect
a retardation of X/10800. If I be the thickness traversed, v and v + 8v the
two velocities of propagation, the relative retardation is /8v/t?. To take
an example, suppose that / = 20 inches, X = Io ^ 00 th inch : so that if Bv/v
exceed 10"*, the fact might be detected*. It appears therefore that we
are able to observe extraordinarily minute relative differences in the
velocities of propagation of the two circularly polarised rays,
3. The laws of the phenomenon were investigated in detail by Verdet,
who proved experimentally that in a given medium the rotation between
any two points on a ray of light of given kind is proportional to the
difference of magnetic potential at those points. When the path of the
ray is singly or doubly curved, the rotation is to be estimated upon
principles similar to those applicable to tinst^ in curved rod?*.
4. Absolute determinations of magnetic rotation in bisulphide of carbon
have been made by Gordon , and by H. Becquerel ,., whose results differ,
however, by about 9 per cent. The former obtained his magnetic force by
means of an electric current circulating a great manv times round the
column of CSj. This column being a good deal longer than the coil,
the electro-magnetic effect is approximately determined by the strength
of the current and the number of turns. Of these data the first was
found by a comparison with H (the horizontal component of terrestrial
magnetism). The number of windings in the coil was determined, not by
a simple counting, but a posteriori by an electrical process.
In M. Becquerel's experiments the magnetic force was that of the
earth acting on a column of CSa more than 3 metres in length. The
very small effect (obtained by reversal of the apparatus in azimuth) was
augmented by causing the light to pass the tube 3 or 5 times, but even
with 5 passages the double rotation amounted to only about 30 minutes.
M. Becquerel regards his determination for sodium light as accurate to
* Camb. Nat. Sci. Trip. Ex., 1883.
t Thomson and Tait's Xatmral Philosophy, 119123.
+ When polarized light passes from one medium to another, e.g., from air to glass, the plane
of polarisation is in general twisted without the operation of any magnetic force. This effect,
however, depends upon a part of the light being diverted by reflection, and would disappear if the
transition from one medium to the other were gradual, i.e., occupied a stratum a few wave-lengths
thick. (See Proe. Math. Soe. vol. n. No. 159.) [Art. 63, vol. i. p. 460.]
Phil. Tnuu. 1877, p. 1.
I Ann. d. Chimit, 1882.
362
ON THE CONSTANT OF MAGNETIC
[118
within 1 per cent., which would be indeed a wonderful result considering
the smallness of the rotation.
5. It is important to observe that great care is required in order to
define with sufficient accuracy the kind of light employed. Since the
rotation is approximately proportional to \~- } a change from one sodium
line to the other would make a difference of two parts per thousand.
Both of the above-mentioned experimenters started with white light.
Gordon threw a spectrum upon a screen perforated with a slit, the posi-
tion of which was adjusted to correspond with the thallium line ; while
Becquerel corrected his results indirectly by a subsequent comparison
between the effects of the more mixed light used by him and that
emitted by sodium.
Considering that the employment of white light involved very elaborate
arrangements for analysis (according to wave-length), in order to avoid errors
exceeding in magnitude those likely to be encountered in the polarimetric
or electric determinations, I decided to use light actually emitted from
sodium vapour. The sodium chloride was held by a spoon of platinum
gauze in the flame of a small ordinary Bunsen burner (fig. 1, A). As in
Fig. l.
c B
A. Bunsen burner
B. Mirror with slit.
C. Back mirror.
D. Direct vision prism.
E. Collimating lens.
F. Polarising Nicol.
G. Sugar cell.
H. Tube of bisulphide of carbon.
I. Screen (blackened inside).
J. Analysing Nicol.
Mr Glazebrook's optical investigations, the evaporation of the salt and the
temperature of the flame were stimulated by a jet of oxygen gas brought
in laterally and caused to play round the gauze*.
* [1899. Fox Talbot's early optical work is so little known that I am tempted to quote in full
his short note, in which probably this valuable method "of obtaining homogeneous light of great
intensity " is first described.
" As it is a desideratum in optical science to procure perfectly homogeneous light of sufficient
brightness for many important experiments, I am glad to be able to communicate a method
which in a satisfactory manner supplies that deficiency.
"It is only requisite to place a lump of common salt upon the wick of a spirit-lamp and
to direct a stream of oxygen gas from a blow-pipe upon the salt. The light emitted is quite
homogeneous, and of dazzling brightness. If instead of common salt we use the various salts
of strontian, barytes, &c., we obtain the well-known coloured flames, which are characteristic of
those substances, with far more brilliancy than by any other method with which I am acquainted."
(Phil. Mag. in. p. 35, 1833.)
118] ROTATION OF LIGHT IN BISULPHIDE OF CARBON. 363
At the close of the experiments I examined the light thus obtained
with a powerful spectroscope, and found that under the influence of the
oxygen the originally narrow bright lines dilate almost to the point of
contact, thus forming a bright field upon which the dark D-lines are seen
with beautiful definition. Although the distribution of light appeared to
be tolerably symmetrical, it is a question to what degree of accuracy the
mean quality of this light can be identified with that coming from midway
between the D-lines. Probably we shall be safe in estimating that the error
from this cause is well below
The bright part of the flame being much larger than is required, a
screen (B), perforated with a slit, may conveniently be interposed. In
this course there are two advantages. It allows us to purify the light
from rays of other refrangibilities (of which there is always a sensible
accompaniment, both red and blue) by use of a direct-vision prism (D).
Again, by making this screen of looking-glass, from which a narrow strip
of silvering is removed, and by backing the flame with a parallel mirror
(C), we gain by repeated reflections to and fro, an important increase of
illumination. The success of the polarimetry is very dependent upon the
intensity of the light, but there must be also a reasonable steadiness.
Several arrangements of flame w r hich at first promised well failed in the
latter requirement.
6. The rays from the slit, after purification by the direct vision prism,
are rendered parallel by a collimating lens (E) and pass into the polarising
Nicol (F). The polarimeter employed is on the principle of Laurent, but
according to a suggestion of Poynting* the half-wave plate of quartz is
replaced by a cell ((?) containing syrop, so arranged that the two halves
of the field of view are subjected to small rotations differing by about 2 : .
The difference of thicknesses necessary is best obtained by introducing
into the cell a piece of thick glass, the upper edge of which divides the
field into two parts. The upper half of the field is thus rotated by a
thickness of syrop equal to the entire width of the cell (say ^ inch), but
in the lower half of the field part of the thickness of syrop is replaced by
glass, and the rotation is correspondingly less. With a pretty strong syrop
a difference of 2 may be obtained with a glass ^ inch [inch = 2 - 54 cm.]
thick. For the best results the operating boundary should be a true
plane nearly perpendicular to the face. The pieces used by me, however,
were not worked, being simply cut with a diamond from thick plate glass;
and there was usually no difficulty in finding a part of the edge suffi-
ciently flat for the purpose, i.e., capable of exhibiting a field of view
sharply divided into two parts. I had expected to be troubled with
depolarisation, especially in the thick glass, but a small piece thus cut
* Phil Xag., July, 1880.
364 ON THE CONSTANT OF MAGNETIC [118
out of a large plate is relieved from most of the strain to which it was
originally subject. Probably more care would be required in experiments
where a strong white light could be used; but by previously testing the
rather thin plates used for the sugar cell and for closing the CS 2 tube,
I was able to secure a field of view either half of which under the actual
circumstances could be made quite dark by suitable orientation of the
analysing Nicol.
By this use of sugar, half-shade polarimeters may be made of large
dimensions at short notice and at very little cost. The syrop should
be filtered (hot) through paper, and the cell must be closed to prevent
evaporation.
7. On leaving the sugar cell the light entered the column of bisulphide
of carbon (H). To contain the liquid two tubes of brass were employed at
various times, the ends being closed with plates of worked glass cemented
to the metal with a mixture of glue and treacle. Near one end these tubes
were provided with a lateral (vertical) branch, closed with a cork, through
which passed the stem of the thermometer used for observing the tempera-
ture of the CS 2 . The length of the larger tube (used in Series I. and II.)
was 31'591 inches, and the diameter about 1| inch. The length of the smaller
tube (used in Series III.) was 29*765 inches, and the diameter 1 inch.
When, as in Series I., it was wished to cause the light to traverse the
tube more than once, mirrors were necessary at the ends of the tube.
They consisted of plates of thin looking-glass, from which part of the
silvering was removed, and by means of a little glycerine they were brought
into optical contact with the plates by which the tube was closed. This
arrangement was simple, and had the further advantage of practically
annulling some troublesome reflections; but the want of means of adjust-
ment rendered it necessary that the closing plates should themselves be
pretty accurately parallel.
8. The internal diameter of the ebonite tube, upon which the helix
was wound ( 13), was about 1| inch, and it was intended to utilise the
annular space between the ebonite and the brass as a jacket, through
which water at the temperature of the room might be made to circulate.
This arrangement, however, failed utterly. Within about 10 minutes of
the closing of the circuit of the helix, the definition was lost, and nothing
further could be done until after a long interval of repose. The water-
jacket was then abolished, and the available space filled with paper
wrapped pretty tightly round the tube. This effected a great improve-
ment, enhanced still further in the later experiments of Series III., in
which, by reduction of the diameter of the tube, a wider space became
available for heat insulation. The disturbance by conduction of heat from
the wire to the CS 2 remained, however, the worst feature of the experi-
118] BOTATIOX OF LIGHT IS BISULPHIDE OF CARBON. 365
ments, and could not be obviated without a fundamental alteration in the
apparatus. Probably the best arrangement would be a water-jacket next
the wire, and a good thickness of paper or other insulator between the
water and the C&,.
9. The bisulphide of carbon was purified by treatment with corrosive
sublimate and grease with subsequent distillation (according to the pro-
cedure advocated by Becqnerelj), until most of the unpleasant odour had
disappeared. The transparency is much greater than is readily (if at all)
obtainable with water, provided proper precautions are taken to avoid ex-
posure to light. After being acted upon by light, the OS, attacks brass
and becomes rapidly opaque. In this respect it would be an advantage
to replace the metal tube by one of glass.
10. The analyser consisted, in some experiments, of a Xicol (J, and
in others of a double image prism, and was mounted in a circle made by
the Cambridge Scientific Instrument Company. In order that a rotation
of the plane of polarisation may be correctly indicated by the difference
of the two circle readings, it is necessary that the axis of rotation should
coincide with the direction of the light. This requirement is, however,
not very easily satisfied. At the commencement of a series of experi-
ments the adjustment was made with the aid of a telescope and cruss
wires temporarily substituted for the Xicol, but during the course of a
set of readings the passage of heat into the liquid tended to make the
upper strata warmer than the lower, and thus to bend the rays into a
different direction. It is known* that the error arising from maladjust-
ment in this respect is in great part eliminated by reading the Nioo!
always in both the positions (differing by about ISO") which give extinc-
tion, or (in the half-shade arrangement) equality of Ulununarion. This
plan was constantly followed, but it is not dear that the whole error ran
be thus got rid of. It occurred to me that another term in the harmonic
expansion of the error would be destroyed by use of a double image prism
read in four positions distant about 90". Experiment showed that in spite
of the glare of the nnextingnished image, good readings oonld be obtained
after a little practice., and the comparison of the results arrived at in this
way tends to show that the error is not wholly eliminated in the mean
of two readings taken in positions differing by 180~. But the matter
could be much better investigated with a simplified apparatus and the
use of a strong white light.
In Series IL and III., when the light traversed the tube but once, no
magnification was necessary, and the eye was applied immediately behind
366
ON THE CONSTANT OF MAGNETIC
[118
the analyser. In Series I., the apparent magnitude of the field was much
less, and an opera-glass, magnifying about twice, was employed between
the analyser and the eye.
11. The setting of the Nicol (or double-image prism) by adjustment
of the match between the two parts of the field presented by the half-
shade apparatus was facilitated -by a device that may be found useful.
"In addition to the principal helix, the tube was embraced by an auxiliary
coil of insulated wire, through which could be led the current from a
Leclanche cell. This current was controlled by a reversing key under the
hand of the observer, who was thus able to rock the plane of polarisation
backwards and forwards through a small angle about its normal position.
The amount of the rocking being suitably chosen, the comparison of the
three appearances (two with auxiliary current, and one without) serves to
exclude some imperfect matches that might otherwise have been allowed
to pass*."
12. Apart from the effect of heat upon the CS 2 , the working of the
optical parts was fairly satisfactory. The following zero readings taken
without the current on June 4, 1884, will give an idea of the sort of
accuracy attained. The analyser was a double image prism, and was read
in all four positions, the circuit being made three times.
TABLE I.
103 2
193 4
283
13 2
102 55
193 5
283 2
12 59
102 58
193 3
283 2
13 4
Mean ....
102 58
193 4
283 1
13 2
Subtract . . .
90
180
270
12 58
13 4
13 1
13 2
It appears that an error of 3 or 4 minutes may occur in a single
setting.
13. I now pass to the description of the electrical arrangements. The
magnetic force depends upon the helix and upon the strength of the
current, and we will take these elements in order.
* " Preliminary Note on the Constant of Electro-magnetic Eotation of Light in Bisulphide of
Carbon," Proc. Roy. Soc. vol. xxxvn. p. 146 (June 19, 1884).
118] ROTATION OF LIGHT IN BISULPHIDE OF CARBON. 367
The helix.
The wire is wound upon an ebonite tube, the outside surface of which
was turned true in the lathe, and is kept in its place laterally by ebonite
flanges screwed upon the tube. The distance between the flanges, equal
to the length of the helix, is 9'990 inches ; but the tube itself projects
some inches beyond the flanges, and when it was desired to use an in-
ternal water-jacket, could be further prolonged by additional lengths of
brass tube.
In order to give better opportunity for testing the insulation, on which
the correctness of the final results is entirely dependent, it was decided to
wind on two wires simultaneously, which should be in contact with one
another throughout their entire lengths. The operation was performed on
December 14-15, 1883, with triply-covered wire of diameter about ^ inch,
and no particular difficulty was experienced. The revolutions of the ebonite
tube, mounted in the lathe, were taken with all care by an engine counter,
and amounted to 1842, so that the total number of windings is 3684. The
internal diameter of the helix is 2'188 inches, and the external diameter
is 4'13 inches, [inch = 2'54 cm.]
By endeavouring to force a current from one wire to the other we obtain
a very severe, though of course not absolutely complete, test of the insula-
tion. The resistance between the two wires varied with the hygrometric
condition of the silk, which was not impregnated with paraffin. At first it
was not much over 2 megohms, but latterly reached 6 or 8 megohms, and
was thus abundantly sufficient.
14. As a further test observations were made of the external effect
of the helix upon a suspended magnet, when a powerful current was
passed in one direction through the first wire, and in the opposite direc-
tion through the second. If the positions of the two wires could be
treated as identical, the external effect ought everywhere to vanish. In
consequence, however, of the fact that one wire lies throughout on the
same side of the other, the compensation could not be expected to be
complete, except when the suspended magnet is equidistant from the two
ends. Experiment with the magnet of a reflecting galvanometer showed
that the effect, in fact, varied as the magnet was displaced, but even in
the symmetrical position there was a perceptible outstanding differential
effect. In order to eliminate the influence of other parts of the circuit,
the readings referred only to the deflection of the needle as the current
was reversed in the helix; and the scale of sensitiveness was obtained
by repeating the observations after altering the connexions of the two
wires, so that the current circulated the same way round both, and after
insertion of a high resistance by which the intensity of the current was
368 ON THE CONSTANT OF MAGNETIC [118
reduced in a known proportion. From this it appeared that the differ-
ential effect of the two wires (with a given current) was ^-gW f the
combined effect.
This fraction is tolerably small, but I had expected to find it smaller
still. It seems probable that the incompleteness of compensation is due
to a small difference Cs^oo) in the mean diameter of the windings in the
two cases. To throw light upon this I took careful measures of the re-
sistances of the two wires. Although they had originally formed one
length, their resistances differed by as much as 7 ^th part, that of the
wire which had shown itself least effective being 7-075 B.A., and of the
other 6'965. If, as it seems plausible to do, we attribute the difference
of resistance to difference of diameter, this actual difference must amount
to -y-^ inch. The mean diameter of the windings is about three inches ;
and if the two wires were wound upon a smooth cylinder of this diameter,
the difference in the diameter of the windings would be ^TM f the
whole. As this estimate would be increased were we to take into account
the fact that each winding really sits upon two windings of the layer
underneath, and that these cannot be practically in actual contact, we
may perhaps consider the small anomalous differential effect upon the
external magnet to be sufficiently explained by the observed difference
of resistances.
Correction for finite length.
15. If the tube were infinitely long, the difference of potentials at
its ends due to the unit current in one winding would be 4?r. But on
account of the finiteness of the length a
correction is required, whose approximate
amount is given in Gordon's paper.
Considering, in the first place, one layer
of windings of radius Aa, we know that the
external effect is the same as would be produced by a uniform distribu-
tion of imaginary magnetic matter over the ends, positive (say) over Aa
and negative over Bb, the superficial density being equal to the number (m)
of windings per unit length. The potential at L of the matter on Aa is
27rra(Za LA), or approximately
Ad 2 , Aa*
Similarly the potential at L for the matter on Bb is
Aa 2 , Aa 4
11>~ ROTATION OF LIGHT IN BISULPHIDE OF CARBON.
so that altogether the potential at L for this layer of windings is
Aa- Aa 4 LB- + LA.LB+LA
irmAB T -. r -^ --
4 LA 3 . LB 3
in which mAB denotes the whole number of windings in the layer. This
result has now to be integrated so as to represent the effect of the helix,
whose inner and outer radii we may call Aa^ and Aa*. The mean value
of A a- is
Aaf-Actf
t
and that of Aa 4 is
Thus, if n be the whole number of windings on the helix, the difference
of potential from L to M corresponding to the unit current is
*"
a?( 1
, \LA
120,0, LALB MA. MB
_ Aaf-Aaf (LB* + LA .LB+ LA- MA- + MA . MB + MB- \
80a sfll V LA*.LB* " MB*. MA*
In the present case
Aa= 2-065 (inches), Aa, = 1-094, a 2 a, = 971,
from which we get
^-^-'-em **-**' -ten.
120,0! SOaaOj
In the remainder of the calculation we have to distinguish the two
tubes. For the first
LA=MB = 10-800 inches, LB = MA = 20 790 inches :
and for the second
LA = MB = 9-887 inches, LB = MA = 19 877 inches.
Hence for the first tube we have
4/j7r(l - -00573 + -00006) = 47r x '994:33*;
and for the second
4n-n- (1 - -00655 + -00008) = 4n7r x "99353,
the correction for finite length thus somewhat exceeding one-half per cent.
* In the Preliminary Note the reducing factor for this tnbe was given as -9W49. The
alteration is due to the use of more precise data in place of some quite rough measurements in
round numbers on which, by an oversight, the first calculation was founded.
24
370 ON THE CONSTANT OF MAGNETIC [118
16. We have now obtained the difference of potential at the ends of
the column of CS 2 due to the passage through the helix of unit current.
It yet remains to describe the means adopted for the measurement of the
actual current in absolute measure.
In a former paper, " On the Electro-chemical Equivalent of Silver, and
on the Absolute Electromotive Force of Clark Cells*," it was shown how
the E.M.F. of a Clark cell was obtained by comparison with the difference
of potentials at the extremities of a wire of known resistance, due to the
passage of a current known either directly from its effect upon a current
measuring apparatus, or indirectly through the deposition of silver. For
the purposes of the present investigation this process was reversed, the
Clark cell itself being treated as a standard of E.M.F., by which to de-
termine the value of the current, which traversed the known resistance,
and also the helix by which the magnetic rotation was produced. The
arrangements differed so little from those elaborately described in the
paper referred to, that it seems unnecessary to enter into the matter at
length. If the reader will refer to Fig. (1), [p. 285], he will understand
the electrical connexions, and he may suppose the current-measuring
apparatus, EOF, replaced by the magnetising helix. In point of fact this
helix was situated in another room at a distance from the E.M.F. com-
pensator and its galvanometer T. The direction of the current in the
helix was reversed by a mercury key of the rocker pattern, and care had
to be taken that at this moment the galvanometer contact Q was open.
The general nature of the arrangement will be sufficiently understood
when it is said that the want of balance between the E.M.F. of the Clark
and that at the terminals of the resistance R was made up by E.M.F., taken
from an auxiliary circuit, the value of which was afterwards expressed in
terms of the Clark. Denoting the force thus added or subtracted by r,
upon a scale according to which the force of the Clark was p, the actual
difference of potential at the terminals of .R may be written
Clark.
17. As it was intended to use currents of about one ampere, the
resistance R was made about [1] ohms. The construction was somewhat
similar to that of the [4] described in 33 of the former paper, but on
account of the increase in the current to be carried, three wires of German
silver were used in parallel. The amount of heating was unimportant for
the purposes of the present investigation.
The value of the [1J] was determined by comparison with a combi-
nation of three standard units, one (taking the whole current), and two
in parallel (giving the ). At 13 the resistance is 1*4945 B.A. At 15 C ,
* Phil. Trans. 1884, Part H. 35, 36, 38. [Art. 112, vol. u. p. 278.]
118]
ROTATION OF LIGHT IX BISULPHIDE OF **
5 i
5 i
3
.5
= !
; c
=1 Z =
II I ? ? f f
5 ri 5
I I
-"-
I ^
242
372 ON THE CONSTANT OF MAGNETIC [118
which was adopted as the standard temperature for R and for the Clark,
we have
R = 1-4958 B.A.
18. In consequence of the heating of the copper wires, the current
(usually obtained from secondary cells) fell off somewhat rapidly during
a set of observations, and it was found convenient to take readings of
the E.M.F. compensator simultaneously with the adjustment of the polari-
meter. The former readings were taken by myself and the latter by
Mrs Sidgwick, while the flame (at which the optical observer should
not look) was regulated by an assistant, who also recorded the circle
readings.
The procedure will be most easily explained by an example, for which
purpose I take at random the observations of July 25, recorded in Table II.
It will be seen that the cycle consisted of eight readings, four with
positive and four with negative rotation of the plane of polarisation, and
that this cycle is repeated three times.
The three readings under any one head vary in consequence of the
diminution of the current as well as from errors of observation. The
value of p was
at the beginning p = 7018
at the end p = 7016
Mean p = 7017
Thus in the first observation at 6 h 3| m , when the circle reading was
261 44', the difference of potentials at the extremities of the [1^] was
+ j x Clark I., the temperature of Clark I. and of the [1^] being
'6.
For the mean double rotations in the four positions of the double-
image prism we have
269 17-7 - 261 45'3 = 7 324
359 23-0-351 56'3 = 7 267
89 19-3- 81 49-7 = 7 29'6
179 20-0-171 52-3 = 7 277
Mean 7 29'1
Since all the effects are proportional to the current, it is sufficient
to compare the mean rotation with the mean value of r, viz., 1413 ; so
that the double rotation 7 29''1, or 449'"1, corresponds to a difference of
potentials equal to
/. 1413\ 8430
(1 + 7Q17 j x Clark I. = -x Clark I.
118] BOTATIOX OF LIGHT IX BISULPHIDE OF CAKBOX. 373
The double rotation that would have been found if the current had been
just strong enough to balance Clark L (at die actual temperature) is
19. This result is a function of the temperatures of the cell and of R
as well as of the CS, : and it is rather unfortunate that all three tempera-
ture corrections tell in the same direction. A rise of the thermometer
involves a rise in R and a Ml in the force of the standard cell, so
that on both accounts the current giving the balance is diminished. At
the same time the smaller current acts less advantageously in producing
rotation in consequence of the properties of the C&*. It will be con-
venient to postpone the last correction, and take first the corrections for
temperature in R and the ELILF. of Clark, which relate rather to the
machinery for measuring the current, and which can be made from data
obtained in previous investigations. For this purpose 15 r C. is adopted
as the standard temperature: and the proportional corrections per degree
are -OQOS2 for the KJLF. of Clark and -00044 for the R. makiBg alto-
gether -00126 per degree. For the observations of July 2-5, the correction
is therefore
+ 2-6 x -00126 x 373"-8 = -8-2 x-471 = + l"-
If we take as a standard current that which in traversing; R at 1 > :
would balance Clark L at 15% the double rotation of July 2-3 reduced so
as to correspond with the standard current will be
This rotation corresponds to the temperature 18~U of the OS,. To
obtain comparable results we must reduce to a standard temperature, for
which purpose we will select 18% According to Bichat the rotation at f
may be expressed by
1- -001041 --OOOOUf*
the rotation at being taken as unity. To obtain a more convenient
formula, applicable in the neighbourhood of 18% we may write f = 18 + f.
Thus
1 - DOlOftf - WOOl-lF = i>767 - -00154r = ^767 (1 - -001580 :
so that the coefficient for the correction is -00158. Hence T if the CSj on
July 25 had been at 18% we should have had
3T5'-0 + 375'0 x TO158 x 3 = 375*1) + '592 x "3 = 375U + ^ = 375'*.
- Thus reduced the results for the observations of different days should
agree together.
374
ON THE CONSTANT OF MAGNETIC
[118
lag
1*1
+ + +I++I l + l 1 1 I+ +
;
^lUSb-W
OOOfH
1 1 1 -f
;
ll
.S|||||Sg||K||SS
i
cp (N p O5
t-
U5
cb
*i
S
JL
.,*OC..._.* n o*<
0,0^=0
o
1 1 1 + T + +. 1+ , 1 ,
OrH rH rH
1 1 1 1
2
M
_____,
t~
>O CO O O5
12
it-t-6oo5OO5Oc~t~dodo>'oioiffl
t~
o o >o o
o
H
J|||
p
'P <? ? <?
O CC O 00
I
l-
"
a
o
OOOCOi-IOOOiUSCOt^OOOOrHODO
(M TH O CO
2 o
"000<MC^CO^5r-l(M(NCOOOt-l
-H 000
fc
If
.2
1
kft CO rH CO
i)>ra-^ot-t-oooooit-t--*-*.b
t-10 kOHH
SH
k
WCO-lt-Oir-.iraOSCOO^COOrlOS
COCOTflp
o- -H
' i s a gggg g i g
J
0.
II :I!
;
Ills
I 8 *
(M <N rH O3 i-H " C^ CO
IP!
"Jg fl
Ot^ OpO5CpO<N 000 ^J* b-
C<ICO .COOO^-^GOOOCOiC .OO^t 1
p p >p 35
5 O CO O
Sj
S .
^og^ S a8gJ
CO 1000
cS
g>
S
a
M
3 " " "
118]
ROTATION OF LIGHT IN BISULPHIDE OF CARBON.
375
< $5
H OQ
IP
2--0
o
II
2
1"=^
8,51
o c o
7 ~ -r
I
<x co 5c
*
o o o z si S =
t- t- t- t- t- t-
x o o r: -
3 3
376 ON THE CONSTANT OF MAGNETIC [118
20. The results of all the observations (other than preliminary) which
were thought worthy of reduction are exhibited in the accompanying
tables, grouped in three series. In Series I., II. the first tube was em-
ployed ; the principal difference between them being that in Series I. the
light traversed the tube three times, while in Series II. the light passed
but once. It will be seen that in Series I. the actual double rotation
varied from about 9 to 19, and the currents from about ampere to
1 ampere. In Series II. stronger currents were usually passed, amounting
to about 1^ ampere, but the rotation was only about 9. The extreme
deviation from the mean is only about '4 per cent., if we exclude the
observations of May 29, which owing to interruptions and other causes
were marked as unsatisfactory before reduction.
The Nicol was used as analyser in Series I., and on June 3 of Series II.
The remaining observations of Series II. and the whole of Series III. were
taken with a double-image prism, read in all four positions as already
explained by the example of July 25.
For the observations of Series III. the second tube was employed, with
some improvements in the provision against the communication of heat.
The diminished diameter of the tube was the inducement to pass the
light but once, though it would have been possible to work with three
passages. But when the rays skirt the walls of the tube, there is more
disturbance from heat ; and, indeed, generally the advantage of augmented
rotation is in great measure paid for by greater sensitiveness to deviation
from optical uniformity.
Not only does the communication of heat disturb the definition, but
it tends also to render the actual temperature uncertain. During some
of the more protracted sets of readings with the stronger currents there
was a rise of nearly 2 in the temperature of the CS 2 ; and, although
this rise was carefully watched, it is difficult to feel confident that the
effective mean temperature can be determined with a less error than
say of a degree. Such an error would correspond to about ^ m in
the final number. To avoid increasing the uncertainty under this head
the readings were often concluded, although the definition still remained
satisfactory.
If the apparatus were to be designed afresh I should endeavour to guard
more adequately against these disturbances, and it might then be possible
to use five passages with advantage, more especially if by increasing the
weight of the coil it were practicable to bring the double rotation up to
about 90. The determination of such a rotation with the double-image
prism would be free in high degree from the polarimetric errors considered
in 10. But it is doubtful whether in the present state of science the
additional accuracy would repay the labour involved.
118] ROTATION OF LIGHT Of BISULPHIDE OF CARBON. 377
21. It only remains now to work oat the results in absolute measure.
And first as to the value of the standard current, defined as that, which,
flowing through the [1] at 15 : , balances Clark L at the same temperature.
This value in amperes is expressed by dividing the ELJLF. of Clark L in
RA. volts (see Table XL of former paper [p. 324]) by the resistance of die
[1|] in BLA. units. Hence the standard current is
= -9722 ampere = -09722
If the tube were infinitely long, the difference of magnetic potentials
at its ends would be 4i*Tir; but in the case of the actual tubes we have
to introduce the correcting factors -99433 and '993-33 < 15). Thus for
the first tube, if x be the (single) rotation in minutes corresponding to
difference of potential 1 C.GJ&, the whole actual double rotation for a single
passage of the light will be
2 x -09722 ***** -99433 x x.
From Series L at 1S : this quantity is found to be A x 1128-3. or 376
so that
376~2
* = 2 x -09722 x 4*-n x -99433
In like manner from Series LL we get
* = 2 x-09722 x 4m x^9433
For the second tube used in Series ELL we have to employ a slightly
different correction for finite length. We have
375-8
2 x -09722 x 4rn x i93-53
'.-/. i
The results of Series L and LTL are thus in precise agreement, while that
of Series LL is about y^ lower. Ascribing a somewhat less importance to
Series II. in consequence of the smaller number of sets of observations, we
may take as the final result of the investigation
which gives the rotation in minutes in bisulphide of carbon at 1S : , corre-
sponding to a difference of potential equal to 1 C.GJ&. It should be noticed
that the mean temperature of the observations was so nearly IS" that the
result as given depends scarcely at all upon Bichat's formula for the
dependence of the rotation upon temperature.
22. M. Becquerel gives as his result for 0" C. "0463 minute. To find
the rotation at 18", this must be multiplied by -9767 according to Bichat 's
378 ON THE CONSTANT OF MAGNETIC [118
formula : and as BecquereFs observations were in fact made at about 18,
this reduction does not introduce, but rather removes, an extraneous element.
Thus according to Becquerel
x '0452 minute,
differing by about 7 per cent, from the value found by me.
The comparison with Gordon is more uncertain, inasmuch as his obser-
vations were made on light of the refrangibility of the thallium line. The
corrected* result for this light is in circular measure T5238 x 10~ 5 , or '05238
minute. To pass to sodium we may use a formula given by Becquerel f
and Verdet, according to which the rotation for different wave lengths (A,)
is proportional to /* 2 (/i 2 1) \~~ 2 , p, being the refractive index. At this
rate the '05238 minute for thallium would be '04163 minute for sodium.
The temperature was not directly observed by Gordon, but was esti-
mated to be about 13 C. Assuming this to be correct, the value for 18
would be '0413 minute, or about 2 per cent, less than according to my
determinations.
APPENDIX.
Notes on Polarimetry in general.
The problem of the polarimeter is how best to render evident the
rotation through a small angle 6 of the plane of polarisation of light of
brightness h. The effect of the rotation is to introduce light of ampli-
tude h* sin 6, or h/>6, polarised in the perpendicular plane, and it is this
which must be made to produce a recognisable change. By the use of a
Nicol, or double-image prism, adjusted to the original plane, the light of
brightness hd 2 may be isolated, but, as will be proved presently, this is
not the best method of rendering its existence evident.
From the preceding mode of statement it is clear that the accuracy
obtainable in determining the plane of polarisation increases indefinitely
with the brightness of the light, arid is in fact proportional to the square
root of that brightness^:. Again we see that little is to be expected from
such devices as that of Fizeau, in which the rotation is magnified by
causing the light to pass obliquely through a pile of glass plates. The
brightness of the light polarised in the perpendicular plane (h& 2 ) can only
be diminished by such treatment, arid the increase of rotation, being due
merely to weakening of the first component, is of no value.
* Mr Gordon's result was originally given at double its proper value,
t Ann. d. Chim. t. xn. 1877, p. 78.
This point is insisted upon in an excellent paper by Lippich (Wien. Ber. 85, 9 Feb. 1882),
which has lately come to my notice.
118] ROTATION OF LIGHT IN BISULPHIDE OF CARBON. 379
The arrangements to be adopted depend for their justification upon the
physiological law of the perception of differences of brightness. If dE denote
the difference of sensations, corresponding to two degrees of brightness, H
and H + dH, we have*
, dH
in which H t is a certain constant brightness, supposed to depend chiefly
upon the proper or internal light of the eye, but to which may be added
the effect of light diffused by imperfect translucency of the optical appa-
ratus. If dE denote the smallest perceptible difference, the value of dEIA
is in favourable circumstances as low as ^ or y^y, which means that with
a sufficient total brightness differences of this amount may be apparent
to observation.
Let us now consider the values of dE corresponding to different methods
of procedure. If the analysing Nicol be adjusted for extinction of the
original light, the comparison is between the brightness which cannot be
got rid of (HJ) and (Hg + hff 1 )^. Near the limit of discrimination, to which
case we may confine our attention, hff 3 is small relatively to H 6 , and thus
we may take
The procedure just considered is that which would naturally be adopted
to render evident a small quantity of light of given amount, viz., to isolate
it and compare it with the best attainable darkness. But in the present
problem the circumstances are peculiar in that we are able to deal with
phases. Now if we regard the amplitude (a) of the feeble light as given,
putting a* = hB 3 , we may produce more effect from it by combining it with
other light in the same phase of amplitude (/3) than by isolating it. The
comparison is then between brightnesses (a + /3Y and ft*, or as a is very
small, between /8 s +2o and 0*. Thus
in which /8 e s is written H Q .
The light of amplitude ft is obtained in the simplest possible manner
by merely rotating the analysing Nicol through a small angle, and the
only question is how to exhibit the comparison light, which shall not be
affected when y8 is changed to 08 + a). For this purpose we may divide
the field of view into two halves with an oblique mirror in which is seen
* Helmholtz : Physiologische Optik, 27.
t We may imagine the presentation of the two brightnesses to be consecutive, or more
favourably that both are seen at once, half the field of view being occupied by a black body seen
after reflection in an oblique mirror, whose edge forms the dividing line.
380 ON THE CONSTANT OF MAGNETIC [118
by reflection a feeble light, of the same colour and coming ultimately from
the same source.
It is possible that an instrument upon this principle might be made to
work satisfactorily*, but the half-shade polarimeters of Jellet and Laurent
seem to be in most respects preferable. In them the comparison is between
(13 + a) 2 and (/3 - a) 2 , so that
dE-A
representing twice as great a sensibility. The only thing to be said upon
the other side is that the division line in these instruments can hardly be
made as invisible as the sharp edge of a mirror may be.
In these formula /3 may be chosen at pleasure by suitable adjust-
ments of the polarising arrangements. In order to get the best result,
dE must be made a maximum by variation of ft, a and /3 being treated
as constants. The maximum occurs when /3 = y8 , and its value in the
last case is
Taking dEJA = -%, which is probably about as small as can be expected
in practice, we have for the least perceptible value of a.
whereas without the half-shade arrangement, and with a Nicol simply set
to extinction of the original light,
so that
According to these numbers the half-shade arrangement would have a
tenfold superiority, a result not fully borne out in practice. In explana-
tion of this it is important to notice that the procedure in the absence
of a half-shade arrangement would in reality be very different from what
we have tacitly supposed. The experienced operator, in setting a Nicol
to the position of maximum extinction, does not judge merely by the
degree of darkness attained in the final position, but displacing the
analyser alternately in opposite directions, he estimates the position which
lies midway between those which give similar revivals of light on the
two sides; or, endeavouring to retain in his memory a certain degree of
* Headings would of course be taken in both the positions (one on either side of extinction)
which give a match with the comparison light.
118] ROTATION OF LIGHT IN BISULPHIDE OF CARBON. 381
brightness, he may take actual readings on both sides, of which the mean
will correspond to the desired position. In this way the fundamental
advantage of the half-shade method is in a sense attained, the only differ-
ence being that the brightnesses to be compared are seen consecutively
after a short interval of time, instead of almost simultaneously: and even
this difference becomes less important when the line dividing the field of
view of the half-shade apparatus is so coarse that it cannot be rendered
invisible.
The carrying out of this method is facilitated by a device which is
worthy of trial The Nicol may be mounted loosely, so as to be capable
of turning through a small angle (2 or 3 degrees) between two stops.
These stops are rigidly attached to a rotating piece carrying the vernier,
and it is to the position of this piece (and not that of the Nicol) to which
the readings relate. In taking an observation the piece is turned until
the degree of brightness is unaltered, when the Xicol is put over from
the one stop to the other. It is probable that under these advantageous
conditions more favourable results than hitherto would be obtained with
an undivided field of view.
In the application of the polarimeter. with which the present paper is
mainly concerned, the free play of the Nicol is advantageously replaced
by an equivalent rocking of the plane of polarisation itself through a
small angle on either side of its normal position, produced by the action
of an auxiliary electric current, embracing the experimental tube a mode-
rate number of times, and reversed at pleasure by a suitable key under
the hand of the observer.
In these discussions it has been convenient to take as a basis the
fractional difference of brightnesses which can be recognised on simple
presentation to the eye*, but it must be remembered that if suitable
precautions are taken to avoid asymmetry, there is no theoretical limit
of final accuracy. Thus in ordinary photometry with a divided field (e.g^
Bunsen's grease-spot photometer), the match must not be approached from
one side only. By combining a large number of observations in which
the match is approached as much from one side as from the other, a
degree of accuracy may be practically attained far beyond that corre-
sponding to the difference of brightness which can be directly recognised
by the eye. It is not necessary actually to take readings on the two
sides, though it is sometimes desirable to do so: the essential point is to
secure symmetry. Time may be saved by the plan of providing means
for instantaneous displacements of given amount on either side, as was
* August, 1885. I find that the sensitiveness of the eye to small differences of brightness ii
subject to rery rapid fatigue. Even a few seconds' gazing is often enough to obliterate a
distinction quite apparent at first, and appreciable again after a little repose. This defect is
a great obstacle to the further improvement of photometric methods.
382 ON THE CONSTANT OF MAGNETIC [118
done in the experiments of the present paper by the auxiliary reversible
current.
In practical applications of the polarimeter we have almost always to
determine, not so much a particular plane of polarisation as the rotation
of this plane, due to electromagnetic action, to the substitution of syrop
for water, etc., and it appears that the measurement of this angle must
be affected with a possible error, double of the error possible in the
determination of a single plane. M. Becquerel, indeed, in his interesting
memoir upon the rotation in bisulphide of carbon under the terrestrial
magnetic force*, describes a procedure by which, as he considers, the error
may be reduced. By the introduction of a half- wave plate, adjusted so
that its principal section coincides nearly with the plane of first polarisa-
tion, the angle of rotation is, as it were, reflected by the former plane,
and the difference of readings taken with and without the plate is the
double of the real angle of rotation. If e be the greatest angular error
possible in determining a single plane, M. Becquerel shows that the error
in setting the plate cannot exceed e, from which he argues that the whole
error possible in determining the double angle of rotation is only 3e, or f e
upon the single angle. It appears, however, that the error of adjustment
of the half-wave plate enters doubly into the result, so that the whole
error possible in determining the double angle of rotation rises to 4e, and
the use of the half-wave plate gives no advantage.
One other point may be considered in conclusion. In determinations
of rotation by magnetic force, the effect to be measured may be multi-
plied (as Faraday showed), by causing the light to be reflected backwards
and forwards at the ends of the tube. Against this augmentation of the
angle of rotation we must set the loss in the section of the beam, and
the waste of light in reflection and by absorption. Putting out of sight
for the moment the alteration in the section of the beam, we may easily
determine the most advantageous number of passages as dependent upon
magnitude of rotation and intensity of light. If r be the factor by
which the original intensity must be multiplied, in order to express
the intensity after a single passage and reflection, r n will express the
intensity after n such passages and reflections. The accuracy of the de-
termination will thus be proportional to nr ln , which is a maximum when
r=e~ 2/n . The values of r corresponding to n equal to 1, 3, 5, 7,..., are
'135, '514, '670, '752,..., so that 3 or 5 passages will usually give the
best result.
The argument in favour of a moderate use only of the principle of
reflection is strengthened when we take into account the diminution in
the section of the beam. The already contracted aperture is seen at a
* Ann. d. Chim. t. cci. p. 323 ; 1882.
118] ROTATION OF LIGHT IN BISULPHIDE OF CARBON. 383
greater distance (proportional to n), so that the apparent magnitude of
the field of view is rapidly narrowed. Under these circumstances the
comparisons cannot be made with the usual accuracy. If we have re-
course to a telescope we can indeed restore the apparent magnitude, but
(usually) only at expense of the illumination, since the aperture of the
telescope is limited. If the available aperture do not exceed inch, any
degree of magnification involves a loss of brightness. The importance
of these considerations depends upon the length and diameter of the
tube : but the tendency of the discussion is to show that more than five
passages can rarely be desirable, and that in man\- cases three passages
ought to be preferred to five. If there is any exception, it will be when
powerful white light (as from the sun) is available, or when it is possible
by use of a larger number of passages to bring the whole rotation up to
90 or 180, in which cases, as has already been noticed, the angle may be
determined with peculiar advantage.
POSTSCRIPT.
(October, 1885.)
An important paper* has recently been communicated to the French
Academyf- by M. Becquerel, in which he abandons his former result ( 4),
obtained with the aid of terrestrial magnetic force, in favour of a number
agreeing more nearly with that given by Gordon and myself. In the
new experiments a long column of CS 2 was employed, encompassed by
a spiral conveying a current, the effect of which is shown to depend
upon the magnitude of the current and upon the number of turns, in
approximate independence of other circumstances. M. Becquerel speaks
of this method as new, but it is in reality that employed by Gordon
in 1877^. Most of the complication in Gordon's memoir relates to the
determination of the current, and especially to the circumstance that
the number of turns in the spiral was not ascertained (as it should have
been) during construction, but subsequently by electrical processes. When
the number of turns and the current are known, there is no difference
between the procedure of Gordon and Becquerel and that of the present
memoir.
There is a pretty close resemblance between M. Becquerel's recent
work and mine. In both a soda flame is used as the source of light,
and in both the number of windings on the helices is ascertained during
construction. In the current determinations, M. Becquerel used a gal-
vanometer as an intermediate standard, while I employed for the same
purpose a Clark's cell, the ultimate standard being a silver voltameter
* Ann. d. Chim. Oct. 1886. + C. R., June 2, 1885.
* See his equation (24), p. 15.
384 ON THE CONSTANT OF MAGNETIC ROTATION OF LIGHT, ETC. [118
(and in my case a current-weighing apparatus). Inasmuch as M. Becquerel
uses the same number as that which I obtained for the electro-chemical
equivalent of silver, there should be no difference between us in the
estimation of currents.
In M. Becquerel's experiments the temperature of the CS 2 was usually
about C., and he reduces his results to that standard temperature. He
regards Bichat's formula as confirmed by his observations. According to
this my result for 18 would become '04302' ; whereas M. Becquerel obtains
'04341', nearly 1 per cent, higher. I am at a loss to understand the cause
of this discrepancy. M. Becquerel estimates that his result should be
correct to g^, about the same degree of accuracy which I also had hoped
to have attained. So far as I can judge, I should consider that in respect
of current measurement the advantage lay with me, but that on the optical
side M. Becquerel's arrangements were probably superior.
M. Becquerel repeats his proposal* to found upon his value of the
constant a method for current measurement. I had considered this ques-
tion at (I believe) an earlier date ; and the less sanguine view expressed
in the following paragraph seems to be justified by the discrepancies
between the results of various observers at various times as to the value
of the constant in bisulphide of carbon :
"Another method, available with the strong currents which are now
common, depends upon Faraday's discovery of the rotation of the plane
of polarisation by magnetic force. Gordon found 15-f* as the rotation
due to the reversal of a current of 4 amperes circulating about 1000
times round a column of bisulphide of carbon. With heavy glass, which
is more convenient in ordinary use, the rotation is somewhat greater.
With a coil of 100 windings we should obtain 15 with a current of
40 amperes; and this rotation may easily be tripled by causing the light
to traverse the column three times, or what is desirable with so strong
a current, the thickness of the wire may be increased and the number
of windings reduced. With the best optical arrangements the rotation
can be determined to one or two minutes, but in an instrument intended
for practical use such a degree of delicacy is not available. One difficulty
arises from the depolarising properties of most specimens of heavy glass.
Arrangements are in progress for a redetermination of the rotation in
bisulphide of carbon J."
* C. R. t. xcvin. p. 1253 ; 1884.
t Jan. 1884. In a note recently communicated to the Royal Society (Proceedings, Nov. 15,
1883), Mr Gordon points out that, owing to an error in reduction, the number given by him for
the value of Verdet's constant is twice as great as it should be. The rotations above mentioned
must therefore be halved, a correction which diminishes materially the prospect of constructing a
useful instrument upon this principle.
J From the Proceedings of the Cambridge Philosophical Society for Nov. 26, 1883. See also
Nature, Dec. 13, 1883.
119.
OPTICS.
[Encyclopedia Brita*xica T XTIL
OPTICS, Geometrical. The subject of optics is so extensive that some
subdivision of it is convenient if not necessary. Under the head of LIGHT will
be found a general sketch accompanied by certain development*. The wave
theory and those branches of the subject which are best expounded in
connexion with it are reserved for treatment in a later volume. The object
of the present paper is to give some account of what is generally called
geometrical optics, a theoretical structure based upn the laws ->f reri*-xkn
and refraction. We shall, however, find it advisable not to exclude altogether
the conceptions of the wave theory, for on certain most important and prac-
tical questions no conclusions can be drawn without the use of facts which
are scarcely otherwise interpretable. Indeed it is IK* to be denied that the
too rigid separation of optics into geometrical and physical has done a
deal of harm, much that is essential to a c x
proper comprehension of the subject
having Mien between the two stools.
Systems of Bays IN General In the
investigation of this subject a few prelimi-
nary propositions will be useful.
If a ray AB (fig. 1) travelling in a
homogeneous medium suffer reflexion at a plane or curved surface BD, the
total path between any two points A, C on the ray is a minimum, t>.
AB + BC is less along the actual path than it would be if the point B were
slightly varied
For a variation of B in a direction perpendicular to the plane of reflexion
(that of the diagram) the truth of this statement is at once evident. For a
small variation BB in the plane of reflexion we see that the difference
K. IL 25
386 OPTICS. [119
AB' AB is equal to the projection of BB' upon AB, and that the difference
GB CB' is equal to the projection of BB' upon BC. These projections are
equal, since by the law of reflexion AB and BG are equally inclined to BB' ,
and thus the variation of the total path, AB' + B'G (AB + BC}, vanishes.
A corresponding proposition holds good in the case of refraction. If we
multiply the distances travelled in the first and second media respectively by
the refractive indices appropriate to the media, the quantity so obtained is a
minimum for the actual path of the ray from any point to any other. It is
sufficient to consider the case of a variation of the point of passage in the
plane of refraction.
In the first medium (fig. 2) fiAB' fj.AB = n.BB'cQ^ABD, and in the
second medium // CB - // CB' = // BB' cos CBD.
The whole variation of the quantity in question
is therefore
BB' O cos ABD - // cos CBD).
Now by the law of refraction the sines of the
angles of incidence and refraction are in the
ratio p.' : p., and accordingly
yu, cos ABD- p! cos CBD = 0.
In whichever direction, therefore, the point of
Fig. 2. transition be varied, the variation of the quantity
under consideration is zero. It is evident that the second proposition
includes the first, since in the case of reflexion the two media are the
same.
The principle of the superposition of variations now allows us to make an
important extension. If the quantity, which we may denote by S/AS, be a
minimum for separate variations of all the points of passage between con-
tiguous media, it is also a minimum even when simultaneous variations are
admitted. However many times a ray may be reflected or refracted at the sur-
faces of various media, the actual path of the ray between any two points of its
course makes S/*s a minimum. Even if the variations of refractive index be
gradual instead of sudden, the same principle holds good, and the actual path
of the ray makes fads, as it would now be written, a minimum.
The principle itself, though here deduced from the laws of reflexion and
refraction, is an immediate consequence of the fundamental suppositions of
the wave-theory of light, and if we are prepared to adopt this point of view
we may conversely deduce the laws of reflexion and refraction from the
principle. The refractive index /i is inversely proportional to the velocity of
propagation, and the principle simply asserts that in passing from any point
to any other the light follows the shortest course, that is, the course of
earliest arrival.
119]
OPTICS.
Fig. 3.
If two points be such that rajs issuing from one of them, and ranging
through a finite angle, converge to the other after any
number of reflexions and refractions, the value of 5^*s from
one focus to the other must be the same for all the raya
Thus, in order to condense rays issuing from one point
S upon a second point H by a single reflexion (fig. 3), the
reflecting surface must be such that SP + HP= const., i.e. must be an ellip-
soid of revolution with S and H foci
Again, if it be required to effect the same operation by a single refraction
at the surface of a medium whose index
is p., we see that the surface (fig. 4)
must be such that ^~H J s
= const.
Fig. 4.
If S be at an infinite distance, i.e. if the incident rays be parallel, the surface
is an ellipsoid of revolution with H for focus, and of eccentricity pr l (/* > 1).
Another important proposition, obvious from the point of view of the
wave-theory, but here requiring an independent proof, was enunciated by
Mains. It asserts that a system of rays, emanating originally from a point.
retains always the property of being normal to a surface, whatever reflexions
or refractions it may undergo in traversing singly-refracting media.
Suppose that ABODE, A'RO'D^E' . . . (fig. 5) are rays originally normal
to a surface A A', which undergo reflexions ---__A A'
or refractions at BR, CO', &c. On every
ray take points E, E r , Sue., such that 2/*s is
the same along the courses AE, A'E, &c,
We shall prove that the rays in the final
medium are normal to the surface EE'.
For by hypothesis 2/i* along ABODE is
the same as along A'RC'DE', and, by the
property proved above to attach to every
ray, S/A* reckoned along the neighbouring
hypothetical course A'EGDE is the same
as along A'RC'D'E'. Hence S/AS along
A'BCDE" is the same as along ABODE, or (on subtraction of the common
part) the same along A'B, DE' as along AB, DE. But since AB is perpen-
dicular to A A', the value along A'B is the same as along AB, and therefore
the value along DE" is the same as along DE; or, since the index is the
same, DE = DE, that is, EE is perpendicular to DE. The same may be
proved for every point E which lies infinitely near E, and thus the surface
EE is perpendicular to the ray DE, and by similar reasoning to every other
ray of the system. It follows that reflexions and refractions cannot deprive
252
Fig. 5.
388 OPTICS. [119
a system of rays of the property of being normal to a surface, and it is
evident that a system issuing from a point enjoys the property initially.
Consecutive rays do not in general intersect one another; but if we
select rays which cut the orthogonal surface along a line of curvature, we
meet with ultimate intersection, the locus of points thus determined being a
caustic curve to which the rays are tangents. Other lines of curvature of
the same set give rise to similar caustic curves, and the locus of these curves
is a caustic surface to which every ray of the system is a tangent. By con-
sidering the other set of lines of curvature we obtain a second caustic surface.
Thus every ray of the system touches two caustic surfaces.
In the important case in which the system of rays is symmetrical about
an axis, the orthogonal surface is one of revolution. The first set of lines of
curvature coincide with meridians. The rays corresponding to any one
meridian meet in a caustic curve, and the surface which would be traced out
by causing this to revolve about the axis is the first caustic surface. The
second set of lines of curvature are the circles of latitude perpendicular to
the meridians. The rays which are normal along one of these circles form a
cone of revolution, and meet in a point situated on the axis of symmetry.
The second caustic surface of the general theorem is therefore here repre-
sented by a portion of the axis.
The character of a limited symmetrical pencil of rays is illustrated in
fig. 6, in which BAG is the orthogonal surface, and HFI the caustic curve
having a cusp at F, the so-called geometrical focus. The distance FD
between F and the point where the extreme ray BHDG cuts the axis
is called the longitudinal aberration. On
account of the symmetry FD is an even func-
tion of AB. If the pencil be small, we may
in general consider FD to be proportional to
AB' 2 , although in particular cases the aberra-
tion may vanish to this order of approxima-
tion. Let us examine the nature of the
sections at various points as they may be
exhibited by holding a piece of paper in the
solar rays converging from a common burning-
glass of large aperture. In moving the paper
towards the focus nothing special is observed
up to the position HI, where the caustic surface is first reached. A bright
ring is there formed at the margin of the illuminated area, and this gradually
contracts. At D the second caustic surface DF is reached, and a bright spot
develops itself at the centre. A little farther back, at EG, the area of the
illuminated patch is a minimum, and its boundary is called the least circle of
aberration. Farther back still the outer boundary corresponding to the
119] OPTICS. 389
extreme rays begins to enlarge, although the circle of intersection with the
caustic surface continues to contract. Beyond F the caustic surfaces are
passed, and no part of the area is specially illuminated.
As a simple example of a symmetrical system let us take the case of
parallel rays QR, OA (fig. 7), incident upon a
spherical minor AR, By the law of reflexion
the angle ORq = angle ORQ = angle qOR.
Hence the triangle RqO is isosceles, and if we
denote the radius of the surface OA by r, and
the angle AOR by a, we have
If F be the geometrical focus, OF = AF = ir.
If a be a small angle, the longitudinal aberra-
tion Fq = 0q OF= |r (sec a 1) = \afr, in which AR = ra.
Focal Lines. In the general case of a small pencil of rays there is no one
point which can be called the geometrical focus. Consider the corresponding
small area of the orthogonal surface and its two sets of lines of curvature.
Of all the rays which are contiguous to the central ray there are only two
which intersect it, and these will in general intersect it at different points.
These points may be regarded as foci, but it is in a less perfect sense than in
the case of symmetrical pencils. Even if we limit ourselves to rays in one of
the principal planes, the aberration is in general a quantity of the first order
in the angle of the pencil, and not, as before, a quantity of the second order.
If, however, we neglect this aberration and group the rays in succession
according to the two sets of lines of curvature, we see that the pencil of rays
passes through two focal lines perpendicular to one another and to the
central ray, and situated at the centres of curvature of the orthogonal
surface. At some intermediate place the section of the pencil is circular.
It happens not unfrequently that the pencil under consideration forms
part of a symmetrical system, but is limited in such a manner that the
central ray of the pencil does not coincide with the axis of the system. The
plane of the meridian of the orthogonal surface is called the primary plane,
and the corresponding focus, situated on the caustic surface, the primary
focus. The secondary focus is on the axis of symmetry through which every
ray passes. The distinction of primary and secondary is also employed when
the system, though not of revolution, is symmetrical with respect to a plane
passing through the central ray, this plane being considered primary.
The formation of focal lines is well shown experimentally by a plano-
convex lens of plate-glass held at an obliquity of 20 J or 30 C in the path of
the nearly parallel rays, which diverge from a small image of the sun formed
by a lens of short focus. The convex face of the lens is to be turned towards
390
OPTICS.
[119
the parallel rays, and a piece of red glass may be interposed to mitigate the
effects of chromatic dispersion.
To find the position of the focal lines of a small pencil incident obliquely
upon a plane refracting surface of index JJL.
The complete system of rays issuing from Q (fig. 8) and refracted at the
plane surface CA is symmetrical about the line QC drawn through Q perpen-
dicularly to the surface. Hence, if Q A be the central ray of the pencil, the
Fig. 8.
secondary focus q 2 lies at the intersection of the refracted ray with the axis.
If </> be the angle of incidence, <f>' of refraction, AQ = u, Aq 2 =v,,, then
v 2 _ sn
(1)
u sin <p
To find the position of the primary focus q lt let QA' be a neighbouring ray in
the primary plane (that of the paper) with angles of incidence and refraction
< -f 8<f> and <f>' + S<', Aq l = v l . We have
AA' cos d) = uSd>, A A' cos <4' = Vi8<b' j
moreover, by the law of refraction,
cos o0 = /M cos 0'80' ;
Vj _ fJL COS 2 0'
and thus
(2)
If the refracting surface be curved, with curvature 1/r, we get by similar
reasoning
fl COS 2 <' COS 2 _ /i COS (j)' COS (f)
u r
fji 1 _ fl COS <// COS (j>
~
w 2 u r
in which (1) and (2) are of course included as particular cases.
.(3)
.(4)
119] OPTICSL 391
When the incidence is direct. cos'=l, cosf =1, and i^=^- In
this case 3 and (4i become
C-i.tJ .............................. .(5)
r H r
To jtoi**/ Mtf pjititi&Hf of the fijfal limes of a pencil refracted obliquely
a plate of tkickmess t and index JUL
If 6 be the angle of incidence (and emergence), ' the angle of refraction
of the ray QA$T (fig. 9), &fr = r,. %=c., AQ = x. we get by successive
applications >f li and t2)
If the incidence be direct,
Thns T if we interpose a plate between the eye and an object, the effect is t
bring the object apparently nearer by the amount
49)
On this result is founded a method for determining the refractive index
of materials in the form of plates. A set of
cross wires is observed through a magnifying
glass. On interposition of the plate the
glass must be drawn back through a dis-
tance given by (9) in order to recover the
focus. If we measure this distance and the
thickness of the plate, we are in a position
to determine the refractive index.
Prisms By a prism is meant in optics a
portion of transparent material limited by ***- 9 -
two plane faces which meet at a finite angle in a straight line called the
edge of the prism. A section perpendicular to the edge is called a prin-
cipal section.
Parallel rays, refracted successively at the two faces, emerge from the
prism as a system of parallel rays. The angle through which the rays are
bent is called the deviation.
The deviation depends upon the angles of incidence and emergence ; but,
since the course of a ray may always be reversed, the deviation is necessarily
a symmetrical function of these angles. The deviation is consequently a
392
OPTICS.
[119.
maximum or a minimum when a ray within the prism is equally inclined to
the two faces, in which case the angles of incidence and emergence are equal.
It is in fact a minimum ; and this position of the prism is described as the
position of minimum deviation, and is usually adopted for the purposes of
measurement.
The relation between the minimum deviation D, the angle of the prism i,
and the refractive index jj, is readily
found. In fig. 10 the internal angles
<', ty' are each equal to \i. The ex-
ternal angles <, -^ are also equal, and
are connected with <' by the law of
refraction sin < = p sin <&'. The devia-
tion is 2 (< <'). Hence
Fig- 10. ~sin0'~ sin it
and this is the formula by which the refractive index is usually determined,
since both D and i can be measured with great precision.
The instrument now usually employed for this purpose is called a gonio-
meter or spectrometer. Parallel rays are provided by a collimator, consisting
of an object-glass and telescope-tube, by means of which the subject of
examination, either a fine slit or a set of cross wires, is seen as if it were at
an infinite distance. The parallel rays from the collimator, after reflexion
from a face or refraction through the body of the prism, are received by a
telescope also provided with a set of cross wires at its focus. The table upon
which the prism is supported, as well as the telescope, are capable of rotation
about a vertical axis, and the position of either can be read off at any time
by means of graduated circles and verniers.
As a preliminary to taking an observation it is necessary to focus the
collimator and telescope. The first step is to adjust the eye-lens of the
telescope until the cross wires are seen distinctly and without effort. The
proper position depends, of course, upon the eyesight of the observer, and is
variable within certain limits in virtue of the power of accommodation. It is
usually best to draw out the lens nearly to the maximum distance consistent
with distinct vision. The telescope is now turned to a distant object and
focused by a common motion of the cross wires and eye-lens, until both the
object and the cross wires are seen distinctly at the same time. The final
test of the adjustment is the absence of a relative motion when the eye is
moved sideways across the eye-piece. The collimator is now brought oppo-
site to the telescope and adjusted until the cross wires in its focus behave
precisely like the distant object.
119]
opncsu
To measure the angle of a prism it may be placed with its edge vertical
upon the table, in a symmetrical position with
respect to the oollimator (fig. 11). The tele-
scope is then successively brought into such
positions that the cross wires of the telescope
coincide with the cross wires of the collimator
when seen by reflexion in the two faces. The
difference of the readings is twice the angle of
the prism.
Another method is also often employed in Kg- 11-
which the telescope is held fixed and the prism is rotated. The angle
between the two positions of the table found by use in succession of the two
faces is the supplement of the angle of the prism.
Suppose next that we wish to determine D for the given prism and for
sodium light. The slit of the collimator is backed by a sodium flame, the
telescope is adjusted for direct vision of the slit, and the reading taken. The
prism is now placed upon the table, and rotated until the deviation of the
light from its original direction when seen through the prism is a minimum.
The difference of the readings for the two positions of the telescope is the
value of D. The angle to be observed may be doubled by using the devia-
tion in both directions. In this case no direct reading in the absence of the
prism is required.
The following table of indices of refraction is taken from Watt's Diction-
ary of Chemistry, article Light,"
Same of Substance
1-534
1-531 to 1-552
1-532
1-5-27
1-525 to 1-534
1514
1-514 to 1-542
1-503
1-500
1-500
l-44
1-492
1-4x8
1-654
1-438
1-476
1-475
1457
1-436
1-310
1-1115
Cade add
Crown <rlmss
Site .
Ferrous
Tallow; was
Sulphate of magnesium
Iceland spar
Gum
: .- i
Alum
'-'. - :-: :-.:
lee
A selection from some results given by Hopkinson*. relating to Chance s
glaaum, may be useful to those engaged in the designing of optical instruments.
* Pne. Bey. See. June 1877.
394
OPTICS.
[119
D is the more refrangible of the pair of sodium lines ; b is the most refrangible
of the group of magnesium lines ; (G) is the hydrogen line near G.
Hard
Crown
a
Soft
Crown
Extra
Light
Flint
Light
Flint
Dense
Flint
Extra
Dense
Flint
Double
Extra
Dense
Flint
Specific (
Gravity j
2-48575
2-55035
2-86636
3-20609
3-65865
3-88947
4-42162
B
C
1-513625
1-514568
1-510916
1-511904
1-536450
1-537673
1-568558
1-570011
1-615701
1-617484
1-642874
1-644866
1-701060
1-703478
D
1-517114
1-514591
1-541011
1-574015
1-622414
650388
1-710201
E
b
1-520331
1-520967
1-518010
1-518686
545306
1-546166
1-579223
1-580271
1-628895
1-630204
1-657653
659122
1-719114
1-720924
F
1-523139
1-520996
1-549121
1-583886
1-634748
664226
1-727237
(G)
G
1-527994
1-528353
1-526207
1-526595
555863
556372
1-592190
1-592824
1-645267
1-646068
676111
677019
1-742063
1-743204
h
1-530902
1-529359
1-560010
1-597332
1-651840
1-683577
1-751464
HI
1-532792
1-531416
1-562760
1-600727
1-656219
688569
1-757785
To determine the index of refraction of a liquid it must of course be
placed in a hollow prism, whose faces are formed of some transparent
material, usually of glass. The following results of Dale and Gladstone
show the influence of temperature upon the refracting power of some
important liquids. They relate to the soda flame, or the line D in the
solar spectrum.
Tempera-
ture
Bisulphide of
Carbon
Water
Ether
Alcohol
Absolute
1-6442
1-3330
..
10
1-6346
1-3327
1-3592
1-3658
20
1-6261
1-3320
1-3545
1-3615
30
1-6182
1-3309
1-3495
1-3578
40
1-6103
1-3297
1-3536
50
1-3280
1-3491
60
1-3259
1-3437
Refractive Indices of Bisulphide of Carbon for the several Fixed Lines.
Temperature
A
B
D
E
F
G
11
36'5
1-6142
1-5945
1-6207
1-6004
1-6333
1-6120
1-6465
1-6248
1-6584
1-6362
1-6836
1-6600
Difference ...
0-0197
0-0203
0-0213
0-0217
0-0222
0-0236
119] OPTICS. 395
The rapid alteration of refractive power with temperature is a serious
obstacle to the use of bisulphide of carbon prisms for exact purposes. Not
only does the dispersive power vary from day to day, but inequalities of
temperature in the various parts of the liquid at any one moment disturb
the optical uniformity, and are thus the cause of bad definition. A difference
of 1 Cent, alters the index about as much as a change in the light from one
of the two D lines to the other, so that a variation of one degree within the
prism may be expected to prevent the satisfactory resolution of this double
line.
Excellent results have recently been obtained by Liveing with prisms
containing aqueous solution of iodide of potassium and mercury. This
liquid can be brought up to a density as high as three times that of water,
and gives a powerful dispersion. Some difficulty has, however, been expe-
rienced in finding a suitable cement for the faces. Bisulphide of carbon
prisms are usually cemented with a mixture of glue and treacle.
For many purposes the deviation of the light in passing through an
ordinary prism is objectionable. In such cases recourse may be had to direct
vision prisms (fig. 12), in which two mate-
rials, usually flint and crown, are so com-
bined that the refractions are equal and
opposite for a selected ray, while the dis-
persions are as unequal as may be. The direct vision prism may be con-
trasted with the achromatic lens (see LIGHT). In the first the object is to
obtain dispersion without refraction, and in the second to obtain refraction
without dispersion.
Compound prisms, composed of a flint between two crowns, are also made,
in which the action of the crown is not carried so far as to destroy the
deviation due to the flint. By this construction a larger angle is admissible
for the more dispersive material, but it is not clear that any sufficient
advantage is gained.
The principle of the compound prism is carried to its limit by employing
media of equal refracting power for the part of the spectrum under examina-
tion. For this purpose bisulphide of carbon and flint glass may be chosen.
With Chance's " dense flint " the refractions are the same, and the difference
of dispersions is about as great as for " double-extra-dense flint " and crown.
A dozen glass prisms of 90 may be cemented in a row on a strip of glass
and immersed in a tube of bisulphide of carbon closed at the ends by glass
plates. To vary the ray, which passes without deviation, ether may be mixed
with the bisulphide*.
The formation of a pure spectrum, which may be either thrown upon a
screen or photographic plate, or received at once by the eye armed with a
See "Investigations in Optics," Phil. Mag. January 1880. [Vol. i. Ark 62.]
396 OPTICS. [119
magnifier, has been explained under LIGHT. It sometimes happens that the
object is not to see the spectrum itself, but to arrange a field of view
uniformly illuminated with approximately homogeneous light. For this
purpose the pure spectrum is received upon a screen perforated by a narrow
slit parallel to the fixed lines. The light which passes this second slit (eye-
slit) is approximately homogeneous. Suppose that it corresponds to the red
of the spectrum. The eye, placed immediately behind the eye-slit, receives
only red light, and, if focused upon the prism, sees a red field of view whose
brightness is uniform if the light falling in different directions upon the
original slit be uniform. To secure the fulfilment of the last condition we
may use the light from an overcast sky, or that of the sun reflected from a
large surface of white paper. If it be desired to work by artificial light, an
Argand gas flame diffused by an opal globe will be found suitable. When
the adjustments are correct the tint should be perfectly uniform. Any
difference of colour on the two sides of the field of view is an indication that
the screen is not in its proper place.
The most important application of this arrangement is to the investiga-
tion of compound colours, as carried out by Maxwell*. If light be admitted
also through a second slit, displaced laterally from the position occupied by
the first, a second spectrum overlapping the former will be thrown upon the
screen, and a second kind of light will be admitted to the eye. In this way
we may obtain a field of view lighted with a mixture of two or more
spectrum colours, and we may control the relative proportions by varying
the widths of the slits. For instance, by mixing almost any kind of red with
any kind of green not inclining to blue we may match the brightest yellows,
proving what so many find it difficult to believe, that yellow is a compound
colour. In Maxwell's systematic examination of the spectrum, mixtures of
three colours were used, and the proportions were adjusted so as to match
the original white light incident upon the apparatus.
A similar arrangement (with one original slit) was employed by Helm-
holtz in his examination of a fundamental question raised by Brewster. The
latter physicist maintained that there was abundant evidence to show that
light of definite refrangibility was susceptible of further analysis by absorp-
tion, so that the colour of light (even of given brightness) could not be
defined in terms of refrangibility or wave-length alone. The appearances
which misled Brewster have since been explained as the effect of contrast or
of insufficient purity. It is obvious that light, e.g., from the red end of the
spectrum, may be contaminated with light from some other part, say the
yellow, in such proportion that though originally entirely preponderant it
may fall into the second place under the action of a medium very much more
transparent to yellow than to red. To obtain light of sufficient purity for
* " Theory of Compound Colours," Phil. Trans. 1860.
119] OPTICS. 397
these experiments Helmholtz found it advisable to employ a double prismatic
analysis. A spectrum is first thrown upon a screen perforated by a slit in
the manner already described The light which penetrates the second slit,
already nearly pure, is caused to pass a second prism by the action of which
any stray light is thrown aside. Using such doubly purified light, Helm-
holtz found the colour preserved, whatever absorbing agents were brought
into play. Light of given refrangibility may produce a variety of effects,
visual, thermal, or chemical, but (apart from polarization) it is not itself
divisible into parts of different kinds. If yellow light produces the com-
pound sensation of yellow, we are to seek the explanation in the constitution
of the retina, and not in the divisibility of the light.
In all accurate work with the prism the use of a collimating lens to
render the incident light parallel is a matter of necessity. If the incident
rays diverge from a point at a finite distance, the pencil after emergence will
be of a highly complicated character. There are, however, cases in which a
collimator is dispensed with, and thus it is a problem of interest to find the
foci of a thin pencil originally diverging from a point at a moderate distance.
Even when a collimator is employed, the same problem presents itself when-
ever the focusing is imperfect. For the sake of simplicity the pencil is
supposed to pass so near the edge of the prism that the length of path
within the glass may be neglected in comparison with the distances of the
foci
We denote as usual the angles of incidence and emergence by <b, ty, and
the corresponding angles within the glass by <f>', \jr'. The distance AQ from
the edge of the prism to original source is denoted by u- the corresponding
distances for the primary and secondary foci q lf q 3 by r,, v+. By successive
applications of the results already proved for a single refraction, we get
cos 2 d>
so that
-*
In order that the primary and secondary foci may coincide we must have
^r = tf> ; that is to say, the ray must pass with minimum deviation. This is
sometimes given as a reason why this arrangement should be adopted in
spectroscopes; but in reality, since the slit is parallel to the edge of the
prism, a slight elongation in this direction of the image of a point is without
detriment to the definition. Hence a good image will be seen when the
telescope is adjusted for the primary focus; and it is not clear that any
improvement would arise from coincidence of the two foci, the question being
in feet one of aberration. The position of minimum deviation is, however,
usually adopted for the sake of definiteness, and sometimes it is convenient
398
OPTICS.
[119
that the fixed lines and the extremities of the slit (or the markings produced
by dust) should be in focus together.
The deviation is a symmetrical function of </> and i/r, and therefore is not
altered by an interchange of these angles. The corresponding values of v
are thus by (1) reciprocals, and their product is equal to u*. This principle
has been ingeniously applied by Schuster* to the adjustment for focus of the
telescope and collimator of a spectroscope. The telescope is so placed that
the deviation necessary to bring the object upon the cross wires is greater
than the minimum, and the prism is adjusted in azimuth until the effect is
produced, that position being chosen for which the angle of incidence is
greater than the angle of emergence, so that ^ is greater than u. After
focusing the telescope the prism is turned into the other position which gives
the same deviation, and the collimator is focused, the telescope remaining
untouched. The prism is next brought back to the first position, and the
telescope is again focused. A few repetitions of this operation, always
focusing the telescope in the first position of the prism and the collimator in
the second, will bring both into perfect adjustment for parallel rays.
Lenses. The usual formula for the focal length of lenses (Enc. Brit.
vol. xiv. p. 593),
ignores the fact that the various parts of a lens bounded by spherical surfaces
have not the same focus, and is applicable in strictness only when the aper-
ture is small. It is not necessary here to repeat the process by which (1) is
usually obtained, but before passing on to give the formulae for the aberra-
tion of lenses it may be well to exhibit the significance of (1) from the point
of view of the wave-theory.
Taking the case of a convex lens of glass, let us suppose that parallel
rays DA, EG, GB (fig. 13) fall
upon the lens ACB, and are col-
lected by it to a focus at F. The
points D, E, G, equally distant
from ACB, lie upon a front of the
wave before it impinges upon the
lens. The focus is a point at which
the different parts of the wave
arrive at the same time, and that such a point can exist depends upon the
fact that the propagation is slower in glass than in air. The ray EOF is
retarded from having to pass through the thickness (i) of glass by the
amount (fj, l)t. The ray DAF, which traverses only the extreme edge of
the lens, is retarded merely on account of the crookedness of its path, and
* Phil. Mag. February 1879.
119] OFTICSL 399
the amount of the retardation is measured by AF CF. If F is a foots
these retardations must be equal, or
Now if y be the semi-aperture AC of the lens, and/ be the focal length CF,
AF- CF= \ r \f* + !f\ -/= kff approximately,
whence
In the case of plate-glass ji 1 = \ nearly, and then the rule (21 may be
thus stated : the semi-aperture i* a mean proportional between the focal length
and the thickness. The form (2) is in general the more significant, as well as
the more practically useful, but we may of course express the thickness in
terras of the curvatures and semi-aperture by means of
In the preceding statement it has been supposed for simplicity that the
lens comes to a sharp edge. If this be not the case we must take as the
thickness of the lens the difference of the thicknesses at the centre and at the
circumference. In this form the statement is applicable to concave lenses.
and we see that the focal length is positive when the lens is thickest at the
centre, but negative when the lens is thickest at the edge,
To determine practically the focal length of a convex lens we may
proceed in several ways. A conve- . *
nient plan is to set up a source of
light Q (fig. 14) and a screen q at a Q *~~ ~~<^~ ~* 1
distance exceeding four times the focal
length, and to observe the two posi-
tions of the lens A, A' at which the source is in focus upon the screen.
These positions are symmetrically situated, and the distance between them
is observed. Thus
Now
so that
AQ.Ag_,Q<?-AA*
AQ + Aq Qq
From the measured values of Qq and A A, /can be deduced.
If A and A' coincide, the conjugate foci Q and q are as close as possible
to one another, and then/= {Qq.
400 OPTICS. [119
The focal length on a concave lens may be found by combining it with a
more powerful convex lens of known focus.
Aberration of Lenses. The formula (1) determines the point at which a
ray, originally parallel to the axis and at but a short distance from it, crosses
the axis after passage through the lens. When, however, the ray considered
is not quite close to the axis, the point thus determined varies with the
distance y. In the case of a convex lens the ray DH (fig. 15), distant HO
(= y) from the axis, crosses it after refraction at a point F' which lies nearer
V
Fig. 15.
to the lens than the point F determined by (1), and corresponding to an
infinitely small value of y. The distance F'F is called the longitudinal
aberration of the ray, and may be denoted by Bf.
The calculation of the longitudinal aberration as dependent upon the
refractive index (/JL) and the anterior and posterior radii of the surfaces (r, s)
is straight forward, but is scarcely of sufficient interest to be given at length
in a work like the present. It is found that
,~
r, s, and /being related as usual by (1).
The first question which suggests itself is whether it is possible so to
proportion r and s that the aberration may vanish. Writing for brevity
R, S, F respectively for r~ l , s~ l ,f~*, and taking
G= ^j, so that -S = (//*) -.R,
we get
Since n > 1, both terms are of the same sign ; and thus it appears that
the aberration can never vanish, whatever may be the ratio of r to s. Under
these circumstances all that we can do is to ascertain for what form of lens
119] OPTICS. 401
the aberration is a minimum, the focal length and aperture being given.
For this purpose we must suppose that the first term of (4) vanishes, which
gives
2(/* + 2)(u 1) .
r= -^ f. (o)
The corresponding value of s is
so that
In the case of plate-glass / = T5 nearly, and then from (5), (6), (7)
Both surfaces are therefore convex, but the curvature of the anterior surface
(that directed towards the incident parallel rays) is six times the curvature
of the posterior surface. By (3) the outstanding aberration is
<>
The use of a plano-convex lens instead of that above determined does not
entail much increase of aberration. Putting in (3) s = x , and therefore by
(l)r-l/ we get
*/=-!/ .................................. <)
This is on the supposition that the curved side faces the parallel rays. If
the lens be turned round so as to present the plane face to the incident light
we have r = x , s = \f, and then
nearly four times as great.
For a somewhat higher value of /* the plano-convex becomes the form of
minimum aberration. If s= & in (6), 4 + /* - 2/i a = 0, whence /t = 1 69.
If p be very great, we see from (5) and (6) that r and s tend to become
identical with f.
For the general value of /* the minimum aberration corresponding to (7)
is by (4)
* - (11)
26
402 OPTICS. [119
The right-hand member of (11) tends to diminish as //, increases, but it
remains considerable for all natural substances. If p = 2,
Oblique Pencils. Hitherto we have supposed that the axis of the pencil
coincides with the axis of the lens. If the axis of the pencil, though incident
obliquely, pass through the centre of the lens, it suffers no deviation, the
surfaces being parallel at the points of incidence and emergence. In this
case the primary and secondary foci are formed at distances from the centre
of the lens which can only differ from the distance corresponding to a direct
pencil by quantities of the second order in the obliquity. Hence, if the
obliquity be moderate, we may use the same formulae for oblique as for direct
pencils.
The consideration of excentrical pencils leads to calculations of great
complexity, upon which we do not enter.
Chromatic Aberration. The operation of simple lenses is much interfered
with by the variation of the refractive index with the colour of the light.
The focal length is decidedly less for blue than for red light, and thus in the
ordinary case of white light it is impossible to obtain a perfect image,
however completely the spherical aberration may be corrected. From the
formula for the focal length we see that
so that
or the longitudinal chromatic aberration varies as the focal length and as the
dispersive power of the material composing the lens. The best image will be
formed at a position midway between the two foci, and the diameter d of the
circle over which the rays are spread bears the same ratio to the semi-
aperture of the lens (y) that 8f bears to f. Hence
The diameter of the circle of chromatic aberration is thus proportional to the
aperture and independent of the focal length ; and, since the linear dimensions
of the image are proportional to the focal length, the confusion due to
chromatic aberration may be considered to be inversely as the focal length.
Before the invention of the achromatic object-glass this source of imperfect
definition was by far the most important, and, in order to mitigate its
influence, telescopes were made of gigantic length. Even at the present day
the images of large so-called achromatic glasses are sensibly impaired by
OFT1C&. 403
secondary chromatic aberration, the effect of which is also directly as the
aperture and inversely as the focal length.
Object-glasses. It has been shown in Emc. BriL voL xnr.
p. 59-5, that the condition of achromatism for two thin lenses placed dose
together is
in which//' are the focal lengths of the two lenses, and
$p'.<(fL 1| the dispersive powers of the two kinds of glass. In practice
crown and flint glass are used, the dispersive power of the flint being greater
than that of the crown. Thus/' is negative and numerically greater than/
$o that the combination ooBsasts of a convex lens of crown and a concave lens
of flint, the converging power of the crown overpowering the diverging power
of the flint. When the focal length F of the combination is given, the focal
lengths of the individual lenses are determined bj (1) in conj unction -with
Tne matter, however, is not quite so simple as the above account of ii
might lead us to suppose. In consequence of what is called the irrationality
of spectra, the ratio of dispersive powers of two media is dependent npon the
parts of the spectrum which we take into consideration. Whatever two rays
of the spectrum we like to select, we can secure that the compound lens
shall have the same focal length for these rays, but we shall then find that
for other rays the focal length is slightly different In the case of a single
lens the focal length continually diminishes as we pass np the spectrum from
red to violet. By the us* of two lenses the spectrum, formed as it were along
the axis, is doubled upon itself. The focal length is least for a certain ray,
which may be selected at pleasure. Thus in the ordinary achromatic lens,
intended for use with the eye, the focal length is a minimum for the green,
and increases as we pass away from the green, whether towards red or
towards blue. Stokes has shown that the secondary colour gives a sharp test
of the success of the achromatizing process.
The secondary tints in an objective are readily shown by directing the
telescope to a vertical line separating fight from dark, such as the edge of a
chimney seen in the shade against the sky, and covering half the object-glass
whh a screen having a vertical edge. So delicate is this test that, on testing
different telescopes by well-known opticians, a difference in the mode of
achromatism may be detected. The best results are said to be obtained
when the secondary green is intermediate between green and yellow. This
corresponds to making the focal length a minimum for the brightest part of
the spectrum.
404 OPTICS. [119
"To enable me to form a judgment as to the sharpness of the test
furnished by the tint of the secondary green, as compared with the perform-
ance of an object-glass, I tried the following experiment. A set of parallel
lines of increasing fineness was ruled with ink on a sheet of white paper, and
a broader black object was laid upon it as well, parallel to the lines. The
paper was placed, with the black lines vertical, at a considerable distance on
a lawn, and was viewed through two opposed prisms, one of crown glass and
the other of flint, of such angles as nearly to achromatize each other in the
positions of minimum deviation, and then through a small telescope. The
achromatism is now effected, and varied in character, by moving one of the
prisms slightly in azimuth, and after each alteration the telescope was
focused afresh to get the sharpest vision that could be had. I found that the
azimuth of the prism was fixed within decidedly narrower limits by the con-
dition that the secondary green should be of such or such a tint, even though
no attempt was made to determine the tint otherwise than by memory, than
by the condition that the vision of the fine lines should be as sharp as
possible. Now a small element of a double object-glass may be regarded, so
far as chromatic compensation is concerned, as a pair of opposed prisms ; and
therefore we may infer that the tint of the secondary green ought to be at
the very least as sharp a test of the goodness of the chromatic compensation
as the actual performance of the telescope*."
In the case of photographic lenses the conditions of the problem are
materially different. It is usually considered to be important to secure
" coincidence of the visual and chemical foci," so that the sensitive plate may
occupy the exact position previously found by the eye for the ground glass
screen. For this purpose the ray of minimum focus must be chosen further
up in the spectrum. If, however, the object be to obtain the sharpest possible
photographs, coincidence of visual and chemical foci must be sacrificed, the
proper position for the sensitive plate being found by trial. The middle of
the chemically-acting part of the spectrum, which will vary somewhat accord-
ing to the photographic process employed, should then be chosen for mini-
mum focus.
When the focal lengths of the component lenses have been chosen, it
still remains to decide upon the curvatures of the individual faces. Between
the four curvatures we have at present only two relations, and thus two more
can be satisfied. One of these is given by the condition that the first term
in the expression for the aberration that proportional to the square of the
aperture shall vanish for parallel rays. As to the fourth condition, various
proposals have been made. If equal and opposite curvatures are given to
the second and third surfaces, the glasses may be cemented together, by
which some saving of light is effected. Herschel proposed to make the
* Proc. Boy. Soc. June 1878.
119]
OPTICS.
4C5
aberration vanish for nearly parallel, as well as for absolutely parallel, rays.
This leads to a construction nearly agreeing with that adopted by Fraun-
hofer.
The following results are given by Herschel* for the radii of the four
surfaces, corresponding to various dispersive powers, and to mean refractive
indices 1'524' (crown) and 1'585 (flint). The focal length of the combination
is taken equal to 10, and, as well as the radii, is measured in arbitrary units;
so that all the numbers in the table (with the exception of the first column)
may be changed in any proportion.
Ratio of
Dispersive
Powers
Radius of
First
Surface
+
Radius of
Second
Surface
Radios of
Third
Surface
Radius of Focal
Fourth i Length of
Surface Crown Lens
+
Focal
Length of
Flint Lens
50
6-7485
4-2827
4-1575
14-3697 5-0
10-0000
55
6-7184
3-6332
3-6006
14-5353 4-5
8-1818
60
6-7069
3-0488
3-0640
14-2937 4-0
6-6667
65
6-7316
2-5208
2-5566
13-5709 3-5
5-3846
70
6-8279
2-0422
2-0831
12-3154 3-0
4-2858
75
7-0816
1-6073
1-6450
10-5186 25
3-3333
Fig. 16.
The general character of the combination is shown in fig. 16.
The radii of the first and fourth surfaces within practical limits are so
nearly constant that Herschel lays down the following *
rule as in all probability sufficiently exact for use. A
double object-glass will be free from aberration, provided
the radius of the exterior surface of the crown lens be
6720 and of the flint 14'20, the focal length of the com-
bination being lO'OOO, and the radii of the interior
surface being computed from these data, by the formulae given in all elemen-
tary works on optics, so as to make the focal lengths of the two glasses in
the direct ratio of their dispersive powers.
Numerous experiments have been made with the view of abolishing the
secondary spectrum. Theoretically, if three different kinds of glasses are
combined it will generally be possible to make the focal lengths of the com-
bination equal for any three selected rays of the spectrum. Or the ingre-
dients of one of the glasses may be mixed in such proportions as to suit the
requirements of the problem when combined with crown. In this way
Stokes has succeeded in constructing a small object-glass free from secondary
Phil. Trant. 1821.
406 OPTICS. [119
colour, but it is doubtful whether the practical difficulties could be overcome
in the construction of a large object-glass, where alone the outstanding
chromatic aberration is important.
The practical optician is not limited to spherical surfaces, and the final
adjustment of the aberration of large object-glasses is controlled by the
action of the polishing tool. It is understood that some of the best makers
apply a local correction, according to the methods developed by Foucault for
mirrors. The light from a natural or artificial star is allowed to fall upon
the lens. At the focus is placed a small screen, which is gradually advanced
so as to cut off the light. The eye is immediately behind the screen and is
focused upon the lens. If there are no imperfections the illumination falls
off very suddenly, the surface of the mirror passing from light to dark
through a nearly uniform grey tint. If, however, from uniform aberration,
or from local defects, any of the light goes a little astray, the corresponding
parts of the surface will show irregularities of illumination during the passage
of the screen, and in this manner a guide is afforded for the completion of
the figuring.
Topler* has developed the idea of Foucault into a general method for
rendering visible very small optical differences. Instead of a mere point of
light, it is advisable to use as source an aperture (backed by a bright flame)
of sensible size, and bounded on one side by a straight edge. An image of
this source is formed at a considerable distance by a lens of large aperture
and free from imperfections, and in the plane of the image is arranged a
screen whose edge is parallel to the straight edge of the image, and can be
advanced gradually so as to coincide with it. Behind this screen comes a
small telescope through which the observer examines the object placed near
the lens. When the light is just cut off by the advancing screen, the appa-
ratus is in the most sensitive state, and the slightest disturbance of the
course of the rays is rendered evident. To show the delicacy of the arrange-
ment Toppler introduced into the cone of light a small trough with parallel
glass sides containing distilled water. A syphon dipped under the surface
and discharged distilled water from another vessel, and it was found almost
impossible so to control the temperatures that the issuing jet should remain
invisible. Not only were sound-waves in air, generated by electric sparks,
rendered visible, but their behaviour when reflected from neighbouring
obstacles was beautifully exhibited.
An apparatus on this principle may often be employed with advantage in
physical demonstrations, for instance, for the exhibition of the changes of
density in the neighbourhood of the electrodes of a metallic solution under-
going electrolysis. The smallest irregularity that could be rendered visible
* Pogy. Ann. cxxxi. 1867.
119] OPTICS. 4ffj
would be such as would retard transmitted light by a moderate fraction of
the wave-length*.
In objectives for photographic use the requirements are in many Tcapecto
different from those most important in the case of telescopes. A flat ficM.
a wide angle of view in some cases as much as 90 = freedom from dis-
tortion, and a great concentration of light are more important than a high
degree of definition. As a rule, photographs are not subjected to the ordeal
of a high magnifying power. Usually the picture includes objects at various
distances from the camera, which cannot all be in focus at once. That the
objects at one particular distance should be depicted with especial sharpness
would often be rather a disadvantage than otherwise. A moderate amount
of " diffusion of focus " is thus desirable, and implies residual aberration. In
some lenses an adjustment is provided by means of which the diffusion of
focus may be varied according to the circumstances of the case.
For landscapes and general purposes a so-called single lens is usually
employed. This, however, for the sake of achromatism, is compounded of a
flint and a crown cemented together ; or sometimes three component lenses
are used, the flint being encased in two crowns, one on each side. To get
tolerable definition and flatness of field a stop must be added, whose proper
place is some little distance in front of the lens.
For portraiture, especially before the introduction of the modern rapid
dry plates, a brilliant image was a necessity. This implies a high ratio of
aperture to focal length, which cannot be attained satisfactorily with any
form of single lens. To meet the demand, Petzval designed the " portrait-
lens," in which two achromatic lenses, placed at a certain distance apart.
combine to form the image. This construction is so successful that the focal
length is often no more than three times the available aperture. When
stops are employed to increase the sharpness and depth of focus they are
placed between the lenses.
Vision through a Single Lens. A single lens may be used to improve the
vision of a defective eye, or as a magnirying-glass. A normal eye is capable
of focusing upon objects at any distance greater than about 8 inches. The
eyes of a short-sighted person are optically too powerful, and cannot be
focused upon an object at a moderate distance. The remedy is of course to
be found in concave glasses. On the other hand, persons beyond middle life
usually lose the power of seeing near objects distinctly, and require convex
glasses,
* ETCH when the optical differences are not small it is well to remember thai transparent
bodies ue only risible in virtue of a variable illumination. If the light Calls equally in all
directions, as it might approximately do for an observer on a high monument during a thick
fog, the edge of (for example) a perfectly transparent prism would be absolutely invisible. If a
spherical cloud, composed of absolutely transparent material, surround symmetrically a source of
light, the fllumination at a distance would not be diminished by its presence.
408 OPTICS. [119
A not uncommon defect, distinct from mere short or weak sight, is that
known as astigmatism. In such cases the focal length varies in different
planes, and at no distance is the definition perfect. Many people, whose
sight would not usually be considered inferior, are affected by astigmatism to
a certain extent. If a set of parallel black lines ruled upon white paper be
turned gradually round in its own plane, it will often be seen more distinctly
and with greater contrast of the white and black parts in one azimuth than
in another. When the focal line on the retina is parallel to the length of
the bar, the definition (as in the case of the spectroscope) is not much
prejudiced, but it is otherwise when the bars are turned through a right
angle so as to be perpendicular to the focal line.
In extreme cases a remedy may be applied in the form of glasses of
different curvatures in perpendicular planes, so adjusted both in form and
position as to compensate the corresponding differences in the lens of the
eye.
The use of a lens as a magnifier has been explained under MICROSCOPE.
The simplest view of the matter is that the lens, consistently with good
focusing, allows of a nearer approach, and therefore of a higher visual angle,
than would otherwise be possible.
Telescope, &c. In a large class of optical instruments an image of the
original object is first formed, and this image is examined through a magni-
fier. If we use a single lens merely for the latter purpose, the field of view
is very restricted. A great improvement in this respect may be effected by
the introduction of a field-tens. The ideal position for the field-lens is at the
focal plane of the object-glass. The image is then entirely uninfluenced,
and the only effect is to bend round the rays from the margin of the field
which would otherwise escape, and to make them reach the eye-lens, and
ultimately the eye. If the field-lens and the eye-lens have nearly the same
focal length an image of the object-glass will be formed upon the eye-lens
and through this small image will pass every ray admitted by the object-
glass and field- lens.
However, to obtain a sufficient augmentation of the field of view it is not
necessary to give the field-lens the exact position above mentioned, and
other considerations favour a certain displacement. For example, it is not
desirable that dust and flaws on the field-lens should be seen in focus. In
Huygens's eye-piece the field-lens is displaced from its ideal position towards
the object-glass. In Ramsden's eye-piece, on the other hand, the focal plane
of the object-glass is outside the system. This eye-piece has the important
advantage that cross wires can be placed so as to coincide with the image as
formed by the object-glass. The component lenses of a Ramsden's eye-piece
are sometimes achromatic. For further particulars with diagrams, on the
subject of eye-pieces, see MICROSCOPE.
119] OPTIC& 409
In large telescopes the object-glass is often replaced by a mirror, which
may be of speculum metal, or of glass coated chemically with a very thin
layer of polished silver. The mirror presents the advantage (especially
important for photographic applications) of absolute achromatism. On the
other hand, more light is lost in the reflexion than in the passage through a
good object-glass, and the surface of the mirror needs occasional re-polishing
or re-coating. For fuller information see TELESCOPE.
The function of a telescope is to increase the " apparent magnitude " of
distant objects ; it does not increase the " apparent brightness." If we put
out of account the loss of light by reflexion at glass surfaces (or by imperfect
reflexion at metallic surfaces) and by absorption, and suppose that the mag-
nifying power does not exceed the ratio of the aperture of the object-glass to
that of the pupil, under which condition the pupil will be filled with light,
we may say that the " apparent brightness " is absolutely unchanged by the
use of a telescope. In this statement, however, two reservations must be
admitted. If the object under examination, like a fixed star, have no
sensible apparent magnitude, the conception of "apparent brightness'' is
altogether inapplicable, and we are concerned only with the total quantity of
Light reaching the eye. Again, it is found that the visibility of an object
seen against a black background depends not only upon the " apparent
brightness " but also upon the apparent magnitude. If two or three crosses
of different sizes be cut out of the same piece of white paper, and be erected
against a black background on the further side of a nearly dark room, the
smaller ones become invisible in a light still sufficient to show the larger.
Under these circumstances a suitable telescope may of course bring also the
smaller objects into view. The explanation is probably to be sought in
imperfect action of the lens of the eye when the pupil is dilated to the
utmost. The author of this article has found that in a nearly dark room he
becomes distinctly short-sighted, a defect of which there is no trace what-
ever in a moderate light*. If this view be correct, the brightness of the
image on the retina is really less in the case of a small than in the case of a
large object, although the so-called apparent brightnesses may be the same.
However this may be, the utility of a night-glass is beyond dispute.
The general law that (apart from the accidental losses mentioned above)
the "apparent brightness" depends only upon the area of the pupil filled
with light, though often ill understood, has been established for a long time,
as the following quotation from Smith's Optics (Cambridge, 1738), p. 113,
will show.
"Since the magnitude of the pupil is subject to be varied by various
degrees of light, let NO be its semi-diameter when the object PL is viewed
by the naked eye from the distance OP ; and upon a plane that touches the
* Comb. PhiL Proc. voL iv. [Vol. n. Arte. 82, 96.]
410 OPTICS. [119
eye at 0, let OK be the semi-diameter of the greatest area, visible through
all the glasses to another eye at P, to be found as PL was; or, which is the
same thing, let OK be the semi-diameter of the greatest area inlightened by
a pencil of rays flowing from P through all the glasses ; and when this area
is not less than the area of the pupil, the point P will appear just as bright
through all the glasses as it would do if they were removed ; but if the
inlightened area be less than the area of the pupil, the point P will appear
less bright through the glasses than if they were removed in the same
proportion as the inlightened area is less than the pupil. And these propor-
tions of apparent brightness would be accurate if all the incident rays were
transmitted through the glasses to the eye, or if only an insensible part of
them were stopt."
Resolving Power of Optical Instruments. According to the principles of
common optics, there is no limit to the resolving power of an instrument.
If the aberrations of a microscope were perfectly compensated it might reveal
to us worlds within a space of a millionth of an inch. In like manner a
telescope might resolve double stars of any degree of closeness. The magni-
fying power may be exalted at pleasure by increase of focal length and of the
power of eye-pieces; and there are at any rate some objects, such as the sun,
in dealing with which the accompanying loss of light would be an advantage
rather than the contrary. How is it, then, that the power of the microscope
is subject to an absolute limit, and that if we wish to observe minute detail
on the over-lighted disk of the sun we must employ a telescope of large
aperture ? The answer requires us to go behind the approximate doctrine of
rays, on which common optics is built, and to take into consideration the
finite character of the wave-length of light.
A calculation based upon the principles of the wave-theory shows that,
no matter how perfect an object-glass may be, the image of a star is repre-
sented, not by a mathematical point, but by a disk of finite size surrounded
by a system of alternately dark and bright rings. Airy found that if the
angular radius of the central disk (as seen from the centre of the object-
glass) be 0, 2R the aperture, X the wave-length, then
0=1-2197^,
showing that the definition, as thus limited by the finiteness of X, increases
with the aperture.
In estimating theoretically the resolving power of a telescope on a double
star we have to consider the illumination of the field due to the superposition
of the two independent images. If the angular interval between the compo-
nents of the double star were equal to 26, the central disks would be just in
contact. Under these conditions there can be no doubt that the star would
119] OPTICS. 411
appear to be fairly resolved, since the brightness of the external ring systems
is too small to produce any material confusion, unless indeed the components
are of very unequal magnitude. The diminution of star disks with increasing
aperture was observed by \V. Herschel ; and in 1823 Fraunhofer formulated
the law of inverse proportionality. In investigations extending over a long
series of years, the advantage of a large aperture in separating the compo-
nents of close double stars was fully examined by Dawes.
The resolving power of telescopes was investigated also by Foucault, who
employed a scale of equal bright and dark alternate parts ; it was found to
be proportional to the aperture and independent of the focal length. In
telescopes of the best construction the performance is not sensibly prejudiced
by outstanding aberration, and the limit imposed by the finiteness of the
waves of light is practically reached. Verdet has compared Foucault "s results
with theory, and has drawn the conclusion that the radius of the visible part
of the image of a luminous point was nearly equal to half the radius of the
first dark ring.
The theory of resolving power is rather simpler when the aperture is
rectangular instead of circular, and when the subject of examination consists
of two or more light or dark lines parallel to one of the sides of the aperture.
Supposing this side to be vertical, we may say that the definition, or resolv-
ing power, is independent of the vertical aperture, and that a double Hue will
be about on the point of resolution when its components subtend an angle
equal to that subtended by the wave-length of light at a distance equal to
the horizontal aperture.
The resolving power of a telescope with a circular or rectangular aperture
is easily investigated experimentally. The best object is a grating of fine
wires, about fifty to the inch, backed by a soda-flame. The object-glass is
provided with diaphragms pierced with round holes or slits. One of these,
of width equal, say, to one-tenth of an inch, is inserted in front of the object-
glass, and the telescope, carefully focused all the while, is drawn gradually
back from the grating until the lines are no longer seen. From a measure-
ment of the maximum distance the least angle between consecutive lines
consistent with resolution may be deduced, and a comparison made with the
rule stated above.
Merely to show the dependence of resolving power on aperture it is not
necessary to nse a telescope at all. It is sufficient to look at wire-gauze
backed by the sky, or by a name, through a piece of blackened cardboard
pierced by a needle and held close to the eye. By varying the distance the
point is easily found at which resolution ceases ; and the observation is as
sharp as with a telescope. The function of the telescope is in fact to allow
the use of a wider, and therefore more easily measurable, aperture. An
412 OPTICS. [119
interesting modification of the experiment may be made by using light of
various wave-lengths.
In the case of the microscope the wave-theory shows that there must be
an absolute limit to resolving power independent of the construction of the
instrument. No optical contrivances can decide whether light comes from
one point or from another if the distance between them do not exceed a
small fraction of the wave-length. This idea, which appears to have been
familiar to Fraunhofer, has recently been expanded by Abbe and Helmholtz
into a systematic theory of the microscopic limit. See MICROSCOPE.
Similar principles may be applied to investigate the resolving power of
spectroscopes, whether dispersing or diffracting. Consider for simplicity any
combination of prisms, anyhow disposed, but consisting of one kind of glass.
Let a be the width and //, the index of a parallel beam passing through, and
let the thicknesses of glass traversed by the extreme rays on either side be
t z and ,. It is not difficult to see that, if the index be changed to p
the rays will be turned through an angle given by
Now, if the two kinds of light correspond to a double line which the instru-
ment can just resolve, we have 6 = X/a, and thus
t, - t, =
a formula of capital importance in the theory of the dispersing spectroscope.
In a well-constructed instrument, t lt the smaller thickness traversed may be
small or negligible, and then we may state the law in the following form :
the smallest thickness of prisms necessary for the resolution of a double line
whose indices are /A and /*, + 8/j, is found by dividing the wave-length by fyi.
As an example, let it be required to find the smallest thickness of a
prism of Chance's " extra dense flint," necessary for resolution of the soda-
lines.
By Cauchy's formula for the relation between /n and X we have
From the results given by Hopkinson for this kind of glass we find
B = -984 x 10~ 10 ,
the unit of length being the centimetre. For the two soda-lines
X = 5-889 x 10- 5 , ax = -006 x 10~ 5 ;
and thus the thickness t necessary to resolve the lines is
X 4 10 IO X 4
^ = OPS-V = i-Qfigfrx = ^ ^ cen timetre,
OPTICS. 413
the meaning of which is thai/ the soda-tines will be resolved if, and will not
be resolved unless, the difference of thicknesses of glass traversed by the two
sides of the beam amount to one centimetre. In the most favourable
arrangement the centimetre is the length of the base of the prism. It is to
be understood, of course, that the magnifying power applied is sufficient to
narrow the beam ultimately to the diameter of the pupil of the eje ; other-
wise the full width would not be utilized.
The theory of the resolving power of a diffracting spectroscope, or
grating, is even simpler. Whatever may be the position of the grating, a
double line of wave-lengths X and X + $X will be just resolved provided
where m is the total number of lines in the grating, and m is the order of the
spectrum under examination.
If a grating giving a spectrum of the first order and a prisui of extra
dense glass have equal power in the region of the soda-lines, the former
must have about as many thousand lines as the latter has centimetre? of
available thickness.
The dispersion produced by a grating situated in a given manner is
readily inferred from the resolving power. If a be the width of the beam
after leaving the grating, the angle $0, corresponding to the limit of resolu-
tion, is X a, and thus
50 mil
Thus the dispersion depends only upon the order of the spectrum, the total
number of tines, and the width of the emergent beam.
An obvious inference from the necessary imperfection of optical imagoes
is the uselessness of attempting anything like an absolute destruction of
aberration. In an instrument free from aberration the waves arrive at the
focal point in the same phase. It will suffice for practical purposes if the
error of phase nowhere exceeds ^X This corresponds to an error of |X in a
reflecting and |X in a (glass) refracting surface, the incidence in both cases
being perpendicular.
If we inquire what is the greatest admissible longitudinal aberration in
an object-glass according to the above rule, we find
being the angular semi-aperture.
414 OPTICS. [119
In the case of a single lens of glass with the most favourable curvatures,
Bf is about equal to /a 3 ; so that a 4 must not exceed A,//. For a lens of 3-feet
focus this condition is satisfied if the aperture do not exceed 2 inches.
When parallel rays fall directly upon a spherical mirror the longitudinal
aberration is only about one-eighth as great as for the most favourable-
shaped single lens of equal focal length and aperture. Hence a spherical
mirror of 3-feet focus might have an aperture of 2| inches, and the image
would not suffer materially from aberration*.
On general optics the treatises most accessible to the English reader are
Parkinson's Optics (3rd ed., 1870) and Glazebrook's Physical Optics (1883).
Verdet's Lecons d'optique physique is an excellent work. Every student
should read the earlier parts of Newton's Optics, in which are described the
fundamental experiments upon the decomposition of white light.
[1900. To the above references may now be added Preston's Theory
of Light and Mascart's Traite d'Optique.']
* For fuller information on the subject of the preceding paragraphs see Lord Rayleigh's papers
entitled " Investigations in Optics," Phil. Hag. 1879, 1880. [Art. 62, vol. i. p. 415.]
120.
UBER DIE METHODE DER DAMPFUXG BEI DER
BESTIMMUXG DES OHMS.
[Annalen der Pkysik und Chemie, Band XXIY. pp. 214. 215. 1SS5.]
Mil grossem Interesse habe ich aus einer neueren Mittheilung in den
Annalen ersehen, dass Hr. Wild im Anschluss an einen Yorschlag von
Dorn seine Zahl fur diese Werthe der Siemens'schen Einheit in Ohme
0,9462 auf 0,94315 corrigirt hat, wodurch die Differenz zwischen seiner
Zahl und der von mir gefundenen 0,9415 auf etwa ein Drittel reducirt
wild. Die Untersuchung von Wild scheint sehr sorgfaltig ausgefuhrt
worden zu sein, indess mochte ich doch die Aufmerksamkeit derer, welche
an die Yorziige der Dampfungsmethode glauben, auf eiuige Punkt*
lenken.
Bei der theoretischen Untersuchung wind die Wirkung de^ Magnet*
als identisch mit der eines Solenoids angesehen, durch welches ein con-
stanter Strom geleitet wird, wahrend sie in der That mehr mit der eines
mit einem Eisenkern versehenen Solenoides verglichen werden kann. Mir
scheint die Einfuhrung einer grossen Eisenmasse in den Multiplicator sehr
sorgfaltige Erwagungen zu verdienen. Selbst wenn man annimmt, dass
der grossere Theil der Wirkung durch die Aendemngen gewisser Grossen,
wie der Inductionscoefficienten, compensirt werden kann, so kann doch ein
kleines Residuum zuriickbleiben infolge der Abweichung der Magnetisirung
des Eisens von den einfachen Gesetzen. Ich will nicht behaupten, dass dies
in der That der Fall ist, indess miissen diejenigen, welche die Dampfungs-
methode benutzen, das Gegentheil beweisen.
Ferner ist der Magnet ein Leiter der Electricitat. Es ist nicht erwiesen,
dass nicht galvanische Strome von erhebh'cher Starke in einem 36 mm.
langen und 12 mm. dicken Stabe erzeugt werden konnen, welcher in
416 USER DIE METHODE DER DAMPFUNG &C. [120
einer vom Strom durchflossenen Spirale schwingt. Diese Strome wiirden
vvahrscheinlich proportional der Schwingungsweite wirken ; indess fiihlte
sich Wild veranlasst, andere Glieder dieser Art in seine Reductionsformeln
einzufiihren.
Eine unerfreuliche Seite in Wild's Untersuchungen ist die Nothwen-
digkeit einer Correction, welche T ^^ betragt, und durch den Eisengehalt
des Multiplicatordrahtes nothig wird. Dieser Einfluss wird behandelt, wie
wenn seine einzige Wirkung nur eine Kraft ware, welche den Magnet
stets zu der Symmetrieebene des Multiplicators zuriickftihrt. Auch dieser
Punkt scheint mir genauerer Erwagungen zu bediirfen.
Die Wichtigkeit, womoglich die Griinde der Abweichungen in den
Resultaten der verschiedenen Forscher auf diesem Gebiete aufzufinden,
moge die ausgesprochenen Zweifel entschuldigen. Ich hoffe, dass dadurch
nicht die Meinurig erweckt werde, als wenn ich meine eigene Untersuchung
fur unfehlbar hielte.
121.
ON THE THEORY OF ILLUMINATION IN A FOG.
[Philosophical Magazine, xix. pp. 443 446, 1885.]
As a step towards a better understanding of the action of fog upon
light, it seems desirable to investigate what the phenomena would be in
the simplest case that can be proposed. For this purpose we may con-
sider the atmosphere and the material composing the fog to be absolutely
transparent, and also make abstraction from the influence of obstacles,
among which must be included the ground itself.
Conceive a small source of radiation, e.g. an incandescent carbon fila-
ment, to be surrounded by a spherical cloud, of uniform density, or at
any rate symmetrically disposed round the source, outside of which the
atmosphere is clear. Since by hypothesis there is no absorption, whatever
radiation is emitted by the source passes outward through the external
surface of the cloud. The effect of the cloud is to cause diffusion, i.e. to
spread the rays passing through any small area of the surface (which in
the absence of the cloud would be limited to a small solid angle) more
or less uniformly over the complete hemisphere.
Whether the total radiation passing outwards through the small area
on the external surface of the cloud is affected by the existence of the
cloud depends upon the circumstances of the case. If it be laid down
that the total emission of energy from the source is given, then the
presence of the cloud makes no difference in respect of the energy
passing any element of the spherical area. But this supposition does
not correspond to a constant temperature of the source, in consequence
of the energy received back from the cloud by reflection. To keep the
total emission of energy constant, we should have to suppose a rise
of temperature increasing indefinitely with the size and density of the
cloud.
418 ON THE THEORY OF ILLUMINATION IN A FOG. [121
Let us now suppose that the region under consideration is bounded
upon all sides by a distant envelope of perfect reflecting-power. Then,
whatever the density of the clouds which may wholly or partially occupy
the enclosure, we know, by the second law of thermodynamics, that at
every internal point there is radiation in every direction of the full amount
corresponding to the temperature of the source. In one sense this con-
clusion holds good, even although the matter composing the cloud has
the power of absorption. But in that case equilibrium would not be
attained until the clouds themselves to the remotest parts had acquired
the temperature of the source ; whereas under the supposition of perfect
transparency the temperature of the cloud is a matter of indifference ;
and equilibrium is attained in a time dependent upon that required by
light to traverse the enclosure. So far we have made no supposition as
to the distribution of the cloud ; but we will now imagine a layer of such
thickness as to allow only a very small fraction of the incident radiation
to penetrate it, to line the interior of the reflecting envelope. This layer
itself plays the part of a practically perfect reflector ; and it is not diffi-
cult to see that the reflecting envelope hitherto conceived to lie beyond
it may be removed without interfering with the state of things on the
inner side of the layer of cloud. We thus arrive at the rather startling
conclusion that at any distance from the source, and whatever the distri-
bution of clouds, there is always in every direction the full radiation due
to the temperature of the source, provided only that there lie outside a
complete shell of cloud sufficiently thick to be impervious. And this state
of things is maintained without (on the whole) emission of energy from
the source.
Even if the material composing the cloud possesses absorbing-power
for some kinds of radiation, e.g. for dark radiation, but is perfectly trans-
parent to other kinds, e.g. luminous radiation, the general theorem holds
good as respects the latter kinds ; so that in the case supposed the light
would still be everywhere the same as in a clear enclosure whose walls
have throughout the same luminosity as the source. But in order to
compensate the absorption of dark rays, the source must now be supplied
with energy.
Some of the principles here enunciated have an acoustical as well as
an optical application, and indeed first occurred to me some years ago in
connection with Prof. Tyndall's investigations upon fog-signals. The effect
of " acoustic clouds " analogous to fog (and unattended with absorption of
energy), might be very different upon the report of a gun and upon the
sustained sound of a syren, the latter being reinforced by reflection from
the acoustic fog.
The theory presented in the present paper may be illustrated by the
known solution of the comparatively simple problem of a pile of trans-
121] ON THE THEORY OF ILLUMINATION IN A FOG. 419
parent plates*. If p denote the proportion of the incident light reflected
at a single surface, then the proportion reflected $(w), and transmitted
^ (wi), by a pile of in plates is given by
l-p l-t-(2m-l)p"
From these expressions it is evident that, however small p may be, i.e.
however feeble the reflection at a single surface, we have only to suppose
??i large enough in order that the reflection may be as complete, and the
transmission as small, as we please. Such a pile may, under ordinary
conditions, be regarded as impervious.
But now suppose that after passing the pile of m plates, the light is
incident upon a second pile of n plates, and consider the intensity between
the two piles, the original intensity being unity, as before. For the in-
tensity of the light travelling in the original direction we have
* (TO) + + (ro) .*<).* (TO) + * (TO) . [<j> (n) . <f> (>)}
or on summation of the geometric series,
If we introduce the values of <p and -\jr in terms of m, n, p, this becomes
2rop + 1 - p
In like manner, for the light going the other way we have
or in terms of m, n, p,
2 (m + n)p + lp'
When m and n are great, both expressions reduce to n/(m + n); so that
the light passing in the two directions is equally bright. Moreover, and
this is the point to be especially noticed, however great m may be that
is, however impervious the first pile is, the light between the two piles
may be made to approach the original light in brightness as nearly as we
please, by sufficiently increasing the number of plates in the second pile ;
that is, the light between the piles may be made to be the same as
if the first pile were removed. From this example we may understand
more clearly how a very small quantity of light penetrating directly may
be beaten backwards and forwards, as between two reflectors, until the
original intensity is recovered.
* Stokes, Proc. Roy. Soc. voL xi. p. 545 (1862).
272
122.
A MONOCHROMATIC TELESCOPE, WITH APPLICATION
TO PHOTOMETRY.
[Philosophical Magazine, xix. pp. 446, 447, 1885.]
THE purpose of this instrument is to exhibit external objects as they
would be seen either with the naked eye, or through a telescope, if lighted
with approximately monochromatic light ; that is, to do more perfectly
what is done roughly by a coloured glass.
The arrangernent is not new, though I am not aware that it has ever
been described. In 1870 I employed it for determinations of absorption,
and, if my memory serves me right, I heard soon afterwards from Clerk-
Maxwell that he also had used it. It is, indeed, a very slight modification
of Maxwell's colour-box.
In the ordinary form of that instrument, white light admitted through
a slit is rendered parallel by a collimating lens, dispersed by flint-glass
prisms, and then brought to a focus at a screen, upon which accordingly
a pure spectrum is formed. This screen is perforated by a second slit,
immediately behind which the observer places his eye. It is evident that
the light passing the aperture is approximately monochromatic, so that
the observer, if he focuses his eye suitably, will see the prism illuminated
with this kind of light. The only addition now required to convert the
instrument into a monochromatic telescope is a lens placed just within
the first slit, of such power as to throw an image of external objects
upon the prism or diaphragm upon which the eye is focused. If desired,
an eye-lens may be placed at the second slit; but this is not generally
needed.
122] A MONOCHROMATIC TELESOfWR, WITH AFPUKCATTOX TO
In the present instrument a drarecfc-iisaan
that the optical pants earn be affl disposed in a narrow box of nearly 3 feet
in length. The lease* ane al simgUe lesasjesi, and work aunfeieutly vmIL Tne
sifts are of sneh width B&aiJ either (WMnwixdks with the image of the other,
and their rdaniTe prosiinian its *> (cfoodani iti&ut the mean re&an^bcMlty of the
nght is that oj^nre^pofflKiimig to sodium- ObJ^c^,* SMBI through the insHmment
thus appear as if Egfec^i b-j A .soi'iiinuimi tfbtfmxe..
The- prineipaS (ixbjjfwtt wiuwriD I load in TMTT in the eoostrndtioBi of the
iiKinniimieiftU DJI'>W esMlwtiei w^as to *eir wBurtilihar it ionDd be made of amtiae
in nke- ci^cmpauri^offii <of woflmpKwuimid MgiDtte otf stoflmewluatt diff^nenntt^ -.ooilfflinrs a.
pn-jfelerao; joss m^w amrraetomg- atttfflitti<>Dii im cannecttSfin wittBn ^leidtrac flighting:
It is- seaurwe-Ij me^iessafT to say that; a cwmipairaaQffii <otf ttJais kniwd is pflnrsttcaBr
inn>mp8ete unless it extends to aH itftue sf^iuiraD cioflmipiMiaDitt^ sep airattelj ;
bat Ibr CKxmmercial pmrpioises >roc4i an esttemxd^d (ooxnmpaur^ami a* toioi offlmplli-
caEei. and indeed ntsetesm. D^eOTmiinsanioflus alt tw* i-oimite iotf K&<e peptrainm
as prf:s<ii by Capo. Abeev,. wiOTM irtainDjr smlfce for wdranaity pmurpiojae? :
and in view of the conTenienee. of expme^ra^ nSM- iresmHii b>j a em^He mimunmfoer,,
it is not' onEkdly that p^otpDtfr pffadtieaUlhr (Cioffliiinnxe(J im
content themselTe? wittln a enxnnpu^Dm att, ou- poimiiL It-
that some convention sfconM be amriTd alt wiuBaoflnit inmwfin fertBttar dr L;,T .-
ao that two lights should be conadeined to be taofflnuniMffoaMv equuaJi, if :i v T
have the * intenafcv' C &-S-* ^tS^mm. r ^T ttUlMnuom 3igrin.. Ii ill tr'
understood that, ffltr^ a laodc of estimation assmnres ttinait iLr :r.v^-.:-;T
Taries IMMT the snecttronn in a endoal manner: amd iM* ramffindiaratooffli
~ ~
mar tell against the nae of the aaodium ligiuit as a sttamdaitdn ieasaaauKfti as
the component in qoestion often predominates Tnamfaiij m GwdSfc-fasmes
fiom the actual presence of sodinm*
Whatever rf*ffy be made, an instanuunroeiDA Mke ifer- puntseniitt mmsy ixe
employed to make the desired sefccniuum; aimd h a* appicdblle to SET
photometric arrangement. For nnj (o<wm experinBentB I isaT> nased tLSoe
shadow-method, MM! fiiMJ it, poeabie to flxropaunr amij brtgfcii stwrott, ID>OIW-
different in eelooir. line only dnffiBcimlttr arises froffim the roefflessary
of Idle hgfet by sefodtaofflL azad this f.<raiEaciily predtades *)bser-
with standard eandk& Wid& ^a&-iami>es amid gl.j>w-llamip5 n&e igfat
123.
SELF-INDUCTION IN RELATION TO CERTAIN EXPERIMENTS
OF MR WILLOUGHBY SMITH AND TO THE DETERMINA-
TION OF THE OHM.
[Nature, xxxn. p. 7, 1885.]
IN a Lecture delivered by Mr Wil lough by Smith before the Royal
Institution in June last (see Proceedings) some experiments are detailed,
which are considered to afford an explanation of discrepancies in the
results of various investigators relating to the ohm, or absolute unit of
electrical resistance. As having given more attention than probably any-
one else in recent years to this subject, I should like to make a few
remarks upon Mr Willoughby Smith's views, which naturally carry weight
corresponding to the good service done by the author in this branch of
science.
In the first series of experiments a primary circuit is arranged in
connection with a battery and interrupter, and a secondary circuit in con-
nection with a galvanometer and commutator of such a character that the
make and break induced currents pass in the same direction through the
instrument. Under these circumstances it is found that at high speeds
the insertion of a copper plate between the primary and secondary spirals
entails a notable diminution in the galvanometer deflection, and this result
is regarded as an indication that the molecules of copper need to be
polarised by the lines of force an operation for which there is not time
at the higher speeds. The orthodox explanation of the experiment would
be that currents are developed by induction in the copper sheet, which
thus screens the secondary spiral from the action of the primary, and the
result is exactly what might have been anticipated from known electrical
principles. I have the less hesitation in saying this, because as a matter
of fact I did anticipate from theory the action of a combination very
similar in character. The experiment is described in the Philosophical
Magazine for May 1882 [vol. II. p. 99], and differs from Mr W. Smith's
123] SELF-IXDUCTIOX IX RELATIOX TO CERTAIN EXPERIMENTS, ETC. 423
only in the substitution of a telephone for die galvanometer, and of a
microphone for the interrupter, no reverser in the secondary circuit being
required. By the interposition of a thick copper sheet the sound is greatly
enfeebled,
The second series of experiment* were made with Faradavs "new
magneto-electro machine," in which a copper disk rotates about its centre
between the poles of a horse-sh^e magnet. The currents developed are
examined with a galvanometer whose electrodes touch two points upon
the disk in Mr Smith's experiments one at the centre and the other at
the circumference. At low speeds the distribution is symmetrical with
respect i*. that diameter of the disc which is passing at any moment
between the poles; but as the speed is increased, a certain "drag" is
observed, disturbing the symmetry. This drag, or lagging, was noticed by
Xobili in a very similar arrangement as long ago as 1833 ( Wiedemann's
Electricity, third edition, voL IT. | 374). and is no doubt to be attributed
to the induction of the currents upon themselves.
This question of self-induction is indeed a very important one in respect
of certain methods for determining the ohm ; but it certainly cannot be
said to have been neglected, as Mr W. Smith seenis to suggest- Both in
the original experiments of the British Association Commiiiee with a evil
revolving about a vertical axis, and in my own recent repetition of them,
the self-induction of the coil is a most important feature, and may cause
a displacement of the position of the maximum current from the plane of
the magnetic meridian through as much as 20 r . In my paper (Phil.
Trams. 1882, p. 661) [voL IL Art 80] I thought I had discussed the question
at almost tedious length.
It is possible that Mr W. Smith had in his mind rather determina-
tions bj the method of Lorenz, in which Faraday's disk is used. The
arrangement here, however, differs in one very important respect from
that of Mr W. Smith's experiments in that lines of force are symmetrically
arranged in relation to the axis of rotation. The consequence is that,
however great the speed of rotation, there are no currents circulating in
the disk, and therefore no question arises as to the self-induction of such
currents. What is observed is simply the difference of electrical potential
between the centre and the circumference. It is impossible to discuss
the matter fully here, but the reader will find all that is necessary by
way of explanation in the paper published in the Phil. Trans. (" Experi-
ments by the method of Lorenz for the further Determination of the
Absolute Value of the British Association Unit of Resistance," etc.). My
object in writing is to correct the inference, suggested by W. Smith's
remarks, that the question of self-induction has been neglected by workers
upon this subject.
124.
PROFESSOR TAIT'S "PROPERTIES OF MATTER*"
[Nature, XXXIL pp. 314, 315, 1885.]
THE subject of this excellent little book includes the Mechanical Pro-
perties of matter, and much that is usually treated under the head of
Chemical Physics, such as Diffusion and Capillarity. It might be difficult
to give a reason why electric and thermal conductivities of mercury, for
example, should not be included among its properties as much as its
density and its capillarity; but the distinction is convenient, and to some
extent sanctioned by usage.
In the introductory chapters the author expounds some rather peculiar
views with perhaps more insistence than is desirable in an elementary
work. The word " force " is introduced apologetically, and with the ex-
planation that, " as it does not denote either matter or energy, it is
not a term for anything objective." No one will dispute the immense
importance of the property of conservation, but the author appears to
me to press his view too far. As Dr Lodge has already pointed out, if
conservation is to be the test of existence, Prof. Tait himself does not
exist. I forbear from speculating what Dr Lodge will say when he reads
on p. 11 that "not to have its price is conclusive against objectivity."
Chapters IV. to VII. form an elementary treatise on Mechanics in
which even the learned reader will find much that is interesting in the
way of acute remark and illustration. Under the head of Gravitation
are considered Kepler's laws, the experimental methods for determining
the constant of gravitation (" the mean density of the earth "), and the
attempts (such as Le Sage's) which have been made to explain the origin
of gravitation.
The succeeding chapters on the deformation of solids and the com-
pression of solids, liquids and gases, are perhaps the most valuable part
* "Properties of Matter." By Prof. Tait (Edinburgh, Black).
124] PROFESSOR TAIT'S "PROPERTIES OF MATTER." 425
of the work, and will convey a much needed precision of ideas to many
students of physics whose want of mathematical training deters them
from consulting the rather formidable writings of the original workers in
this field. The connection of Young's modulus of elasticity, applicable to
a rod subject to purely longitudinal pull or push, with the more funda-
mental elastic constants expressing the behaviour of the body under
hydrostatic pressure and pure shearing stress respectively is demonstrated
in full. Pro Tait remarks that " Young's treatment of the subject of
elasticity is one of the few really imperfect portions of his great work
(Lectures on Natural Philosophy). He gives the value of his modulus for
water, mercury, air, &c. ! " A deficiency of explanation must be admitted,
but I am not sure that Young's ideas were really confused. The modulus
for solids corresponds to a condition of no lateral force, that for liquid to
no lateral extension. The distinction should certainly have been pointed
out; but the moduli are really comparable in respect of very important
effects, which Young probably had in his mind viz. the propagation of
sound along a bar of the solid in one case, and in the other through a
fluid, whether unlimited or contained in an unyielding tube.
As a great admirer of Dr Young's work, I cannot resist adding that if
in some respects his treatment of elasticity is defective, in others it is
in advance of many modern writings. Witness the following passage*:
"There is however a limit beyond which the velocity of a body striking
another cannot be increased without overcoming its resilience, and breaking
it, however small the bulk of the first body may be, and this limit depends
upon the inertia of the parts of the second body, which must not be dis-
regarded when they are impelled with a considerable velocity. For it is
demonstrable that there is a certain velocity, dependent on the nature
of a substance, with which the effect of any impulse or pressure is trans-
mitted through it; a certain portion of time, which is shorter accordinglv
as the body is more elastic, being required for the propagation of the force
through any part of it; and if the actual velocity of any impulse be in
a greater proportion to this velocity than the extension or compression, of
which the substance is capable, is to its whole length, it is obvious that
a separation must be produced, since no parts can be extended or com-
pressed which are not yet affected by the impulse, and the length of the
portion affected at any instant is not sufficient to allow the required
extension or compression."
The theories of " bending " and of " torsion " are discussed in Chapter XI.
When the section of the rod deviates from the circular form, the torsional
problem becomes rather complicated ; but a statement is given of some of
* [Lectures on Natural Philosophy, vol. i. p. 144.]
426 PROFESSOR TAIT'S " PROPERTIES OF MATTER." [124
the interesting results of Saint Venant's investigations. In his treatment
of the compression of solids and liquids, the author is able to make valuable
contributions derived from his own experimental work.
In the chapter on " gases " a long extract is given from Boyle's Defence
of the Doctrine Touching the Spring and Weight of the Air, in order to show
how completely the writer had established his case in 1662. As to this
there can hardly be two opinions, and Prof. Tait is fully justified in
insisting upon his objections to " Mariotte's law." In Appendix IV. a
curious passage from Newton is discussed, in which the illustrious author
appears to speak of Mariotte sarcastically. It is proper that these matters
should be put right ; but Prof. Tait is hardly impartial enough himself to
succeed in enlisting the complete sympathy of foreigners. Cases of glaring
injustice should be rectified ; but there will always be a tendency (from
which Englishmen cannot claim to be exempt) to give a full measure of
credit to one's own countrymen, if only because one is better informed
concerning their labours.
There is one matter, suitable to an elementary work, which I should
be glad to see included in a future edition, viz., the principle of dyna-
mical similarity, or the influence of scale upon dynamical and physical
phenomena. It often happens that simple reasoning founded upon this
principle tells us nearly all that is to be learned from even a successful
mathematical investigation, and in the very numerous cases in which such an
investigation is beyond our powers, the principle gives us information of
the utmost importance. An example will make this clear. The pitch of
a tuning-fork of homogeneous steel is dependent upon the size and shape
as well as upon the elastic quality of the material ; but the matter is too
difficult for rigorous mathematical treatment. If, however, it be asked,
How does the pitch depend upon the size of the fork, the shape and material
being given ? we need no complicated mathematics at all. The principle
of dynamical similarity tells us at once that the time of vibration is
proportional to the linear dimension.
Another example might be taken from a reaction which Prof. Tait
describes as specially complex viz., collision. A glass ball drops upon a
marble floor from a height of one foot. How does the size of the ball
affect the strains during collision and the danger of rupture ? The prin-
ciple teaches that if the scale of time be altered in the same proportion
as the scale of length, similarity is secured, so that the strains are equal
at corresponding times and at corresponding places. Hence a larger ball
is not more likely to break than a smaller one, unless in consequence of
the greater duration of the strains. I feel sure that in Prof. Tait's hands
this very important and fundamental principle might be made intelligible
to the great mass of physical students.
124] PROFESSOR TAIT'S "PROPERTIES OF MATTER." 427
It would lead us too far to refer in detail to the various subjects
treated in the later chapters under capillarity, diffusion, osmose, tran-
spiration, viscosity, &c., but there is one point that I should like to
mention. The explanation on p. 249 of the behaviour under water of
drops of ink and of solution of permanganate of potash assumes the ex-
istence of a capillary tension in the surface separating the two fluids. In
my own experiments on jets with this very solution, I have never seen
any tendency to break up into drops (as, according to Savart and Plateau,
there would be in air) and have therefore supposed that the capillary
force was nil, or at any rate very small. Moreover theory shows that the
force depends entirely upon the suddenness of transition between two
media, which suddenness must be broken down almost instantaneously
when two miscible liquids come into contact. As the matter stands
there seems to be here some discrepancy, which, perhaps, Prof. Tait
could elucidate.
In his preface the author holds out hopes of further volumes on the
same plan, dealing with dynamics, sound, and electricity. The readers of
the present work will, I am sure, join in the wish that the appearance of
these may be delayed no longer than is absolutely necessary.
125.
A THEOREM RELATING TO THE TIME-MODULI OF
DISSIPATIVE SYSTEMS.
[Report of the British Association, pp. 911, 912. 1885.]
IN the Proceedings of the Mathematical Society for 1873 [Art. 21], it is
shown that the time of vibration of a conservative system fulfils a stationary
condition, so that the time of vibration in any normal mode would remain
unaltered, even though the system, by the application of suitable constraints,
be made to vibrate in a mode slightly different. It is pretty evident that
a similar theorem must obtain for the time-moduli of the normal modes of a
dissipative system, but a formal statement may not be useless.
The class of systems referred to is that of which the mechanical properties
depend upon two functions, one being the dissipation function F and the
other either the kinetic energy T, or the potential energy V. As examples
of the first case may be mentioned the subsidence of the small motion of a
viscous fluid contained in a fixed envelope, and of free electric currents in a
conductor. On the other hand, in the distribution of heat in a thermal
conductor, or of electricity in a cable, the undissipated energy is usually
regarded as potential. The argument is almost exactly the same whichever
case be contemplated ; to fix ideas we will take the former.
By suitable transformation the two quadratic functions T and F may be
reduced to sums of squares of co-ordinates, and these co-ordinates are conse-
quently called normal. Thus :
i 2 + i (2) # + . ..
in which all the coefficients [1] ... (1) ... are positive.
125] THEOREM RELATING TO THE TIME-MODULI OF DKSIPATIYE STSTEMSL 429
The normal modes are those represented by the separate variation of the
co-ordinates, and the corresponding differential equations are of the form :
whence
where
P = (*) M-
If r, be the time-modulus, the time in which the motion is diminished in
the ratio of e : I, ~ t =p~\
Suppose now that by suitable constraints an arbitrary type of morion is
imposed upon the system, so that ^ = A$, = A, ... where J,. JL, &c.
are given (real) coefficients. Then
and the equation of motion
gives as the solution x g~ ft , where
It is evident that the value of p (and therefore of T) is stationary when
all but one of the coefficients A t , A s , &e., vanish, that is when the type
coincides with one of those natural to the system.
From this theorem corollaries may be drawn as from the corresponding
theorem for times of vibration. The greatest time-modulus can only be
reduced by the application of constraint, and where the normal mode is
difficult of calculation a good approximation to the greatest time-modulus
may be had from a hypothetical type chosen so as not to deviate too widely
from the real one. Any increase in T, or diminution in F, as a function of
the co-ordinates entails in general an augmentation in all the time-moduli.
In the case of free electric currents, already referred to as an example, this
augmentation of time-moduli would result from the approximation of iron
(treated as a non-conductor), or from an improvement {however local) in
conductivity.
126.
ON THE ACCURACY OF FOCUS NECESSARY FOR SENSIBLY
PERFECT DEFINITION.
[Phil. Mag. xx. pp. 354358, 1885.]
IN my "Investigations in Optics*" I have examined the effect upon
definition of small disturbances of the wave-surfaces from their proper forms.
It follows, for instance, that the aberration of a plano-convex lens focusing
parallel rays of homogeneous light is unimportant, so long as the fourth
power of the angular serni-aperture does not exceed the ratio of the wave-
length to the focal distance (a 4 < (X//)}, a condition satisfied by a lens of
3 feet focus, provided that the aperture be less than 2 inches. I propose at
present to apply similar principles to the question of focusing.
The most convenient point of view is that explainedf for calculating the
focal length of lenses. If the lens
AB converges parallel rays to a
focus at F, the retardation of the
central ray EF, due to the substi-
tution of a thickness t of glass for
air, is (/JL \)t ; and this must be
equal to the retardation of the extreme rays passing the (sharp) edge of the
lens, i.e. AF - CF. Thus, if A C = y, FC=f t
approximately, which gives the focal length in terms of the semi-aperture
and the " thickness " of the lens.
If we suppose that p, varies,
&f, ........................... (2)
giving the change of focus required to compensate the change of /A. Let us,
however, inquire what is the state of things at the old focus. The secondary
rays from the extreme boundary of the lens arrive with the same phase as
before the change of index ; but the central ray undergoes a relative retarda-
tion amounting to S/A . t. This quantity tells us the discrepancy of phase ;
* Phil. Mag. 1879 and 1880. [Vol. i. p. 435.]
t Loc. cit. p. [439].
126] ACCURACY OF FOCUS NECESSARY FOR SENSIBLY PERFECT DEFINITION. 431
and we know that if it is less than JX, the agreement of phase is still good
enough to give nearly perfect definition. Hence from (2) we see that a
displacement Bf from the true focus will not impair definition, provided
(3)
It appears that the linear accuracy required is the same whatever the
absolute aperture of the object-glass may be, provided that the ratio of
aperture to focal length be preserved.
In some trials that I have made the diameter of the object-glass was
1| inch, and the focal length 12 inches [inch = 2 54 cm.]. Taking
x = ioioo mch >
we get from (3)
S/< Olio inch,
a result which corresponded very well with observation. The instruments
employed were the collimator and telescope of a spectrometer, the object
under examination being a slit backed with a soda-flame. A high-power
eye-piece was used, and the telescope was adjusted until the edge of the slit
and the wire in the eye-piece were seen well defined together. The instru-
ment was unprovided with an easy focusing motion, so that it was nt
possible to try backwards and forwards conveniently. In this way the
setting corresponded more closely to the suppositions of theory than if it
were the result of comparisons between appearances at equal distances
within and without the point chosen. It will be understood that there is no
theoretical limit to the accuracy with which a focal point may be ultimately
determined, if the lenses are good, and observations are multiplied with
suitable precautions to avoid asymmetry.
In ten settings the extreme difference was only 02 inch, showing that a
displacement of Ol inch from the true focal point was just recognizable.
By using various coloured flames, or by throwing a spectrum upon the
slit of the apparatus, we may determine the focal length for different kinds
of light With proper achromatic lenses the differences should be pretty
small, the minimum focal length corresponding to the yellow-green rays. It
so happens that my instrument is far from properly compensated, and gives
a fair primary spectrum, so that the difference of focus for yellow and green
is very easily recognized. In the case of a single lens this method would
give the dispersive power of the glass with fair accuracy. By comparison
with the theory of the resolving power of prisms, we see that the dispersion
is about as favourably determined with a lens as with a prism of equal thick-
ness. In either case a change of index such that 8/1. t = ^X leaves the phase
agreement nearly unaltered at the original points ; but in other respects the
circumstances are probably rather more favourable in the case of the prism.
432 ACCURACY OF FOCUS NECESSARY FOR SENSIBLY PERFECT DEFINITION. [126
It is generally considered that the most accurate way of focusing a small
telescope is to move the eye across the eye-piece, altering the adjustment
until there seems to be no relative motion of object and cross wires. I have
tried this plan in an improved form in order to see whether a higher degree
of accuracy of adjustment was really attainable, although theory seemed to
show that no great advance was to be looked for. A heavy pendulum,
executing complete vibrations in about two seconds, was fitted up in front of
the telescope, and carried with it a screen perforated by a slit. The width
of the slit was about a quarter of the entire aperture, and the oscillations
were at first of such amplitude as just to bring the extreme edges of the lens
into play. In the earlier experiments the slit of the collimator was backed
by the clouds, a piece of green glass being interposed. This was before I had
discovered the remarkably unachromatic character of the instruments, and I
was puzzled to interpret the appearances presented. On one side of the
focus the relative motion of the image was (as it should be) in the same
direction as that of the pendulum, and on the other side in the opposite
direction ; but the transition was not well defined, and the image executed
evolutions very visible to the observer, who at the same time was not able to
describe them as swinging in one direction or the other. The effect upon
the eye was remarkably unpleasant and fatiguing to watch ; it disappeared
when recourse was had to sodium light, and doubtless depended upon the
variation of quality in the light. It may be noticed that spherical aberration
.would show itself by a swinging of the image in a period half that of the
pendulum.
With the soda-flame the adjustment to focus by getting rid of the
swinging motion was pretty accurate ; but not much advantage was gained
in comparison with a setting by simple inspection under full aperture. As
before, the extreme difference in a set of ten was about '02 inch.
The substitution of white for monochromatic light was instructive. In
either extreme position of the oscillating slit the light was seen to be spread
into a spectrum of moderate length, the blue and red being interchanged
after each half period. Under these circumstances the cross wires can be
made to maintain their position in that part of the spectrum only for which
the telescope is focused. If, for example, it be the green of the spectrum, we
may bring the cross wires to this position when the pendulum is at rest, and
then, in spite of the oscillation, the position will be maintained. If, without
altering the focus, we move the cross wires to another part of the spectrum,
then, when the pendulum oscillates, the wires will be seen on a different part
of the spectrum after each half period. In order to fix the new part of the
spectrum upon the cross wires, a change of focus is demanded. This experi-
ment would hardly succeed with properly compensated object-glasses, but it
could be imitated with the aid of single lenses.
127.
ON AX IMPROVED APPARATUS FOR CHRISTIANSENS
EXPERIMENT.
[Pkilewpkical Magazimc, xx. pp. 358360, 1885.]
THE very beautiful experiment in question, described by C. Christiansen
in Wiedemann's ArnRalfR for November 18S4 T consist* in immersing glass-
powder in a mixture of benzole and bisulphide of carbon of such, pro por: ions
that for one part of the spectrum the indices of the solid and of the fluid are
the same. Being interested in this subject from having employe*! me same
principle for a direct- vision spectroscope (PhiL Mag. January lSSO r p. 5->
[voL L p. 4-56], I have repeated Christiansen's experiment in a somewEM
improved form, which it may he worth while briefly to describe, as the
matter is one of great optical interest.
I must premise that the beauty of the effect depends upon die corre-
spondence of index being limited to one part of the spectrum. Rays lying
within a very narrow range of refrangibility traverse the mixture freely, bat
the neighbouring rays are scattered laterally, much as in paadng ground
glass. Two complementary colours are therefore exhibited, one by direct,
and the other by oblique, light. In order to see these to advantage, there
should not be much diffused illumination ; otherwise the directly transmitted
monochromatic light is liable to be greatly diluted. The prettiest colours
are obtained when the undisturbed rays are from the green ; bat the greatest
general transparency corresponds to a lower point in the spectrum.
The improvement referred to relates merely to the use of a fiat-sided
bottle to contain the preparation. In order to get a satisfactory result it is
necessary that the sides of the containing vessel be pretty good optically.
This condition may be satisfied with a built-up cell, but on account of the
difficulty of <Bi^r a suitable cement, it is rarely that such cells remain in
good order for any length of time. It occurred to me that a bottle might be
434 ON AN IMPEOVED APPARATUS FOR CHRISTIANSEN'S EXPERIMENT. [127
made to answer the purpose, provided the precaution were taken of using the
same kind of glass for the bottle and for the powder. The outer surfaces of
the glass sides of the bottle can be worked flat, while the unavoidable irregu-
larities of the inner surfaces are compensated by the liquid, which, being
adjusted to have the same index as the powder, will have also the same
index as the glass of the bottle.
The bottles that I have used* are about 3 inches high, 1^ inch wide, and
about | inch thick, outside measurement. The outer surfaces are worked
(like plate-glass), and not merely flattened upon a wheel, as is usual with
ordinary perfume bottles. For my earlier trials I was provided with a piece
of flint glass from the same pot as the bottles ; but although the experiment
succeeded well enough as regards the elimination of the internal irregu-
larities of the walls, the glass-powder itself did not behave as well as I had
seen plate-glass powder do. It appeared ultimately that the flint was not
sufficiently homogeneous for the purpose, and another specimen of flint was
also a partial failure, from the same cause ; but a sample of optical flint,
kindly supplied to me by Dr Hopkinson, gave excellent results.
It is more important that the powder should be homogeneous in itself
than that it should correspond very accurately with the glass of the bottle.
For ordinary purposes plate-glass powder (all, of course, from one piece) may
be used in a bottle of soda-glass, or even of ordinary low flint. In preparing
the powder great care is required to exclude dirt. With respect to the
coarser grades there is no great difficulty, but the finer powder is apt to be
contaminated with the substance ~of the mortar. I prefer to use one of iron,
so that a magnet will remove the foreign matter. The elimination of fine
dust is also facilitated by a blast of wind from bellows.
In order to get good definition it is necessary not only that the powder be
homogeneous, but that the temperature be uniform ; for, as Christiansen has
shown, the transmitted ray rises rapidly in refrangibility with temperature.
In order to secure homogeneity it is sometimes necessary to shake up the
preparation, which (to prevent the formation of air-bubbles) is best done
with a rather gentle motion while the bottle is held nearly horizontal. The
proportion of liquids necessary varies with the temperature and with the
kind of glass. Flint will require a higher proportion of bisulphide of carbon
than plate-glass. If the transmitted light is too high in the spectrum, add
more bisulphide of carbon ; if too low, more benzole.
The experimenter, working in a room lighted in the ordinary manner,
will be disposed at first to underrate the degree of homogeneity of the trans-
mitted light. If the shutters be nearly closed, so that light finds entrance
through a narrow slit only, a better judgment can be formed, which may be
tested by prismatic analysis.
* Supplied by Messrs Harvey and Peak, Beak Street, Regent Street.
127] ON AN IMPROVED APPARATUS FOR CHRISTIANSENS EXPERIMENT. 435
In an otherwise dark room lighted by a powerful soda-flame, it is
interesting to remark how very slight a change in the critical colour mani-
fests itself in the general appearance of surrounding objects seen through the
preparation. When the ray of maximum transmission corresponds closelv to
that of soda, the powder is almost invisible, and objects are seen as through
a clear medium. But so slight a change of temperature brings with it a
hazy appearance, that it requires some care to obtain the best effect It is
desirable also to exclude by absorbing media the blue light which usually
attends a soda- flame in very sensible degree.
[1900. In subsequent experiments (Xalvre, LX. p. 64, 1899) it has been
found that the range of refrangibility of the light transmitted by a layer
15 to 20 mm. thick is but 2 times that embraced by the two /)-lines.]
a
128.
OPTICAL COMPARISON OF METHODS FOR OBSERVING
SMALL ROTATIONS.
[Philosophical Magazine, xx. pp. 360, 361, 1885.]
IN order to measure very small rotations, e.g. of the suspended parts of a
galvanometer or magnetometer, two methods are commonly employed. We
may either observe with a magnifier the motion of a material pointer ; or.
following Gauss, cause the rotating parts to carry round a mirror in which a
scale is seen by reflection. In a modification of Gauss's method, well known
from Sir W. Thomson's galvanometers, the image of a dark or bright line is
thrown objectively upon the scale. In deciding which arrangement to adopt
in any particular case, various circumstances would have to be taken into
account, but still a comparison of capabilities from a purely optical point of
view is not without interest.
In the mirror method the optical limit depends upon the horizontal
breadth of the mirror itself. The easiest road to the desired conclusion, as
well as the most instructive, is by a direct application of the principles of the
wave theory. To take the simplest case, we will suppose the mirror rect-
angular. Consider, then, a luminous point, and its image after reflection,
whether in the focal plane of a telescope, or formed directly upon a scale.
The optical work being perfect, the secondary rays from every part of the
mirror agree in phase at the focal point. Now suppose that the mirror
rotates through such an angle that one vertical edge advances a quarter of a
wave-length (JX), while the other retreats to the same amount, and consider
the effect on the phase-relations at the point in question. It is evident that
one extreme wave is accelerated and the other retarded by \, and that the
phases are now distributed uniformly over a complete cycle. The result is
therefore darkness ; and the effect of the rotation has been to shift the image
through half the width of the central bright band which, with accompanying
128] CQXPARISOX OF METHODS FOB OBSERVING SMALL R0TATIOS&. 437
fiingies, is the representative in the image of a math^M*^! 1^*. Such a
motion would be visible (with proper arrangements as to magnifying), but
the limits of resolving power are being approached. It is to be noticed that
the conclusion K independent of the focal length of the minor and of the
employment of a telescope. Provided of course that the rail width of the
mirror is really used, a motion of its Tertical edges through \ may be made
evident.
A iramparison with the method by direct observation of a pointer is now
easy : for. as Las been proved by the researches of Abbe and Helmholtz, a
motion of |\ may be rendered evident in a very similar degree by direct
appEeati">n of a perfect microscope to the moving object. If, therefore, we
^ppoise the length of die pointer to be equal to the half-width of the mirror,
the two methods are optically upon a level It is needless to say that it
would be easy to give the pointer a great advantage in this respect : but the
direct use of the microscope would often be interfered with by motions in the
line of sight, making it impossible to preserve the focus. And besides this,
it is sometimes necessary for the observer to remain at a distance.
* Set - Imw*ttgriMff in OftiesT PML Mm*. Is-TSL IttO, j[A--: -
129.
ON THE THERMODYNAMIC EFFICIENCY OF THE
THERMOPILE.
[Philosophical Magazine, xx. pp. 361363, 1885.]
DURING the last few years the thoughts of many electricians have turned
to the question of the possibility of replacing the dynamo by some develop-
ment of the thermopile ; and it is, I believe, pretty generally recognized that
the difficulty in the way is the too free passage of heat by ordinary conduc-
tion from the hot to the cold junction. The matter may perhaps be placed
in a clearer light by an actual calculation, accompanied by a rough numerical
estimate applicable to the case of German silver and iron.
If t, t denote the temperatures of the hot and cold junctions respectively,
e the electromotive force per degree Centigrade, the whole electromotive
force for n pairs in series will be represented approximately by
ne(t-t ).
The magnitude of the current ((7) is found by dividing this by the sum of
the internal and external resistances (R + R) ; and the useful work done
externally per second is RC*. It reaches a maximum when the external
resistance is equal to the internal ; and its amount is then
The value of the internal resistance R depends upon the dimensions and
specific resistances of the bars. Denoting the latter quantities by r lt r 2 , and
taking <r 1} <r 2 to represent the areas of section, the common length being I,
we have
129] OX THE THERMODYXAJOC EFFICIENCY OF THE THERMOPILE. 439
so that the external work per second is
We will now compare this with the work dissipated by ordinary conduc-
tion of heat along the bars.
If Q be the amount of heat conducted by the n pairs, r/, r 2 ' the thermal
resistances, then
The fraction of this heat, supplied at temperature t, which might be con-
verted into work by a perfect engine working between the absolute tempera-
tures t and f t , is (t - f,)/< ; so that the work dissipated per second is
tl W r _
where J denotes the mechanical equivalent of heat.
The ratio of this to the useful work is
independent of (t ^), of n, and of /. It is further evident that the ratio in
question does not depend upon the absolute values of the sections, or of the
electrical and thermal resistances, but only upon the ratios of these quan-
tities. Thus the efficiency of the thermopile is independent of the absolute
dimensions of the bars, and even of the difference of temperatures at the
junctions. The power is increased by diminishing the length and increasing
the sections to a limit not indicated by these expressions (in which the
terminal temperatures are regarded as given), and probably determined in
practice by the necessity of conveying the heat to the scene of action.
The resistances being given, the ratio of sections a., o^ is to be determined
so as to make our ratio a minimum. This happens when
and thus we get
440 ON THE THERMODYNAMIC EFFICIENCY OF THE THERMOPILE. [129
To turn this into numbers for the case of German silver and iron, we have
the following approximate numbers in C.G.S. measure* :
7\ = 2-0 x 10 4 , r/ = 1-0 x 10 1 ,
r 2 = 1-0 x 10 4 , r/ = 5-0 x 10,
e = 3-0x 10 3 .
The value of J is 4'2 x 10 7 , and for t we will assume 500 (absolute
measure). The use of these gives, as the ratio of the work that would be
obtained by a perfect engine from the conducted heat to that actually
obtained from the thermo-electric force, the number 300 ; from which we
may conclude that the steam-engine and dynamo are not likely to be super-
seded by a German-silver and iron thermopile, even though considerable
allowance be made for the admitted roughness of the numerical estimate.
As regards other materials, it is interesting to note that the ratio contains
as divisor the square of the electromotive force per degree.
* Everett's " Units and Physical Constants "; Landolt's Tabellen.
130.
ON WAVES PROPAGATED ALONG THE PLANE SURFACE
OF AN ELASTIC SOLID.
[Proceedings of the London Mathematical Society, xvn. pp. 411, 1885.]
IT is proposed to investigate the behaviour of waves upon the plane tree
surface of an infinite homogeneous isotropic elastic solid, their character
being such that the disturbance is confined to a superficial region, of thick-
ness comparable with the wave-length. The case is thus analogous to that
of deep-water waves, only that the potential energy here depends upon
elastic resilience instead of upon gravity*.
Denoting the displacements by a, ft, 7, and the dilatation by 6, we have
the usual equations
> &c., ..................... (1)
in which
da d{3 dy
If a, /3, 7 all vary as e**, equations (1) become
7/J
, &c ................... (3)
Differentiating equations (3) in order with respect to x, y, z, and adding,
we get
(V 2 + /i 2 )0 = 0, ................................. (4)
in which
(5)
* The statical problem of the deformation of an elastic solid by a harmonic application of
jpiiiiiuni to its surface has been treated by Prof. G. Darwin, Phil. Mag. Dec. 188*2. Jan. 1886.
See also Camb. Math. Trip. Ex. Jan. 20, 1875, Question IT.
442 ON WAVES PROPAGATED ALONG [130
Again, if we put
equations (3) take the form
(VMJ4'-{l-)fi.' &c ...................... (7)
A particular solution of (7) is*
1 dO Id6 IdB
a = -h?dx> P = -h?dy' ^-tedz' ............ (8)
in order to complete which it is only necessary to add complementary terms
u, v, w satisfying the system of equations
= 0, (V- + k-)v = 0, (V 2 + k*)w = 0, (9)
du dv dw
5S + ^ + & = (10)
For the purposes of the present problem we take the free surface as the
plane z = 0, and assume that, as functions of x and y, the displacements are
proportional to e ifx , e i{ry . Thus (4) takes the form
(<^ 2 + A 2 -/ 2 -<7 2 )0 = 0;
so that
6 = Pe~ rz + Qe +rz , (11)
where
r 2 =/ 2 + <7 2 -/* 2 (12)
In (11), r is supposed to be real; otherwise the dilatation would penetrate
to an indefinite depth. For the same reason, we must retain only that term
(say the first) for which the exponent is negative within the solid -f*. Thus
Q = 0, and we will write for brevity P = 1, or rather P = e ipt e ifx e^ ; but the
exponential factors may often be omitted without risk of confusion, so that
we may take
e = e~" (13)
At the same time the particular solution becomes
f
For the complementary terms, which must also contain e ifx , e^ y as factors,
equations (9) become
f*-g*)u = 0, &c. ; .................. (15)
* Lamb on the Vibrations of an Elastic Sphere, Math. Soc. Proc. May 1882.
t By discarding these restrictions we may deduce the complete solution applicable to a plate,
bounded by parallel plane free surfaces ; but I have not obtained any results which seem worthy
of quotation.
130] THE PLANE SURFACE OF AN ELASTIC SOLID. 443
whence, as before, on the assumption that the disturbance is limited to a
superficial stratum,
u = Ae~ a , v = Be~ K , w = C<r tz , ............... (16)
where
s=/ + <f-* .............................. (17)
In order to satisfy (10), the coefficients in (16) must be subject to the
relation
ifA+igB-sC=0 ............................ (18)
The complete values of a, /?, 7 may now be written
a = - fetT" + Ae-**, p = - l jL t e- n + Be-* 2 , 7 = ^ e - TZ + Ce~ a ,
............ (19)
in which A, B, C are subject to (18); and the next step is to express the
boundary conditions for the free surface. The two components of tangential
stress must vanish, when z = 0, and these are proportional to
dft + <h *y + ^
dz dy ' dx dz
respectively. Hence
s B = ^+igC, sA = 2t ff + ifV. ............... (20)
Substituting from (20) in (18), we find
................ (21)
We have still to introduce the condition that the normal traction is zero at
the surface. We have, in general,
or, if we express X in terms of /*, h, k,
so that the condition is
fc 2 - 2A 2 - 2 (+ r 2 + h-sC) = 0,
or, on substitution for r* of its value from (12),
tf-2(f* + g*)-2h*sC = ......................... (22)
By eliminating C between (21) and (22), we obtain the equation by which
the time of vibration is determined as a function of the wave-lengths and of
the properties of the solid. It is
or, by (17),
(23)
444 ON WAVES PROPAGATED ALONG [130
If we square (23), and introduce the values of r 2 and s 2 from (12), (17), we
get
{2 (/ 2 + g*) - k*}< = 16 (/ 2 + gj (/ 2 + f- A 2 ) (/ 2 + f - & 2 ).
As f and g occur here only in the combination (f 2 + # 2 ), a quantity homo-
geneous with h? and A; 2 , we may conveniently replace (f- + g 2 ) by unity.
Thus
k* - 8k e + 24& 4 - 16& 2 - 16A 2 & 2 + 16A 2 = (24)
Since the ratio A 2 : k 2 is known, this equation reduces to a cubic and deter-
mines the value of either quantity.
If the solid be incompressible (A, = oo ), A 2 = 0, and the equation becomes
& 6 -8& 4 + 24& 2 -16 = (25)
The real root of (25) is found to be '91275, and the equation may be written
(k 2 - -91275) (k 4 - 7-08725& 2 + 17-5311 ) = 0.
The general theory of vibrations of stable systems forbids us to look for
complex values of k 2 , as solutions of our problem, though it would at first
sight appear possible with them to satisfy the prescribed conditions by
taking such roots of (12), (17), as would make the real parts of the exponents
in e~ rz , e~ sz negative. But, referring back to (23), which we write in the
form
(2 - kj = 4rs,
or, in the present case of incompressibilit^ by putting r = 1,
(2 - & 2 ) 3 = 4s,
we see that we are not really free to choose the sign of s. In fact, from the
complex values of k 2 , viz., 3'5436 2 230 K, we find
4s = -27431 + 6-8846 1;
so that the real part of s is of the opposite sign to r, and therefore e~ rz , e~ sz
do not both diminish without limit as we penetrate further and further into
the solid.
Dismissing then the complex values, we have, in the case of incompressi-
bility, the single solution
> = E' = -91275 (/ 2 + # 2 ) (26)
From (19), (20), (21), we get in general
130] THE PLAXE SURFACE OF AN ELASTIC SOLID. 445
In the case of incompressibility, we have If given by (26), and
r =/*+?, # = -08725 (/ J + 5 ).
Hence
-3433 e~ K
(30)
If we suppose the motion to be in two dimensions only, we mav put g =
so that = 0, and
in which
Jt=-9554/, s = -2954/. ........................ (32)
For a progressive wave we may take simply the real parts of (31). Thus
AV/- = (e-'* - -5433e-) sin (pt +fx) )
K~/!f = (e~ fi - l'840e-) cos (pt +/r) J
The velocity of propagation is p'f, or -955 4- \ f (/jL p), in which \'(jJi'p) is the
velocity of purely transverse plane waves. The surface waves now under
consideration move, therefore, rather more slowly than these.
From (32), (33), we see that a vanishes for all values of x and t when
g*-/" = -5433, i.e., when fz = '8659. Thus, if X' be the wave-length (2-irf),
the horizontal motion vanishes at a depth equal to '1378 X'. On the other
hand, there is no finite depth at which the vertical motion vanishes.
To find the motion at the surface itself, we have only to put z = in (33).
We may drop at the same time the constant multiplier (h-f) which has no
present significance. Accordingly,
i = '4567 sin (pt +, 7 = - "840 CDS ( pt +/r), ......... (34)
showing that the motion takes place in elliptic orbits, whose vertical axis is
nearly the double of the horizontal axis.
The expressions for stationary vibrations may be obtained from (30) by
addition to the similar equations obtained by changing the sign of p, and
similar operations with respect to f and g. Dropping an arbitrary mul-
tiplier, we may write
a = / { er n + 5433 e~ a ] cospt sin/r cos^y j
= -{-e- r * + -5433e- c }cos/rfco8/.rsin0y | , ......... (35)
y = r {+ e~ n - 1-8400-**} cos pt cos/r cos^ry )
in which
(36)
446 ON WAVES PROPAGATED ALONG
As before, the horizontal motion vanishes at a depth such that
[130
We will now examine how far the numerical results are affected when we
take into account the finite compressibility of all natural bodies. The ratio
of the elastic constants is often stated by means of the number expressing
the ratio of lateral contraction to longitudinal extension when a bar of the
material is strained by forces applied to its ends. According to a theory
now generally discarded, this ratio (<r) would be \ ; a number which, how-
ever, is not far from the truth for a variety of materials, including the
principal metals. In the extreme case of incompressibility cr is ^, and there
seems to be no theoretical reason why <r should not have any value between
this and 1 *.
The accompanying table will give an idea of the progress of the values of
& 2 /(/ 2 + # 2 ) as dependent upon \/p, or upon <T. It will be observed that
the value diminishes continuously with X, in accordance with a general
principle f.
X
ff
/i 2 /& 2
W+0 2 )
W(/ 2 + f7 2 )
00
i
9127
9554
M
4
i
8453
9194
k
7640
8741
-*M
1
1
4746
6896
As an example of finite compressibility, we will consider further the second
case of the table. From (12), (17),
r 2 - -7182 (/ 2 + g*\ r = "8475 V(/ 2 + <f),
s 2 = 1 547 (/" + <7 2 ), s = -3933 V(/ 2 + # 2 )-
Hence, from (27), (28), (29), in correspondence with (30), we have
h 2 a = if {- e~ rz + -57730-^} &** e ifx e \
h*$ = ig {- e~ rz + -5*773 er"} e** <P* e t ....... (37)
/i 2 7 = -8475 V(/ 2
* Prof. Lamb, in his able paper, seems to regard all negative values of <r as excluded a priori.
But the necessary and sufficient conditions of stability are merely that the resistance to com-
pression (\ + AI) and the resistance to shearing (n) should be positive. In the second extreme
case of a medium which resists shear, but does not resist compression, \= -|,u, and <r= -1.
The velocity of a dilatational wave is then f of that of a distortional plane wave. (Green, Camb.
Trans. 138.) The general value of a is X/(2X + 2,u).
t Math. Soc. Proc. June 1873, vol. iv. p. 359 [vol. i. p. 171]. Theory of Sound, t. i. p. 85.
Lamb, loc. cit. p. 202.
130] THE PLANE SURFACE OF AN ELASTIC SOLID. 447
For a progressive wave in two dimensions, we shall have
- -57736-*) sin (pt +fx) }
)
-) cos (pt +f
At the surface,
Kalf = 4- -4227 sin (pt +/*) ) (39)
h^ff= - -6204 cos (pt -h/r) )
so that the vertical axes of the elliptic orbits are about half as great again as
the horizontal axes.
It is proper to remark that the vibrations here considered are covered by
the general theory of spherical vibrations given by Lamb in the paper
referred to. But it would probably be as difficult, if not more difficult, to
deduce the conclusions of the present paper from the analytical expressions
of the general theory, as to obtain them independently. It is not improbable
that the surface waves here investigated play an important part in earth-
quakes, and in the collision of elastic solids. Diverging in two dimensions
only, they must acquire at a great distance from the source a continually
increasing preponderance.
131.
ON PROF. HIMSTEDT'S DETERMINATION OF THE OHM.
[Philosophical Magazine, xxi. pp. 10 13, 1886.]
As there is still some discrepancy in the values of the ohm obtained
by able workers using various methods, it seems desirable to put forward
any criticisms that may suggest themselves, in the hope that the causes
of disturbance may thus come to be better understood. I propose accord-
ingly to make a few remarks upon the paper of Professor Himstedt,
translated in your November number, not at all implying that his results
may not be as good as any other, but rather in order to raise discussion
on certain points which the author may be able to treat satisfactorily when
he publishes a more detailed account of his work.
The leading feature in the method of Prof. Himstedt is the use of a
commutator, or separator, by which the make- and break-induced currents
are dissociated, one or the other passing in a stream at equal small inter-
vals of time through a galvanometer, by whose aid their magnitude is
appreciated. The instrument works with mercury contacts. When I first
considered the methods available for the solution of this problem at
Cambridge in 1880, I found ready to my hand an ingenious apparatus,
contrived by Prof. Chrystal for this very purpose. The contacts were
effected by metallic dippers, controlled by eccentrics, and passing in and
out of mercury cups. What determined me against this method*, not-
withstanding its obvious advantages in respect of sensitiveness, was the
recollection of unavailing attempts of my own in 1870 to make satis-
factory mercury contacts with dippers carried by electrically maintained
tuning-forks. Even when silver was the metal employed, the contacts were
* I may remark that Brillouin used a commutator of this nature in his researches on the
comparison of coefficients of induction: Theses presentees a la Faculte des Sciences de Paris,
1882.
131] ON PROF. HIMSTEDT'S DETERMINATION OF THE OHM. 4.4,9
uncertain, and no trustworthy galvanometer deflection could be obtained.
It may be mentioned, in passing, that the object was to obtain, through
the galvanometer, a stream of charges of a condenser, separated from the
discharges, with a view to the determination of v (the ratio of electrical
units). Dippers in electrical connexion with the body of the fork (and
by means of a wire attached to the stalk with one pole of the condenser)
were carried on both the upper and lower prongs. Underneath these,
mercury cups were so arranged that the vibrating-fork was in contact
with them alternately, but never with both at the same time. One of
the cups was connected with the insulated pole of the battery and the
other with the earth. The fork was driven by a current entirely insu-
lated from it. It was found, however, that the contacts could not be
made perfect, and the direct use of the fork was abandoned in favour of
a commutator with platinum contacts driven by the fork*. This form
of the apparatus is unsuitable as a separator of induced currents; and
I was inclined to favour the observation of a single induced current with
a ballistic galvanometer as carried out by Rowland, and afterwards by
Glazebrook.
It is to be presumed that the contact difficulty has been overcome by
Prof. Himstedt ; and my principal reason for mentioning it is that I found
it particularly capricious and insidious. The galvanometer indication would
often remain steady for minutes together, and then suddenly change. It
would be interesting to know whether Prof. Himstedt has met with any
behaviour of this sort.
The next question that I wish to raise relates to the measurement by
the galvanometer of a series of induced currents, each of short duration.
On page 421 there is a reference to "cross magnetization" that I do
not quite understand. I have myself -f objected to the use of a ballistic
galvanometer, on the ground of the tacit assumption that the needle at
the moment of the impulse, when subject to a powerful cross-magnetizing
force, retains its axial magnetization unaltered ; but in the method of
Prof. Himstedt the question assumes a different shape. In this case the
needle stands in an oblique position, and we have to consider whether the
axial magnetization does not alter under the action of a force having a
sensible axial component*. In all probability Prof. Himstedt has considered
this matter. It admits of a very simple test, all that is necessary being
to deflect the needle into its oblique position with an external permanent
magnet, and then to allow, the induced currents to pass, suppressing the
* "On the Determination of the Number of Electrostatic Units in the Electromagnetic Unit
of Electricity," J. J. Thomson, Phil. Trans. 1883, p. 719.
t Phil. Trans. 1882, p. 670. [Vol. n. p. 48.]
J "On a Permanent Deflection of the Galvanometer Needle, Ac." Brit. Assoc. Report, 186&
Phil. Mag. Jan. 1877. [Vol. i. p. 310.] Chrystal, Phil. Mag. Dec. 1876.
450 ON PROF. HIMSTEDT'S DETERMINATION OF THE OHM. [131
interruption of the secondary contact. Both make- and break-induced cur-
rents would then pass, whose mean value is zero ; and any deflection of
the needle under these conditions would be a sign that its magnetism
fluctuated and that the evaluation of either stream alone would be
vitiated.
An interesting feature in Prof. Himstedt's work is the arrangement of
the primary and secondary coils, of which the former is a long solenoid
embraced by the latter. The fact that as regards the secondary the
induction-coefficient depends sensibly upon the number of turns only,
without regard to radius, is much in its favour. Any one who has had
to do with the measurement of coils will appreciate too the advantage
of reducing the primary to a single layer. There are, however, disad-
vantages in this arrangement which must be kept in sight. I will not
dilate upon the use of a wooden core on which to wind the primary,
though I should think it hardly safe. But assuming that there is no
important uncertainty as to the value of R (the mean radius of the
primary), though it should be remarked that it occurs in the formula
as a square, nor in the data relating to the secondary, we have still to
consider the factor K, expressive of the number of turns per unit length
in the primary. So far as appears, the value of this quantity is obtained
by simply dividing the whole number of turns, 2864, by the measured
length, 135-125 cm. Now there is here a tacit assumption either that
the wire is wound with perfect uniformity, or that we have to deal only
with the mean value. The latter alternative is manifestly incorrect, since
the central parts lying nearly in the plane of the secondary are necessarily
more effective than the remoter parts. In point of fact the simplicity of
this arrangement is more apparent than real, relating rather to calcula-
tion than to measurement, as I have already had occasion to remark* in
connexion with a somewhat similar use of a long solenoid in Mascart's
determination of the electrochemical equivalent of silver. How far uni-
formity was attained in the present case I have no means of judging ;
but where the successive turns are merely brought into contact with one
another I should not expect a high degree of precision, if only because
the thicknesses of the wire and silk are liable to vary. Again, it may
be possible to verify the uniformity a posteriori, or to obtain data for
the calculation of a correction. But at any rate it seems misleading to
exhibit the result as determined by the average number of turns per unit
length, when it really depends also upon the ratios of the rates of winding
at the various parts of the length.
* Phil. Tram. 1884, p. 413. [Vol. n. p. 280.]
132.
OX THE CLAKK CELL AS A STANDARD OF
ELECTRO-MOTIVE FORCE,
Tntmmetia>m* T O^XXYL. pp. 781 $MO.
89. THE importance of a convenient standard df eBecftro-m&tive :': ?>=
is now fully recognised. It gives the most avalkWe meams i>f m^^imrLmz
currents, especially of large amount, and has been used tor chis ptnirpoi^ by
several experimenters. I may refer to my mTestigatiiGn cm the- Cjusraniti
of Magnetic Rotation of Light in EsulplhMe of Carbon*, in whk-h trie
currents were all measured by reference to a Clark celt wfowjee vafae was
originally obtained by ahsohrte measaremenlts and rerii^ an m-^n;^ b j
the sflver voltameter. Clark cells are exeeedingiy cioevemeni^ in -:.&. anxd
would doubtless be generally empfoyed. ooold confideaffice be :'-.: in :i-;:?
permanence^ and in the equality of eels set ncp by liEifejrtcL: pers:-L?
the same recipe. To these questions I feave gi^em HDiQctu attentoom :
the result of a large experience is very ikTio-mrafc-Le- &>> tine
ness of the cells, if reasonable precautions be observed in ehargim^:
I believe that any one who takes the tre-rnble tt> .^et up three cc :':-ir
cells and compares them occasionally, will be in possesso*! of a Standard
of F.M.F. which he may trust to about -j^tth part.
The present memoir is to be regarded as, sntppleme&iitary t> tiuat *im ttine
Electro-chemical Equivalent of Silver, and on tine Absolute Electro-motive
Force of Clark Cellsf, and the paragraphs are numbered accordingly. The
total number of cells experimented upon is large. Of my own eonstruction
there have been about 60 of the ordinary kind (with sdliid zincs*, and about
30 of the J7-pattern Q 28) with zinc amalgam. In addition to these some
f PiOL Tram. Put H. WML pToL a. ft .]
452 ON THE CLARK CELL AS [132
40 cells made by others have been compared, with very interesting results
to be given later.
Before entering into details it may be convenient to summarise the
principal sources of error. The E.M.F. may be too high, (1) because the
paste is acid, (2) because the paste is not saturated with zinc sulphate.
The first fault tends to cure itself, and is rarely found after the cells are
a month old. The second is the origin of the more serious discrepancies
that have been met with in commercial cells. If the E.M.F. is too low, the
cause may be, (1) that the cell has become dry, in which case the drop
will probably be progressive, (2) the solution is super- saturated with zinc
sulphate, (3) the mercury is impure.
Believing that these cells are capable of affording standards of a high
degree of precision, and that they ought to be in general use, I have
gone into considerable detail as to the procedure which may be adopted.
This may give the impression that the preparation is troublesome, but
in reality the method that I propose is much simpler than those hitherto
employed and thought to be necessary. To show how easy it is to set
up these cells, I may refer to two large ones, contained in glass cylinders
of about 4 inches diameter and provided with wooden covers by which
the electrodes are carried. Enough common mercury was poured in to
cover the bottom, contact being made with it by means of a platinum
wire sealed in a glass tube. The jar was then filled to a height of about
4 inches with saturated solution of commercial zinc sulphate with which
some mercurous sulphate had been rubbed up in a mortar. The zinc
electrode was cut from ordinary sheet metal, and was suspended hori-
zontally near the top of the liquid by a projecting tail. After the first
few weeks these large cells have never deviated from the standard by
much more than y^y, and have been found very convenient for certain
purposes on account of their comparatively small resistance. They have
also been used for preliminary comparisons with cells whose value was
unknown, in which case there was danger of more current passing than
it is desirable to allow through delicate standards.
40. The method followed for making the recent comparisons is the
same in substance as that described in 28. The use of a high resistance
galvanometer gave a greater facility of reading, a change of y^ w in the
E.M.F. under measurement giving a motion of the spot of light which could
be seen without a telescope from across the room.
The accompanying table (XIII.) gives the values of most of the older
cells in continuation of that contained in the note to 30 [vol. II. p. 331].
Cells (4), (8), (9) were, I think, left at Cambridge; (18) and (19) were
observed at intervals during 1885, but the E.M.F. was found to fall. When
about three parts per thousand too low, they were removed for examination,
132] A STANDARD OF ELECTRO-MOTIVE FORCE.
153
z
Ilisliiillss I s s
= =! ! S z S z = =
X ' .'
^
i = ? =
= = =
s
.
jl
454 ON THE CLARK CELL AS [132
and found to be dry. The water had exuded, or evaporated, through
cracks in the paraffin wax. The cells of the .//-pattern, H s , H 1S were
broken in a manner to be presently explained. On the other hand some
new cells of the H- pattern, a,b,...f, are included. They are those referred
to in the previous paper as having been fitted up by Mr Threlfall, and
are more than a year old.
The agreement exhibited in Table XIII. is very remarkable. In many
cases the cells may be depended on not to vary relatively more than 2 or
3 parts in 10,000, notwithstanding considerable changes of temperature.
It is, indeed, doubtful whether even the whole of the small variations
recorded are real. 1 C. influences the E.M.F. about 8 parts in 10,000,
and differences of temperature of two or three-tenths of a degree may
well have occurred, since the cells were variously mounted, and no par-
ticular precautions were taken beyond the avoidance of readings at times
when the temperature of the room (immediately under the roof) was
changing rapidly.
It may be convenient to recall that cells 1, 5, were made in Oct., 1883 ;
10-13 and 14-19 in May, 1884; H 6 in March, 1884; H w , H u also in
March, 1884.
41. In cells of the ordinary type the principal source of weakness
is imperfect sealing at the top, due to cracks in the paraffin wax. As
pointed out by Dr Alder Wright*, a better result is obtained if the
whole cell be imbedded in a large mass of wax than when (as in my
cells) the wax is applied merely inside the tube, above the cork sus-
taining the zinc. During the last year I have replaced paraffin by
marine glue, which, so far as can be judged at present, may be relied
upon to effect a complete seal. The procedure will be described presently
more in detail.
The cause of failure in the .//-cells is of a different nature. Many of
the earlier cells had been found to break in the amalgam leg, and the
trouble was attributed to a hardening and expansion of the contents
( 29). Such a hardening had, in fact, been observed in one or two
cases. More recent experience, however, has proved that the cause must
be looked for elsewhere, several cells having failed in which no trace of
solid amalgam was to be found. Nevertheless the amalgam is the cause
of the trouble, for out of a large number of breakages not one has
occurred in the leg containing pure mercury. It would appear that some
alloying takes place with the platinum wire in contact with the amalgam,
and that this gradually extends itself with fatal results to the part of
the platinum sealed into the glass, from which place the cracks are
* Phil. Mag. July, 1883, p. 32.
132]
A STANDARD OF ELECTRO-MOTIVE FORCE.
455
always observed to radiate. It is hoped that a cure will be found in a
plan, adopted for some recent cells, of melting in a little cement (marine
glue has been used) so as to protect from the amalgam the part of the
platinum which lies nearest to the glass; but it is too soon to speak
with certainty.
42. The H-foTrn lends itself to hermetical sealing, and at one time
I anticipated advantage from this course. There is, however, such a
large amount of spare liquid that there is no likelihood of trouble from
desiccation, even if the corks allow a little evaporation. Indeed, by with-
drawing the corks a fresh supply of liquid could be introduced at any-
time. It happened on one occasion that an H-ce\\ to which a large
excess of salt had been added, was so far crusted up next the metallic
surfaces that it began to show signs of failing E.M.F., much as if it were
going dry. The mass was so compact that no impression could be made
upon it with a glass rod ; but it was bored through with a steel reamer,
when the E.M.F. at once recovered its normal value. In such cases the
accessibility is advantageous, especially for purposes of experiment. It is
well, however, to avoid such a large excess of salt as was present in this
case. By alternate melting and oystallisation as the temperature rise>
and falls, there is a tendency to aggregation, of which the cell above
referred to affords an extreme example.
In the construction of cells with solid zinc electrodes, I have fallen
back upon a simplified pattern nothing more in fact than a small tube
with a platinum wire sealed through its closed end. See figure.
G.P. Covered Wire
-Marine Glue
Saturated \
Solution Zn. SOJ
Pure Hg..~
Sealing Wax
:vr|l Paste of Hg. SO 4
-,-Pt. Wire
456 ON THE CLARK CELL AS [132
The only objection to this form is that the cells cannot, without
precaution, be supported from underneath. Most of mine are held at
the centre by a spring (cut from sheet metal) against a piece of board
mounted on its edge. In this case the copper electrodes are secured in
sealing-wax to the wooden stand. For single cells, when portability is
desired and convenience of immersion in water or ice, it is a good plan
to enclose the whole in a rather long and narrow test-tube. A little
cotton wool supports the cell and prevents it from shaking about laterally.
The gutta-percha covered leads pass through a piece of cork inserted near
the top of the test-tube, and a little marine glue poured over the cork
makes all tight. In order to give mechanical support to the platinum
wire, which is liable to break where it passes through the glass, the ex-
ternal application of sealing-wax is recommended a precaution applicable
also to the J^T-cells.
43. In charging the cells the first step is to pour in sufficient pure*
mercury to cover the platinum effectively. The paste (of which more
presently) is next introduced, with the aid of a small funnel, care being
taken not to soil the sides above the proper level. The zincs, cut from
rods of pure zinc, as supplied by Hopkins and Williams, and not recast,
are soldered to copper wires and cleaned in the lathe. Just before use
they are dipped in dilute sulphuric acid, washed in distilled water, and
dried with a clean cloth or filter paper. Each zinc is mounted in a short
piece of cork fitting the tube (but not too tightly), and nicked in order
to allow of the passage of air. The cork is pushed gradually down until
its lower face is almost in contact with the paste. The object is to leave
but little air, and at the same time to avoid squeezing up the paste between
the cork and the glass. The whole is now made tight by pouring marine
glue over the cork high enough to cover the zinc and soldering, and leave
only the wire projecting. The tube should rise high enough to receive
the glue, and thus secure a good adhesion.
The marine glue is melted over the gas flame in a small pot or basin,
and stirred, until uniform, with a small stick. It should be fluid enough
to pour by its own weight. If necessary, a little benzole may be added,
but the cement should be pretty hard when cold.
In the operation of pouring in the marine glue the glass is heated by
the glue sufficiently for adhesion ; but this heat does not extend appre-
ciably below the cork. Neither in this, nor any other stage of the process
of charging, is heat applied to the paste.
* Except when the contrary is stated, mercury distilled in vacua has been used for Clark cells.
There is, I believe, a difficulty in purchasing mercury thus treated ; but every physical laboratory
should be provided with an apparatus for this purpose. That employed by me was distilled
at Cambridge in an apparatus set up by Mr W. N. Shaw.
132] A STANDARD OF ELECTRO-MOTIVE FORCE, -15?
44. The earlier cells, prepared with paste, which was doubtless
strongly acid, frequently gave irregular results far several weeks. Ex-
treme cases are afforded by 15 and 16, which are shown by Table YUL
of the former paper to have been alt first more than 2 per cent, too
strong. Moreover, as appears from the continuation of this table in the
notes [p. 331 j, it took nearly two months for these cells to settle down to
their normal values. The cause of irregularity is to be sought rather at
the mercury than at the zinc (or amalgam) electrode.
In order to examine this question, 17-cells were 
