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Full text of "Scientific papers"

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 n