
THE LIBRARY
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OF CALIFORNIA
LOS ANGELES
SCIENTIFIC PAPEBS
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SCIENTIFIC PAPERS
BY
JOHN WILLIAM STEUTT,
BARON RAYLEIGH,
O.M., D.Sc., F.R.S.,
CHANCELLOR OP THE UNIVERSITY OF CAMBRIDGE,
HONORARY PROFESSOR OF NATURAL PHILOSOPHY IN THE ROYAL INSTITUTION.
VOL. V.
19021910.
CAMBRIDGE :
AT THE UNIVERSITY PRESS
1912
Thou hast ordered all things in measure and number and weight.
WISDOM xi. 20.
Engineering
Library
0X3
CONTENTS.
ART. PAGE
273. Interference of Sound I
[Royal Institution Proceedings, xvn. pp. 17, 1902; Nature, LXVI. pp. 42 44, 1902.J
274. Some General Theorems concerning Forced Vibrations and
Resonance . . . it// ... ........ . 8
Work Done . . ?" ." ,'^ , '. '. " . . 10
One Degree of Freedom . ... . . . ... . 11
Several Degrees of Freedom . ., . ... 13
Action of Resonators. ... ...... . . . . . 15
Energy stored in Resonators . . . . . . 23
[Philosophical Magazine, ill. pp. 97117, 1902.]
275. On the Law of the Pressure of Gases between 75 and 150
millimetres of Mercury . ....'. . . . 27
The Manometers . ,, .,'.,;.. ... 28
General Arrangement of Apparatus . . .. '\ ... 29
The Side Apparatus . .... ^. . . 30
General Sketch of Theory .31
Thermometers . . . .'.."'.. . 32
Comparison of Large Bulbs ... . . . 33
Comparison of Gauges . , .; , . ..,., .,.. . 33
The Observations . . ..... 35
The Reductions . . . . . . . . 37
The Results . . ','. : V; p, . . 39
[Phil. Trans. CXCVIII. A, pp. 417430, 1902.]
276. On the Pressure of Vibrations . .' ' . . . . . . . 41
[Philosophical Magazine, in. pp. 338346, 1902.]
277. On the Question of Hydrogen in the Atmosphere . . 49
Spectroscopic Evidence . . .',;'"'. . 49
Determinations by Combustion . . . . . 51
[Philosophical Magazine, in. pp. 416422, April 1902.]
278. Does Chemical Transformation influence Weight? ... 56
[Nature, LXVI. pp. 58, 59, 1902.]
799411
vi CONTENTS
ART. PAGE
279. Is Rotatory Polarization influenced by the Earth's Motion? . 58
[Philosophical Magazine, iv. pp. 215220, 1902.]
280. Does Motion through the ^Ether cause Double Refraction ? . 63
[Philosophical Magazine, IV. pp. 678683, 1902.]
281. On the Distillation of Binary Mixtures 68
Distillation of a Pure Liquid ...... 68
Two or more Liquids which press independently . . 68
Liquids which form true Mixtures ..... 69
Konowalow's Theorem . . . . . . . 71
Calculation of Residue 73
Observations . . . . . . . . . 74
Alcohol and Water 75
Hydrochloric Acid and Water 77
Ammonia and Water ....... 79
Sulphuric Acid and Water 80
Acetic Acid and Water 80
A new Apparatus with uniform Regime .... 81
[Philosophical Magazine, IV. pp. 521537, 1902.]
282. Note on the Theory of the Fortnightly Tide .... 84
[Philosophical Magazine, v. pp. 136141, 1903.]
283. On the Free Vibrations of Systems affected with Small
Rotatory Terms 89
[Philosophical Magazine, v. pp. 293297, 1903.]
284. On the Vibrations of a Rectangular Sheet of Rotating Liquid . 93
[Philosophical Magazine, v. pp. 297301, 1903.]
285. On the Spectrum of an Irregular Disturbance ... 98
[Philosophical Magazine, v. pp. 238243, 1903.]
286. Considerations respecting the Combustion of Modern Propel
lants in Closed Vessels and in Guns . . . . .103
Closed Vessels 103
Combustion in Guns ........ 105
[Minutes of Explosives Committee, 1903.]
287. On the Bending of Waves round a Spherical Obstacle . . 112
[Proceedings of the Royal Society, LXXII. pp. 4041, 1903.]
288. On the Proportion of Argon in the Vapour rising from Liquid
Air 115
[Philosophical Magazine, v. pp. 677680, 1903.]
CONTENTS Vll
ART.
289. On the Theory of Optical Images, with Special Reference to
the Microscope. (Supplementary Paper) . . . .118
[Journ. R. Micr. Soc. pp. 474482, 1903.]
290. On the Production and Distribution of Sound . . .126
Theory of Conical Trumpet 126
Data respecting FogSignals 128
Comparison with Musical Instruments, &c. 128
Cones and Resonators . . . . . . .130
Vibration Indicator 132
Reeds '. 132
Trumpets of Elongated Section ..... 133
Work done by Detached Sources . v. .,,.,. ; ; 135
Continuous Distributions . .1 ... . : ,. ..,. v ,. . 138
Experimental Illustrations. . . ,.,. *. . 139
[Philosophical Magazine, VI. pp. 289 305, 1903.]
291. On the Work done by Forces operative at one or more Points
of an Elastic Solid . . . ; . l . , " ;' . . 142
[Philosophical Magazine, vi. pp. 385392, 1903.]
292. On the Acoustic Shadow of a Sphere . . ;. . . 149
Appendix. By Professor A. LODGE . ; / . . 162
Note by LORD RAYLEIGH . . ,..*. . . 164
[Phil. Tram. 203 A, pp. 87110, 1904.]
293. Shadows .... ( . . . '. ''_',, '. ' 166
[Royal Institution Proceedings, Jan. 15, 1904.]
294. Sir George Gabriel Stokes, Bart., 18191903 . .. .., . 173
[Royal Society YearSoot, 1904.]
295. On the Measurement of certain very short Intervals of Time . 190
[Nature, LXIX. pp. 560, 561, 1904.]
296. Note on the Application of Poisson's Formula to Discontinuous
Disturbances. . . s . ' ~ . " . ' . . . . 193
[Proceedings of the London Mathematical Society, Ser. 2, n. pp. 266 269, 1904.]
297. Fluid Friction on Even Surfaces . . . . .,.. . . . 196
[Note to a paper by Prof. Zahm, Phil. Mag. vni. pp. 66, 67, 1904.]
298. On the Electrical Vibrations associated with thin terminated
Conducting Rods . . . . . . . , .198
[Philosophical Magazine, vni. pp. 105107, 1904.]
VJii CONTENTS
ART. PAGE
299. On the Density of Nitrous Oxide 201
[Proceedings of the Royal Society, LXXIV. pp. 181183, 1904.]
300. Note to a Paper by Prof. Wood on the Achromatization of
approximately Monochromatic Interference Fringes by a
highly dispersive Medium 204
[Philosophical Magazine, vni. pp. 330, 331, 1904.]
301. On the Open OrganPipe Problem in Two Dimensions. . 206
[Philosophical Magazine, vm. pp. 481 487, 1904.]
302. Extracts from Nobel Lecture 212
[Given before the Royal Academy of Science at Stockholm, 1904.]
303. On the Compressibility of Gases between One Atmosphere and
Half an Atmosphere of Pressure 216
The Manometers 218
General Arrangement of Apparatus ..... 220
The Side Apparatus 221
General Sketch of Theory 221
Thermometers ......... 222
The Large Reservoirs ....... 223
Comparison of Manometers 224
The Observations * . 225
The Reductions 227
The Results 228
[Phil. Trans. A, cciv. pp. 351372, 1905.]
304. On the Pressure of Gases and the Equation of Virial . . 238
[Philosophical Magazine, IX. pp. 494505, 1905.]
305. The Dynamical Theory of Gases and of Radiation . . 248
[Nature, LXXI. p. 559; LXXII. pp. 54, 55; pp. 243, 244, 1905.]
306. An Optical Paradox 254
[Philosophical Magazine, ix. pp. 779781, 1905.]
307. The Problem of the Random Walk 256
[Nature, LXXII. p. 318, 1905.]
308. On the Influence of Collisions and of the Motion of Molecules
in the Line of Sight, upon the Constitution of a Spectrum
Line 257
[Proceedings of the Royal Society, A, LXXVI. pp. 440444, 1905.]
309. On the Momentum and Pressure of Gaseous Vibrations, and on
the Connexion with the Virial Theorem .... 262
[Philosophical Magazine, x. pp. 364374, 1905.]
CONTENTS ix
ART. PAGE
310. The Origin of the Prismatic Colours 272
[Philosophical Magazine, x. pp. 401407, 1905.]
311. On the Constitution of Natural Radiation .... 279
[Philosophical Magazine, XI. pp. 123127, 1906.]
312. On an Instrument for compounding Vibrations, with Application
to the drawing of Curves such as might represent White
Light 283
Note on the Principle of the SandClock . . .285
[Philosophical Magazine, XI. pp. 127 130, 1906.]
313. On Electrical Vibrations and the Constitution of the Atom . 287
[Philosophical Magazine, xi. pp. 117 123, 1906.]
314. On the Production of Vibrations by Forces of Relatively Long
Duration, with Application to the Theory of Collisions . . 292
[Philosophical Magazine, xi. pp. 283291, 1906.]
315. On the Dilatational Stability of the Earth .... 300
[Proceedings of the Royal Society, A, LXXVII. pp. 486 499, 1906.]
316. Some Measurements of WaveLengths with a Modified
Apparatus . ... . . '. . . . 313
[Philosophical Mayazine, xi. pp. 685703, 1906.]
317. On the Experimental Determination of the Ratio of the
Electrical Units . 330
[Philosophical Magazine, xn. pp. 97 108, 1906.]
318. On the InterferenceRings, described by Haidinger, observable
by means of Plates whose Surfaces are absolutely Parallel . 341
[Philosophical Magazine, xn. pp. 489493, 1906.]
319. On our Perception of Sound Direction ..... 347
[Philosophical Magazine, xm. pp. 214232, 1907.]
320. Acoustical Notes . . . . . ...... ... 364
Sensations of Right and Left from a revolving Magnet and
Telephones 364
Multiple Harmonic Resonator . . . . . 366
TuningForks with slight Mutual Influence . . . 369
Mutual Reaction of Singing Flames . . .. .371
Longitudinal Balance of TuningForks .... 372
A TuningFork Siren and its Maintenance . . . 376
Stroboscopic Speed Regulation ...... 377
Phonic Wheel and Commutator 377
[Philosophical Magazine, xm. pp. 316333, 1907.]
X CONTENTS
ART. PAGE
321. On the Passage of Sound through Narrow Slits . . . 380
Appendix 386
[Philosophical Magazine, xiv. pp. 153161, 1907.]
322. On the Dynamical Theory of Gratings 388
[Proceedings of the Royal Society, A, LXXIX. pp. 399416, 1907.]
323. Note on the remarkable case of Diffraction Spectra described
by Prof. Wood 405
[Philosophical Magazine, xiv. pp. 6065, 1907.]
324. On the Light dispersed from Fine Lines ruled upon Reflecting
Surfaces or transmitted by very Narrow Slits . . . 410
[Philosophical Magazine, xiv. pp. 350359, 1907.]
325. On the Relation of the Sensitiveness of the Ear to Pitch,
investigated by a New Method 419
[Philosophical Magazine, xiv. pp. 596604, 1907.]
326. Effect of a Prism on Newton's Rings 426
[Philosophical Magazine, xv. pp. 345351, 1908.]
327. Further Measurements of WaveLengths, and Miscellaneous
Notes on Fabry and Perot's Apparatus .... 432
Thirty Millimetre Apparatus 433
Magnifying Power 435
Adjustment for Parallelism ...... 436
Behaviour of VacuumTubes ...... 436
Control of the figure of the glasses by bending . . 437
Figuring by Hydrofluoric Acid ...... 438
Effect of Pressure in Fabry and Perot's Apparatus . . 440
[Philosophical Magazine, xv. pp. 548558, 1908.]
328. On the Aberration of Sloped Lenses and on their Adaptation to
Telescopes of Unequal Magnifying Power in Perpendicular
Directions 442
[Proceedings of the Royal Society, A, LXXXI. pp. 2640, 1908.]
329. Hamilton's Principle and the Five Aberrations of von Seidel . 456
[Philosophical Magazine, xv. pp. 677687, 1908.]
330. Vortices in Oscillating Liquid 465
[Proceedings of the Royal Society, A, LXXXI. pp. 259271, 1908.]
CONTENTS XI
ART. PAGE
331. Acoustical Notes. VIII . . .478
Partial Tones of Stretched Strings of Variable Density . 478
Maintenance of Vibrations by Impact of Water Drops . 481
Discrimination between Sounds from directly in front and
directly behind the Observer 483
The Acousticon 483
Pitch of Sibilants 486
Telephones 486
[Philosophical Magazine, xvi. pp. 235 246, 1908.]
332. On Reflexion from Glass at the Polarizing Angle . . . 489
[Philosophical Magazine, xvi. pp. 444449, 1908.]
333. Note on Tidal Bores . .495
[Proceedings of the Royal Society, A, LXXXI. pp. 448, 449, 1908.]
334. Notes concerning Tidal Oscillations upon a Rotating Globe . 497
Plane Rectangular Sheet 498
Spherical Sheet of Liquid 506
[Proceedings of the Royal Society, A, LXXXII. pp. 448464, 1909.]
335. On the Instantaneous Propagation of Disturbance in a Dis
persive Medium, exemplified by Waves on Water Deep and
Shallow 514
[Philosophical Magazine, xvm. pp. 1 6, 1909.]
336. On the Resistance due to Obliquely Moving Waves and its
Dependence upon the Particular Form of the Forepart of a
Ship 519
[Philosophical Magazine, xvm. pp. 414416, 1909.]
337. On the Perception of the Direction of Sound . . . 522
[Proceedings of the Royal Society, A, LXXXIII. pp. 6164, 1909.]
338. The Theory of Crookes's Radiometer . . " ". v> . ' j . 526
[Nature, LXXXI. pp. 69, 70, 1909.]
339. To determine the Refractivity of Gases available only in
Minute Quantities 529
[Nature, LXXXI. p. 519, 1909.]
340. Note as to the Application of the Principle of Dynamical
Similarity 532
[Report of the Advisory Committee for Aeronautics, 1909 10, p. 38.]
341. The Principle of Dynamical Similarity in Reference to the
Results of Experiments on the Resistance of Square Plates
Normal to a Current of Air ...... 534
[Report of Advisory Committee, 191011.]
xii CONTENTS
ART.
342. Note on the Regularity of Structure of Actual Crystals . 536
[Philosophical Magazine, xix. pp. 9699, 1910.]
343. Colours of Sea and Sky 540
[Royal Institution Proceedings, Feb. 25, 1910; Nature, LXXXIII. p. 48, 1910.]
344 The Incidence of Light upon a Transparent Sphere of Dimen
sions comparable with the WaveLength .... 547
Experimental ......... 567
[Proceedings of the Royal Society, A, LXXXIV. pp. 25 46, 1910.]
345. On Colour Vision at the ends of the Spectrum . . .569
[Nature, LXXXIV. pp. 204, 205, 1910.]
346. Aerial Plane Waves of Finite Amplitude .... 573
Waves of Finite Amplitude without Dissipation . . 573
Waves of Permanent Regime ...... 583
Permanent Regime under the influence of Dissipative
Forces 587
Resistance to Motion through Air at High Velocities . 608
[Proceedings of the Royal Society, A, LXXXIV. pp. 247284, 1910.]
347. Note on the Finite Vibrations of a System about a Configura
tion of Equilibrium . . . . . . . .611
[Philosophical Magazine, xx, pp. 450456, 1910.]
348. The Problem of the Whispering Gallery .... 617
[Philosophical Magazine, xx. pp. 10011004, 1910.]
349. On the Sensibility of the Eye to Variations of WaveLength in
the Yellow Region of the Spectrum ..... 621
[Proceedings of the Royal Society, A, LXXXIV. pp. 464468, 1910.]
Figure 5 to face p. 377
ERRATA.
VOLUME I.
Page 144, line 6 from bottom. For D read D l .
442, line 9. After ^^ insert y.
443, line 9. For (7) read (8).
,, 443, line 10. For i) read .
446, line 10. For <f> read <f>'.
448, line 5. For v read c.
459, line 17. For 256, 257 read 456, 457.
,, 524. In the second term of equations (32) and following for &K~ l read A/*" 1 .
528, line 3 from bottom. For e int read e*^*^).
,, 538, line 11 from bottom. This passage is incorrect.
VOLUME II.
,, 197, line 19. For nature read value.
240, line 22. For dp[dx read dp/dy.
241, line 2. For dujdx read du/dy.
244, line 4. For kjn read n/k.
,, 414, line 5. For favourable read favourably.
551, first footnote. For 1866 read 1886.
VOLUME III.
,,""92, line 4. For Vol. I. read Vol. II.
129, equation (12). For e^^dx read e u ( i ~ x )du.
314, lineal. For (38) read (39).
522, equation (31). Insert as factor of last term 1/R.
548, second footnote. For 1863 read 1868.
569, second footnote. For alcohol read water.
,, 580, line 3. Prof. Orr remarks that a is a function of r.
VOLUME IV.
277, equation (12). For dz read dx.
299, first footnote. For 1887 read 1877.
,, 369, footnote. For 1890 read 1896.
400, equation (14). A formula equivalent to this was given by Loreuz in 1890.
418. In table opposite 6 for 354 read 324.
556, line 8 from bottom. For reflected read rotated.
273.
INTERFERENCE OF SOUND.
[Royal Institution Proceedings, xvn. pp. 1 7, 1902;
Nature, LXVI. pp. 4244, 1902.]
FOR the purposes of laboratory or lecture experiments it is convenient
to use a pitch so high that the sounds are nearly or altogether inaudible.
The wavelengths (1 to 3 cm.) are then tolerably small, and it becomes
possible to imitate many interesting optical phenomena. The ear as the
percipient is replaced by the high pressure sensitive flame, introduced for
this purpose by Tyndall, with the advantage that the effects are visible
to a large audience.
As a source of sound a " birdcall " is usually convenient. A stream of
air from a circular hole in a thin plate impinges centrically upon a similar
hole in a parallel plate held at a little distance. Birdcalls are very easily
made. The first plate, of 1 or 2 cm. in diameter, is cemented, or soldered,
to the end of a short supplytube. The second plate may conveniently be
made triangular, the turned down corners being soldered to the first plate.
For calls of medium pitch the holes may be made in tin plate. They may
be as small as ^ mm. in diameter, and the distance between them as little
as 1 mm. In any case the edges of the holes should be sharp and clean.
There is no difficulty in obtaining wavelengths (complete) as low as 1 cm.,
and with care wavelengths of '6 cm. may be reached, corresponding to about
50,000 vibrations per second. In experimenting upon minimum wavelengths,
the distance between the call and the flame should not exceed 50 cm., and
the flame should be adjusted to the verge of flaring*. As most birdcalls
are very dependent upon the precise pressure of the wind, a manometer in
immediate connection is practically a necessity. The pressure, originally
somewhat in excess, may be controlled by a screw pinchcock operating on
a rubber connecting tube.
* Theory of Sound, 2nd ed. 371.
INTERFERENCE OF SOUND
[273
In the experiments with conical horns or trumpets, it is important that
no sound should issue except through these channels. The horns end in
short lengths of brass tubing which fit tightly to a short length of tubing (A)
soldered airtight on the face of the front plate of the birdcall. So far there
is no difficulty ; but if the space between the plates be boxed in airtight,
the action of the call is interfered with. To meet this objection a tinplate
box is soldered airtight to A, and is stuffed with cottonwool kept in position
by a loosely fitting lid at C. In this way very little sound can escape except
through the tube A, and yet the call speaks much as usual. The manometer
is connected at the side tube D. The wind is best supplied from a gas
holder.
With the steadily maintained sound of the birdcall there is no difficulty
in measuring accurately the wavelengths by the method of nodes and loops.
Fig. 1.
A glass plate behind the flame, and mounted so as to be capable of sliding
backwards and forwards, serves as reflecting wall. At the plate, and at any
distance from it measured by an even number of quarter wavelengths there
are nodes, where the flame does not respond. At intermediate distances,
equal to odd multiples of the quarter wavelength, the effect upon the flame
is a maximum. For the present purpose it is best to use nodes, so adjusting
the sensitiveness of the flame that it only just recovers its height at the
minimum. The movement of the screen required to pass over ten intervals
from minimum to minimum may be measured, and gives at once the length
of five complete progressive waves. For the birdcall used in the experiments
of this lecture the wavelength is 2 cm. very nearly.
When the sound whose wavelength is required is not maintained, the
application of the method is, of course, more difficult. Nevertheless, results
1902] INTERFERENCE OF SOUND 3
of considerable accuracy may be arrived at. A steel bar, about 22 cm. long,
was so mounted as to be struck longitudinally every two or three seconds
by a small hammer. Although in every position the flame shows some
uneasiness at the stroke of the hammer, the distinction of loops and nodes
is perfectly evident, and the measurement of wavelength can be effected
with an accuracy of about 1 per cent. In the actual experiment the wave
length was nearly 3 cm.
The formation of stationary waves with nodes and loops by perpendicular
reflection illustrates interference to a certain extent, but for the full develop
ment of the phenomenon the interfering sounds should be travelling in the
same, or nearly the same, direction. The next example illustrates the theory
of Huyghens' zones. Between the birdcall and the flame is placed a glass
screen perforated with a circular hole. The size of the hole, the distances,
and the wavelength are so related to one another that the aperture just
includes the first and second zones. The operation of the sounds passing
these zones is antagonistic, and the flame shows no response until a part of
the aperture is blocked off. The part blocked off may be either the central
circle or the annular region defined as the second zone. In either case the
flame flares, affording complete proof of interference of the parts of the sound
transmitted by the aperture*.
From a practical point of view the passage of sound through apertures
in walls is not of importance, but similar considerations apply to its issue
from the mouths of horns, at least when the diameter of the mouth exceeds
the half wavelength. The various parts of the sound are approximately
in the same phase when they leave the aperture, but the effect upon an
observer depends upon the phases of the sounds, not as they leave, but as
they arrive. If one part has further to go than another, a phase discrepancy
sets in. To a point in the axis of the horn, supposed to be directed horizontally,
the distances to be travelled are the same, so that here the full effect is
produced, but in oblique directions it is otherwise. When the obliquity is
such that the nearest and furthest parts of the mouth differ in distance by
rather more than one complete wavelength, the sound may disappear altogether
through antagonism of equal and opposite effects. In practice the attainment
of a complete silence would be interfered with by reflections, and in many
cases by a composite character of sound, viz. by the simultaneous occurrence
of more than one wavelength.
In the fog signals established on our coasts the sound of powerful sirens
issues from conical horns of circular crosssection. The influence of obliquity
is usually very marked. When the sound is observed from a sufficient
distance at sea, a deviation of even 20 from the axial line entails a considerable
* [1901. See Vol. m. p. 31.]
12
4 INTERFERENCE OF SOUND [273
loss, to be further increased as the deviation rises to 40 or 60. The difficulty
thence arising is met, in the practice of the Trinity House, by the use of two
distinct sirens and horns, the axes of the latter being inclined to one another
at 120. In this way an arc of 180 or more can be efficiently guarded, but
a more equable distribution of the sound from a single horn remains a
desideratum.
Guided by the considerations already explained, I ventured to recommend
to the Trinity House the construction of horns of novel design, in which an
attempt should be made to spread the sound out horizontally over the sea,
and to prevent so much of it from being lost in an upward direction. The
solution of the problem is found in a departure from the usual circular section,
and the substitution of an elliptical or elongated section, of which the short
diameter, placed horizontally, does not exceed the half wavelength ; while
the long diameter, placed vertically, may amount to two wavelengths or
more. Obliquity in the horizontal plane does not now entail much difference
of phase, but when the horizontal plane is departed from such differences
enter rapidly.
Horns upon this principle were constructed under the supervision of
Mr Matthews, and were tried in the course of the recent experiments off
St Catherine's. The results were considered promising, but want of time
and the numerous obstacles which beset large scale operations prevented an
exhaustive examination.
On a laboratory scale there is no difficulty in illustrating the action of
the elliptical horns. They may be made of thin sheet brass. In one case
the total length is 20 cm., while the dimensions of the mouth are 5 cm. for
the long diameter and 1 cm. for the shorter diameter. The horn is fitted
at its narrow end to A (Fig. 1), and can rotate about the common horizontal
axis. When this axis is pointed directly at the flame, flaring ensues; and
the rotation of the horn has no visible effect. If now, while the long diameter
of the section remains vertical, the axis be slewed round in the horizontal
plane until the obliquity reaches 50 or 60, there is no important falling off
in the response of the flame. But if at obliquities exceeding 20 or 30
the horn is rotated through a right angle, so as to bring the long diameter
horizontal, the flame recovers as if the horn had ceased sounding. The fact
that there is really no falling off may be verified with the aid of a reflector,
by which the sound proceeding at first in the direction of the axis may be
sent towards the flame.
When the obliquity is 60 or 70 it is of great interest to observe how
moderate a departure from the vertical adjustment of the longer diameter
causes a cessation of effect. The influence of maladjustment is shown even
more strikingly in the case of a larger horn. According to theory and
1902] INTERFERENCE OF SOUND 5
observation a serious falling off commences when the tilt is such that the
difference of distances from the flame of the two extremities of the long
diameter reaches the half wavelength in this case 1 cm. It is thus
abundantly proved that the sound issuing from the properly adjusted
elliptical cone is confined to a comparatively narrow belt round the horizontal
plane and that in this plane it covers efficiently an arc of 150 or 160.
Another experiment, very easily executed with the apparatus already
described, illustrates what are known in Optics as Lloyd's bands. These
bands are formed by the interference of the direct vibration with its very
oblique reflection. If the birdcall is pointed toward the flame, flaring ensues.
It is only necessary to hold a long board horizontally under the direct line to
obtain a reflection. The effect depends upon the precise height at which the
board is held. In some positions the direct and reflected vibrations cooperate
at the flame and the flaring is more pronounced than when the board is away.
In other positions the waves are antagonistic and the flame recovers as if no
sound were reaching it at all. This experiment was made many years ago
by Tyndall who instituted it in order to explain the very puzzling phenomenon
of the "silent area." In listening to fogsignals from the sea it is not
unfrequently found that the signal is lost at a distance of a mile or two and
recovered at a greater distance in the same direction. During the recent
experiments the Committee of the Elder Brethren of the Trinity House had
several opportunities of making this observation. That the surface of the
sea must act in the manner supposed by Tyndall cannot be doubted, but
there are two difficulties in the way of accepting the simple explanation as
complete. According to it the interference should always be the same, which
is certainly not the case. Usually there is no silent area. Again, although
according to the analogy of Lloyd's bands there might be a dark or silent
place at a particular height above the water, say on the bridge of the Irene,
the effect should be limited to the neighbourhood of the particular height.
At a height above the water twice as great, or near the water level itself,
the sound should be heard again. In the latter case there were some
difficulties, arising from disturbing noises, in making a satisfactory trial ; but
as a matter of fact, neither by an observer up the mast nor by one near the
water level, was a sound lost on the bridge ever recovered.
The interference bands of Fresnel's experiment may be imitated by
a bifurcation of the sound issuing from A (Fig. 1). For this purpose a sort
of Ttube is fitted, the free ends being provided with small elliptical cones,
similar to that already described, whose axes are parallel and distant from
another by about 40 cm. The whole is constructed with regard to symmetry,
so that sounds of equal intensity and of the same phase issue from the two
cones whose long diameters are vertical. If the distances of the burner from
the mouths of the cones be precisely equal, the sounds arrive in the same
6 INTERFERENCE OF SOUND [273
phase and the flame flares vigorously. If, as by the hand held between,
one of the sounds is cut off, the flaring is reduced, showing that with this
adjustment the two sounds are more powerful than one. By an almost
imperceptible slewing round of the apparatus on its baseboard the adjustment
above spoken of is upset and the flame is induced to recover its tall equilibrium
condition. The sounds now reach the flame in opposition of phase and
practically neutralise one another. That this is so is proved in a moment.
If the hand be introduced between either orifice and the flame, flaring ensues,
the sound not intercepted being free to produce its proper effect.
The analogy with Fresnel's bands would be most complete if we kept the
sources of sound at rest and caused the burner to move transversely so as to
occupy in succession places of maximum and minimum effect. It is more
convenient with our apparatus and comes to the same thing, if we keep the
burner fixed and move the sources transversely, sliding the baseboard without
rotation. In this way we may verify the formula, connecting the width of
a band with the wavelength and the other geometrical data of the experiment.
The phase discrepancy necessary for interference may be introduced,
without disturbing the equality of distances, by inserting in the path of one
of the sounds a layer of gas having different acoustical properties from air.
In the lecture carbonic acid was employed. This gas is about half as heavy
again as air, so that the velocity of sound is less in the proportion of 1 : 1'25.
If / be the thickness of the layer, the retardation is '25 I ; and if this be equal
to the half wavelength, the interposition of the layer causes a transition
from complete agreement to complete opposition of phase. Two cells of tin
plate were employed, fitted with tubes above and below, and closed with films
of collodion. The films most convenient for this purpose are those formed
upon water by the evaporation of a few drops of a solution of celluloid in
pearoil. These cells were placed one in the path of each sound, and the
distances of the cones adjusted to maximum flaring. The insertion of carbonic
acid into one cell quieted the flame, which flared again when the second cell
was charged so as to restore symmetry. Similar effects were produced as
the gas was allowed to run out at the lower tubes, so as to be replaced by air
entering above*.
Many vibrating bodies give rise to sounds which are powerful in some
directions but fail in others a phenomenon that may be regarded as due
to interference. The case of tuning forks (unmounted) is well known. In
the lecture a small and thick wineglass was vibrated, after the manner of
a bell, with the aid of a violin bow. When any one of the four vibrating
segments was presented to the flame, flaring ensued ; but the response failed
* In a still atmosphere the hot gases arising from lighted candles may be substituted for the
layers of C0 2 .
1902] INTERFERENCE OF SOUND 7
when the glass was so held at the same distance that its axis pointed to the
flame. In this position the effects of adjacent segments neutralise one another
and the aggregate is zero. Another example, which, strangely enough, does
not appear to have been noticed, is afforded by the familiar open organ pipe.
The vibrations issuing from the two ends are in the same phase as they start,
so that if the two ends are equally distant from the percipient, the effects
conspire. If, however, the pipe be pointed towards the percipient, there is
a great falling off, inasmuch as the length of the pipe approximates to the
half wavelength of the sound. The experiment may be made in the lecture
room with the sensitive flame and one of the highest pipes of an organ, but
it succeeds better and is more striking when carried out in the open air with
a pipe of lower pitch, simply listened to with the unaided ear of the observer.
Within doors reflections complicate all experiments of this kind.
[1910. Some further discussion of interfering sources will be found in
Phil. Mag. Sept. 1903, Art. 290 below.]
274.
SOME GENERAL THEOREMS CONCERNING FORCED
VIBRATIONS AND RESONANCE.
[Philosophical Magazine, in. pp. 97117, 1902.]
THE general equation for the small vibrations ff a system whose con
figuration is defined by the generalized coordinates T/TJ, ^r 2 ,... may be
written*
ddT dF ,dV
where T, F, V, denoting respectively the kinetic energy, the dissipation
function, and the potential energy, have the forms
" ............. (2)
V=
in which the coefficients a, b, c are constants.
If we substitute in (1) the values of T, F, and V, and write I) for d/dt, we
obtain a system of equations which may be put into the form
where e rt denotes the quadratic operator
e rl = a rl ,D' + b rl D + c rt ......................... (4)
And it is to be remarked that since
a ri = agr, b ri = bgr, c n = c.,
it follows that
<? = ..................................... (5)
See Theory of Sound, Vol. i. 82, 84, 104.
1902] FORCED VIBRATIONS AND RESONANCE 9
If we multiply the first of equations (3) by ^j, the second by fa, &c.,
and then add, we obtain
(6)
In this the first term represents the rate at which energy is being stored
in the system ; 2F is the rate of dissipation ; and the two together account
for the work done upon the system in time dt by the external forces M^, ^ 2
fee,
In considering forced vibrations of simple type we take
V^E^W, 2 = # 2 e^, &c., ^ (7)
and assume that fa, fa, &c. are also proportional to &&. The coordinates
are then determined by the system of algebraic equations resulting from the
substitution in (4), (3) of ip for D. The most general motion possible under
the assumed forces would require the inclusion of free vibrations, but (unless
F = 0) these die out as time progresses.
By the theory of determinants the solution of equations (3) may be
expressed in the form
.(8)
where V denotes the determinant of the symbols e. If there be no dissipation,
V, or as we may write it with fuller expressiveness V (ip), is an even function
of ip vanishing when p corresponds to one of the natural frequencies of
vibration. In such a case the coordinates fa, &c. in general become infinite.
When there is dissipation, V (ip) does not vanish for any (real) value of p.
If we write
in which V 1} V 2 are even functions of ip, V 2 depends entirely upon the
dissipation, while if the dissipation be small, V l is approximately the same
as if there were none.
As it will be convenient to have a briefer notation than that of (8), we
will write
...(10)
in which A, a. are real and are subject to the relations
A n =A 9r , ,* = C 11 )
10 SOME GENERAL THEOREMS CONCERNING [274
In order to take account of the phases of the forces, we may suppose similarly
that in (7)
E^R^e*. , = &>; &c (12)
Work Done.
If we suppose that but one force, say acts upon the system, the values
of the coordinates are given by the first terms of the righthand members of
(10). The work done by the force in time dt depends upon that part of
d^i/dt which is in the same phase with it, corresponding to the part of \Jr,
which is in quadrature with the force. Thus, taking the real parts only
of the sjwnbolic quantities, so that
V, = R, COS (pt + 6,\ Vi = A " R i (P* + *i + ")> < 13 >
we have as the work done (on the average) in time t
 pAnRffcoa (pt f 0,) . sin (pt + 0, + a,,) dt,
or bpRfAnsiuan.t (14)
As was to be expected, this is independent of 6 l .
Another expression for the same quantity may be obtained by considering
how this work is dissipated. From (6) we see that
JV&dt = ZfFdt = b n JWdt + b^f^dt + ... + 26,,/M/fc + (15)
Taking again the real parts in (10), we have
(16)
u a).*, (17)
12 a ls ).t; (18)
so that by (15) the work dissipated in time t is
tfRft {Mil 1 + bnAj +... + 2b M A u Acos(a n  o 12 ) + ...]. ...(19)
Equating the equivalent qviantities in (14) and (19), we get
 p~ l A n sin o n = bnA n * + b^A^ 3 + . . . + 2b u A u A M cos (  o ia ) + . . (20)
This assumes a specially simple form when F is a function of the squares only
of dtyi/dt, &c., so that 6 12 , &c. vanish.
In (14) we have calculated the work done by a force ^j acting alone upon
the system. If other forces act, the expression for ^ will deviate from (13) ;
but in any case we may write
, = #,<, ^1 = ^6**, (21)
and the work done in unit of time by the real part of ^ on the real part
of \Jr, will be
^rsin^tf,), (22)
and depends upon the product of the moduli and the difference of phases.
If ^ consist of two or more parts of the form (21), the work done is to
be found by addition of the terms corresponding to the various parts.
1902] FORCED VIBRATIONS AND RESONANCE 11
One Degree of Freedom.
The theory of the vibrations of a system of one degree of freedom, resulting
from the application of a given force, is simple and well known, but it will
be convenient to make a few remarks and deductions.
The equation determining ^ in terms of M* is
so that in the notation of (10)
...(24)

c  ap* + ipb
As in (14), the work done by the force in unit time is
(25)
and it reaches a maximum (6 and p being given) when the tuning is such
that cap* = 0, that is when the natural vibrations are isoperiodic with the
forced vibrations. The maximum value itself is
(26)
Let us now suppose that two forces act upon the system, one of which
^ is given, while the second W is at disposal, and let us inquire how much
work can be withdrawn by M*'. It will probably conduce to clearness if we
think of an electric circuit possessing selfinduction and resistance, and closed
by a condenser, so as to constitute a vibrator. In this acts a given electro
motive force of given frequency. At another part of the circuit another
electromotive force can be introduced, and the question is what work can be
obtained at that point. Of course any work so obtained, as well as that
dissipated in the system, must be introduced by the operation of the given
force .
It will suffice for the moment to take "9 such that ty due to it is unity,
which will happen when ^ = A~ l e~ i<l . If ^ be Re ie , the complete value
of T/T is
y^l+ARe*^* ............................ (27)
The work done (in unit time) by V consists of two parts. That corresponding
to the second term in (27) is the same as if ^' had acted alone and, as in
(14), its value is
%pR*A sin a.
The work done by ^ upon the first part of </r given in (27) is, as in (22),
12 SOME GENERAL THEOREMS CONCERNING [274
The whole work done by * is found by adding these together; and the
work withdrawn from the system by ' is the negative of this, or
kpJ&A sin a  $pR sin (28)
In this expression the first term is negative, and the whole is to be made
a maximum by variation of R and 6. The maximum occurs when
sin0=l, 2 J R4sina=l; (29)
and the maximum value itself is
...(30)
A sin a *
This corresponds to Mod = A~ l ; and as the work abstractable is propor
tional to Mod*^, we have in general for the maximum
sn a
Now, as we see from the values of A and a in (24), or otherwise by (20),
p~ l A sin a = bA* ',
and thus the maximum work that can be abstracted is
It may at first occasion surprise that the work obtainable should be
independent of a and c, upon which the behaviour of the system as a
resonator depends. But the truth is that by suitable choice of ^ we have
in effect tuned the system, and so reduced it to the condition of evanescent
a and c in the electrical illustration to a merely resisting circuit. Had we
assumed the evanescence of a and c from the beginning, we could of course
have arrived more simply at the expression (31).
In the case of maximum withdrawal of energy the complete symbolical
value of >r in (27) becomes
(32)
the part of the complete value which is in the same phase as before being
halved.
It is not difficult to recognise that the result as to the maximum work
abstractable admits of further generalization. So far we have considered the
case of a single degree of freedom, e.g., a single electric circuit. Other degrees
of freedom, e.g., neighbouring electric circuits, do not affect the result, provided
that the forces in them all vanish and that the only dissipation is that already
considered. If in equations (3) all the quantities b except b n vanish, as well
as the forces ^, "V 3 , &c., the second, third, and following equations determine
1902] FORCED VIBRATIONS AND RESONANCE 13
real ratios between all the other coordinates and fa, and virtually reduce the
system to a single degree of freedom. The reaction of the other parts of the
system will influence the force M* 1 / required in order to abstract most work,
but not the maximum value itself.
Several Degrees of Freedom.
Hitherto we have supposed that the force by means of which work is
abstracted is of the same type as that which is supposed to be given and
by means of which work is introduced into the system ; but in the investi
gations which follow it will be our object to trace the effect of voluntary
operation upon one coordinate A/^, while the system is subject to given forces
^2, ^3, &c., operating upon the remaining coordinates. To explain what is
meant the more clearly, let us consider the simple case of two electric circuits
influencing one another by induction. Each circuit may be supposed to be
closed by a condenser, so as to constitute, when considered by itself, a simple
vibrator. If a given periodic electromotive force (^2) act in the second circuit,
the current in the first circuit would depend upon the various elements of
the compound system. Let it be proposed to inquire what work can be
withdrawn at the first circuit by electromotive forces C^) there applied.
For simplicity it will be supposed that the first circuit has no resistance
When ^ acts alone, the i/r 1 due to it is given by
+ 1 = A 12 e { "V 2 ............................... (33)
If we ascribe to ^ 2 the value A 12 ~ 1 e~ ia ^, the ^ due to it will be unity, and
this for the present purpose is the simplest supposition to make. When a
force ^ is introduced, the complete value of ^ will deviate from unity,
but "^2 is supposed to retain throughout the above prescribed value. If
*& l = R l e i6i > as in (12), the complete expression of ^ is
^ = 1 + A n R 1 e i ^+'J ......................... (34)
The work done by "^ upon this is composed of two parts. If the real
components of the expressions be retained, the first is
pfR l cos (pt + 0,) . sin ptdt =
and the second is, as in (14),
a quantity necessarily positive.
Thus altogether the work done by , in unit time is
(35)
14 SOME GENERAL THEOREMS CONCERNING [274
The work that may be abstracted from the circuit is the negative of this,
and it is to be made a maximum by variation of ^ and B t . We must take
sin0, = l, 2/Mii8ina n = l; .................. (36)
and the maximum work abstractable will be
or or  ......... (37)
8A n sin a,,
The symbolic expression for ^ becomes at the same time from (22)
(38)
so that the part of ^ in the same phase as when ^ = is half as great as
before.
So far as the results embodied in (37), (38), and (39) are concerned,
it is a matter of indifference whether the prescribed ^ = 1 when , =
is due to ^ 2 only, or to ^ 2 acting in conjunction with forces V 3 , &c.,
corresponding to further degrees of freedom. But for the present we will
suppose that there are only two degrees of freedom.
The maximum work that can be abstracted by ,, when a is given,
may be expressed as
(39)
where ^ is due to ^ 2 acting alone. By (33)
Mod'
and by (20) with 6 n = 0, b ia = 0,
jr
so that the maximum work obtainable is simply
(40)
That it is independent not only of a,,, c n , a,,, c^, but also of the coefficients
of mutual influence a K , c ]3 , is very remarkable. To revert to the electrical
example, the work abstractable in the first circuit (devoid of resistance) when
a given electromotive force acts in the second, is independent of the value
of the coefficient of mutual induction. If indeed this coefficient be very
small, the supposition of zero resistance becomes more and more unpractical
on account of the large currents which must then be supposed to flow. But
the theoretical result remains true, when 6 n is diminished without limit.
In view of its independence of so many circumstances that might at first
be supposed material, it may now not be surprising to note that (40)
coincides with (31), that is that the work obtainable in the second circuit
is the same as might have been obtained in the first where the given force
itself acts.
1902] FORCED VIBRATIONS AND RESONANCE 15
The existence of further degrees of freedom than those corresponding to
the given force "^ and the disposable force , makes no difference to (39).
And so long as ^2, ^ are the only forces in operation, we have still
122
Max. work =  t ^ ^ ................... (41)
8A U sin er u
If further all the coefficients b vanish, except b x> (40) remains unaffected.
If, however, we suppose that 633, 644, &c. are finite, while 6 U , 6 12 , b w , b&, &c.
still vanish, (20) gives
 p~ l A u sin o n = b^A^ + b u A n * + ..., ............... (42)
and the expression for the maximum work becomes
Since 633, &c. are positive, the value of (43) is less than when 633, &c. vanish.
The expression (43) is necessarily more complicated than (40); but
a simple result may again be stated if we suppose that given forces act
successively of the second, third, and following types, provided they be of
such magnitudes that they would severally (the noncorresponding resistances
vanishing) allow the same work to be abstracted by "*P lt that is provided
Mod 2 ^P, Mod 2 "^ Mod 2 W
= r (44)
On this supposition the sum of the energies abstractable in the various cases
has the value
^^, (45)
of the same form as before.
In the electrical application we have to consider any number of mutually
influencing circuits, of which the first is devoid of resistance. The electro
motive forces acting successively in the other circuits are to be inversely
as the square roots of the resistances of those circuits, i.e. such as would
do the same amount of work on each circuit supposed to be isolated and
reduced (e.g. by suitable adjustment of the associated condenser) to a mere
resistance. The sum of all the works abstractable in the first circuit is then
the same as if there were no other circuits than the first and second ; or,
again, as if the second circuit were isolated and it were allowed to draw
work from it.
Action of Resonators.
We now abandon the idea of drawing work from the system by means
of ^j, and on the contrary impose the condition that , shall do no work,
positive or negative. The effect of ^ is then equivalent to a change in the
16 SOME GENERAL THEOREMS CONCERNING [274
inertia a,,, or spring c n , associated with this coordinate and the operation
may be regarded as a tuning, or mistiming, of the system. If, as before,
^,, due to the given force 3 , be unity, and , = R^e* 9 *, the complete value
of ^ is that given in (34), and (35) represents the work done altogether
by , in unit time. Equating this to zero, we get as the relation between
jR, and 0,,
.4 n .RiSma u = sin0j, ........................... (46)
and the part of ^i due to , is
sin 6^ "+')
sin
The modulus of this is a maximum when sin 0, = 1, and the value of the
maximum is coseca n . In this case (47) becomes
1 + tcota u , .............................. (48)
and the complete value of ^ is
^icotct,,, ................................. (49)
in quadrature with the former value, viz., 1.
We may regard the state of things now defined as being in a sense the
greatest possible disturbance of the original state of things. If the system
be quite out of resonance, forces and displacements are nearly in the same
phase, and a u is small. The altered i/r, is then a large multiple of the
original value.
The work done by ^ on the complete value of fa is zero by supposition ;
but the work done upon the part of fa due to itself is by (14) in unit time
This corresponds to the original fa = 1, or ^ 2 = 4 M  1 r ^. If the prescribed
value of ^o be now left open, we have as the work in question
. ; ........................ (50 )
n n
and this by (20) is the same as
'
6j,, 6,,, &c. being supposed to be zero. This expression differs from (43) only
as regards the numerical factor. If b K , &c. also vanish, (51) becomes
26,7" 81m P'
If in (51) we introduce the suppositions of (44), we get as in (45) for the
sum of all the values
.................................. ()
1902] FORCED VIBRATIONS AND RESONANCE 17
In an interesting paper entitled " An Electromagnetic Illustration of the
Theory of Selective Absorption of Light in a Gas *," Prof. Lamb has developed
a general law for the maximum energy emitted by a resonator situated in
a uniform medium when submitted to incident plane waves. " The rate at
which energy is carried outwards by the scattered waves is, in terms of the
energyflux in the primary waves,
<>
where X is the wavelength, and n is the order of the spherical harmonic
component of the incident waves which is effective." Prof. Lamb remarks
that the law expressed by (53) " is of a very general character, and is
independent of the special nature of the conditions to be satisfied at the
surface of the sphere. It presents itself in the elastic solid theory, and again
in the much simpler acoustical problem where there is synchronism between
plane waves of sound and a vibrating sphere on which they impinge."
The generality claimed by Lamb for (53) seemed to me to indicate
a still more general theorem in the background ; and it was upon this
suggestion that the investigations of the preceding pages were developed.
An initial difficulty, however, stood in the way. The occurrence of n, a
quantity special to the spherical problem, seemed to constitute a limitation ;
and the further question suggested itself as to why the efficiency of the
resonator should rise with increasing n. For example, why in the acoustical
problem should a resonator formed by a rigid sphere, moored to a fixed point
by elastic attachments (n = 1), be three times as effective as the simple
resonator (n = 0), for which the theory is given by my book on Sound,
319?.
The answer may be found in a slightly different presentation of the matter.
In the above example the rigid sphere is supposed to be symmetrically moored
to a fixed point, and the vibration actually assumed is in a direction parallel
to that of propagation of the incident waves. Three degrees of freedom are
really involved here, while the more typical case will be that in which the
motion is limited to one direction. The efficiency of the resonator will then
be proportional to the square of the cosine (//,) of the angle between the
direction of vibration and that of the incident waves ; and the mean efficiency
will bear to the maximum efficiency (^ = 1) a ratio equal to that of
that is of . Thus, if the vibration in the case of n = 1 be limited to one
direction, the mean efficiency of the resonator is the same as when n = ;
and a similar conclusion will hold good in all cases. In this way the factor
* Cambr. Trans. Vol. xvni. p. 348 (1899).
R. v. 2
18 SOME GENERAL THEOREMS CONCERNING [274
is eliminated, and the statement assumes a form more nearly capable
of generalization to all vibrating systems.
Now that a general theorem (52) has been demonstrated, it will be of
interest to trace its application to some case of a uniform medium, for which
purpose we may take the simple acoustical resonator. But this deduction
is not quite a simple matter, partly on account of the extension to infinity,
and also, I think, for want of a more general theory of waves in a uniform
medium than any hitherto formulated. If the object be merely to obtain
a result, it is far more easily attained by a special investigation from the
formulae of the Theory of Sound, on the lines indicated by Prof. Lamb. It
may perhaps be well to sketch the outline of such an investigation.
The time factor e* at being suppressed, the velocitypotential <f> of the
primary waves is ( 334) e acx , or e****, and the harmonic component of the
nth order has the expression
....... (53bis)
while ( 329) the corresponding expression for the divergent secondary
waves is
cos ki i sin kr
...(54)
Accordingly
(a, + 1  ( iy oo.) P.
/ d \ cos kr
(l)foi B P n _^)_ ............................... (55)
Now the only condition imposed upon the appliances introduced at
the surface of the sphere is that they shall do no work. The velocity is
dfyjdr + d\fr/dr, and the pressure, proportional to d<f>/dt + d^/dt, is in
quadrature with (55). All therefore that is required is that (55) and its
derivative with respect to r be in the same phase, or that the ratio of these
symbolic expressions be real. Since P n is a wholly odd or wholly even
function, this requirement is satisfied if
2n + 1 ( l) n ika n ,
be real.
( l) B A;a n
If A n , which may be complex, be written Ae*", we get
kA = (!) (2n+l)sin a. (56)
Thus A is a maximum when
sina = (l) n , (57)
and the maximum value is
A=(2n + l)/k (58)
1902] FORCED VIBRATIONS AND RESONANCE 19
By (57), (58) a n =  ( l)i(2n + I)/*, ........................ (59)
so that in (55) 2n+ I  ( l) n ika n = 0, ........................ (60)
but (f> n + TJr n does not itself vanish.
If the incident plane waves are regarded as due to a source at a great
distance R, we have in correspondence with (53)
/1\
(61)
with which we may compare
^^^PM ............................ (62)
r+i
The work emitted from the primary source being represented by R* I dp,
that emitted, or rather diverted, by the resonator will be
Now
Also Mod 2 a n = , ........................ (63)
so that the ratio of works is
(64)
This agrees with the result given in Theory of Sound, 319 for a symmetrical
resonator (n = 0).
In order to express (64) in terms of the energyflux (per unit area) of the
primary waves at the place of the resonator, we have only to multiply (64)
by the area (4?rjR 2 ) of the sphere of radius R. If we restore 2?r/X for k, we
get as the equivalent of (64)
(2n + I)*, 2 / 71 " ............................... (65*)
If we limit the resonator to one definite harmonic vibration of order n
and suppose that the primary waves may be incident indifferently in all
directions, the mean of the values of (65) is \ 2 /7r simply, as follows from known
properties of the spherical functions.
* It will be observed that (65) is the double of the value (53) above quoted.
Dec. 17. I have since learned that Prof. Lamb's calculations for the acoustical problem have
already been published. See Math. Soc. Proc. Vol. xxxn. p. 11, 1900, where equation (44) is
identical with (65) above. Beference may also be made to Lamb, Math. Soc. Proc. Vol. zzzn.
p. 120, 1900.
22
20 SOME GENERAL THEOREMS CONCERNING [274
Before we can apply the general theorem (52) to an independent
investigation of these results, it is necessary to consider the connexion
between the formulae for plane and spherical waves ; and for this purpose
it is desirable to use a method which, if not itself quite general, is of a
character susceptible of generalization. If </> denote the velocitypotential
due to a " force " 4>dF acting at the element of volume dV and proportional
to the periodic introduction and abstraction of fluid at that place, we may
write
, ........................... (66)
where k = 2?r/X and B is some multiplier, which may be complex. The time
factor e ikat is suppressed. In order to obtain plane waves we may suppose
that 4> acts uniformly over the whole slice between x and x + dx. The effect
may be calculated as in a wellknown optical investigation. If p z = r 2 a?,
the element of volume is 2np dp dx, or 2?rr dr dx ; and for the plane waves
(67)
Here <I> acts at x = 0, and
jj ar ' ' ' , Y'Q /* w.u' 'ar */. .............. .^UO^
Since <j> must be in the same phase as 4>, it follows that B is real.
We have now to consider the work done in generating the plane waves
per unit of time and per double unit area of wavefront. For this we have
rt
(69)
\*
Jo
or since by (67) Mod <f> = 2?r f ^ Mod 4>. (70)
we get for the work propagated in one direction per unit of area of wave
front
Reverting now to (66), we see that for divergent waves
or 47JT 2 Mod 8 <j> = bnB* Mod 2 (< d V).
Accordingly by (71), since at a sufficient distance the distinction between
plane and divergent waves disappears, the work emitted in unit time by a
pointsource QdV is
(72)
1902] FORCED VIBRATIONS AND RESONANCE 21
It may be observed that in order to preserve a better correspondence
between " force " and " coordinate " a somewhat different interpretation is
here put upon 4> from that adopted in Theory of Sound, 277. If we
compare our present (71) with (10), 245, we find that
1
so that according to the present interpretation of 4>, (66) gives
whereas in the notation of 277
1 fihr
, (74)
We are now to some extent prepared for the application of (52), but the
difficulty remains that (52) deals in the first instance with a finite system
subject to dissipative forces; whereas the uniform medium is infinite, and
need not be supposed subject to any forces truly dissipative. There is,
however, no objection to the introduction of a small dissipative force of the
character supposed in the general theorem, that is, proportional everywhere
to d<f>/dt. Under this influence plane waves are attenuated as they advance ;
the law of attenuation being represented by the introduction into (67) of the
factor e~ ax , where a is a small quantity, real and positive.
The connexion between a and b may be investigated by considering the
action of the dissipative force  b<f> operative over a slice Bx at x = in
causing the attenuation. By (67) the effect at x of this force is represented
by
, . 2irBe ik * . . ,
C<f> =  rr  <p OX,
so that B<f)/<f> =  ^TrabBBx.
By supposition this must be the same as aBx', and accordingly
(75)
If we use this result to eliminate B from (72), we get as the work emitted
from a pointsource
~ Mod 2 (<DdF) ........................... (76)
The formula (52) expresses the sum of all the works emitted by the
resonator when submitted successively to all the various forces ^P, subject
themselves to conditions (44). These conditions are satisfied in the present
case if we identify each with the force QdV acting over the various equal
elements dV into which infinite space may be divided, the value of 4> being
22 SOME GENERAL THEOREMS CONCERNING [274
everywhere the same. Each pointsource is regarded as the origin of piano
waves which fall upon the resonator. The efficiency of the sources which lie
in a given direction still depends upon the distance, the waves as they reach
the resonator being attenuated by the resistance and also in the usual manner
according to the law of inverse squares.
Let us compare the efficiency of the element 4>rfF at distance r with the
efficiency of an equal element at distance unity, the value of a being so small
that no perceptible attenuation due to it occurs in distance unity. The
element of volume
dV=d<r.d($r*), (77)
in which for the present d<r [the element of angular area] is kept unchanged.
The efficiency of the element at distance r varies as
and
Hence for the sum of all the elements lying within da we have
a ,~. x efficiency of (3>dV) at distance unity.
This has now to be again integrated with respect to da: The result may
be expressed by the statement that the sum of all the works emitted by the
resonator is
2 VTT x mean work emitted by resonator corresponding to the various
positions of the pointsource on the sphere r = l.
By the theorem (52) this sum of all the works is also expressed by
or in accordance with (76) is equal to
27T
jjry x work emitted by (3>dV) itself.
We see therefore that the mean work emitted by the resonator for
positions of the pointsources distributed uniformly over the sphere r = 1
is equal to the work emitted by each of the pointsources themselves divided
by Id*. If the pointsources are supposed to lie at a distance r in place of
unity, the divisor becomes fcr* in place of If.
Although the above deduction may stand in need of some supplementing
before it could be regarded as rigorous at all points, it suffices at any rate to
show that the general theorem (52) really does include the more special cases
1902] FORCED VIBRATIONS AND RESONANCE 23
which suggested it. In some applications, e.g. to an elastic solid, we should
have at first to suppose the forces introduced at any element of volume d V to
act in various directions, but no great complication thence arises, and the
general result finally takes the same form.
Energy stored in Resonators.
In preceding investigations we have been concerned with energy emitted
from a resonator. We now turn to the consideration of some general theorems
relating to the energy stored, as it were, in the resonator when the applied
forces have frequencies in the neighbourhood of the natural frequency of the
resonator. And we will treat first the simple case of one degree of freedom.
As in (2) we have
T = kafa F=^\ V=%c^\ ............... (78)
giving as the equation of vibration
a$ + hjr + c^ = V = EeW ......................... (79)
The time factor being suppressed, the solution of (79) is
<80)
n, equal to V(c/a), being the value of p corresponding to maximum resonance.
If, as we suppose, b is very small, the important values of Mod 2 ^ are con
centrated in the neighbourhood of p = n, and we may substitute n for p in
the term p 2 b 2 . Also n 2 p z may be identified with Zn (n p). Accordingly
(81) becomes
1 Mod 2
 ................... (82)
We now suppose that Mod 2 V is constant, while p varies over the small
range in the neighbourhood of n for which alone Mod 2 ^ is sensible, and
inquire as to the sum of the values of Mod 2 i/r. Since
_, 1 + a 2 ^ a '
we find [Mod 2 
.(83)
2bc
or again c Mod 2 f dp = ^ Mod 2 (84)
On the left \ c Mod 2 i/r represents the potential energy of the system at the
24 SOME GENERAL THEOREMS CONCERNING [274
phase of maximum displacement, which is the same as the nearly constant
total energy, so that (84) gives the integral of this total energy as p passes
through the value which calls out the maximum and (by supposition) very
great resonance.
The most remarkable feature of (84) is perhaps that the integral is
independent of a and c. Large values of these quantities will increase the
energy of the system at the point where p = n ; but on the other hand this
maximum falls off more rapidly as p departs from the special value.
We pass next to the more difficult considerations which arise when the
force ^2 is of one kind, while the coordinate fa on which the resonance
principally depends is of another. In the first instance we shall suppose
that there are no other than these two degrees of freedom.
If in equation (3) we assume MV fa, fa, &c. to vanish, we get
where e u , e n , e^ have the values given in (4) with ip substituted for D.
We suppose further that 6,. 2 = 0, 6 U = 0, so that the dissipation depends
entirely on 622. With these simplifications the numerator of (85) becomes
e 12 = C K  p 12 , .............................. (86)
and for the denominator (taken negatively)
e u e. a  e^ = (c n  p*a n ) (c^  p^a^)
(cvp'a^ + ipb^Cup'au) ............................ (87)
If n be one of the values of p corresponding to maximum resonance,
the real part of (87) vanishes when p = n ; so that
(Cii  lAiu) (c  w 2 a M )  (c 12  n 2 a 12 ) 2 = 0, ............... (88)
or written as a quadratic in r? 2 ,
CuCa  c 12 2  n (a u cv + a^ u + 2a 12 c 12 ) + n 4 (a,^  a, 2 2 ) = ....... (89)
By subtraction of (89), (87) may be written
2a 12 c 12 )
p 2 a 11 ) ................... (90)
If &J2 were zero, fa would become infinite for p = n. If we assume that
b a , while not actually zero, is still relatively very small, the values of p in the
neighbourhood of n retain a preponderating importance ; and we may equate
p to n with exception of the factor (p  n). Thus (86), (90) become
e,2 = c, 2 n 2 12 , .............................. (91)
1902] FORCED VIBRATIONS AND RESONANCE 25
 e 12 2 =  2n (jo  w) {0^0* + a^c n  2a 12 c, 2  2n 2 (a,,^  a 12 2 )}
= ~ n ~ {c llC22  c 12 2  rc< (ana.  a 12 2 )} + tn k, (c n  n 2 a u ), . . .(92)
use being made of (89).
From (91), with use of (88),
Mod" = (c u !*)(<**%); .................. (93)
and from (92)
Mod 2 (e u e^  e 12 2 ) = * (P ~ n * {c,,^  c 12 2  n KO.  a 12 2 )} 2
,,'a u ) 2 ...................... (94)
If we now. as for (84), carry out the integration with respect to p,
Mod ^ 2 being constant, we find from (85)
Mod 2 ^ 2 2^ {c llC22  c 12 2 
So far we have assumed merely that the compound system is in high
resonance when p = n; but more than this is required in order to arrive
at a simple result. We must further assume that the coefficients of
interconnexion a 12 , c 12 are small (6 12 has been already made zero), so that the
resonating coordinate may vibrate with a considerable degree of independence.
We are also to suppose that n corresponds to these comparatively independent
vibrations, so that c u  n?a u = approximately, while c* n^a^ is relatively
large. These simplifications reduce the bracket in the denominator of (95)
to
CnC22 n'OuOu, or to c n (Ca n^On);
whence we obtain finally
'p_ 7T
'
In this expression, which is of the same form as (84), the numerator
on the left may be considered to represent the integrated energy of the
resonator. It must not be overlooked that the suppositions involved are
to some extent antagonistic. For example, the coefficient of b w in (90) is
treated as constant when p varies, although (c n n s a u ) is small. The theorem
should be regarded as one applicable in the limit when b*> is exceedingly
small.
If there be more than two degrees of freedom, the result is unaffected,
provided that the forces ^ 3 , ^ 4 , &c. of the new types vanish and that the
only dissipation is that represented by 6 ffl . By the 3rd, 4th, &c. of (3) the
26 FORCED VIBRATIONS AND RESONANCE [274
new coordinates may be eliminated. In this process 6 M is undisturbed, and
everything remains as if there were only two coordinates as above.
The idea of the integration with respect to p is borrowed from a paper
by Prof. Planck (Ann. d. Phys. I. p. 99, 1900), in which is considered the
behaviour of an infinitely small electromagnetic resonator under incident
plane waves. The proof of the general theorem covering Prof. Planck's
case would require a process similar to that by which (51) was established.
Subject to the condition
Mod^ 2 = Mod* 3 = = M^15 r (97)
'>: by,
we might expect to find, as in (52),
275.
ON THE LAW OF THE PRESSURE OF GASES BETWEEN
75 AND 150 MILLIMETRES OF MERCURY.
[Phil. Trans, cxcvin. A, pp. 417430, 1902.]
IN a recently published paper* I have examined, with the aid of a new
manometer, the behaviour of gases at very low pressures, rising to 1'5 millims.
of mercury, with the result that Boyle's law was verified to a high degree of
precision. There is, however, a great gap between the highest pressure
there dealt with and that of the atmosphere a gap which it appeared
desirable in some way to bridge over. The sloping manometer, described in.
the paper referred to, does not lend itself well to the use of much greater
pressures, at least if we desire to secure the higher proportional accuracy
that should accompany the rise of pressure. The present communication
gives the results of observations, by another method, of the law of pressure in
gases between 75 millims. and 150 millims. of mercury. It will be seen that
for air and hydrogen Boyle's law is verified to the utmost. In the case of
oxygen, the agreement is rather less satisfactory, and the accordance of
separate observations is less close. But even here the departure from
Boyle's law amounts only to one part in 4000, and may perhaps be referred
to some reaction between the gas and the mercury. In the case of argon too
the deviation, though very small, seems to lie beyond the limits of experi
mental errors. Whether it is due to a real minute departure from Boyle's
law, or to some complication arising out of the conditions of experiment, must
remain an open question.
In the case of pressures not greatly below atmosphere, the determination
with the usual column of mercury read by a cathetometer (after Regnault) is
sufficiently accurate. But when the pressure falls to say onetenth of an
atmosphere, the difficulties of this method begin to increase. The guiding
idea in the present investigation has been the avoidance of such difficulties
* Phil. Trans. Vol. 196, A, p. 205, Feb. 1901. [Vol. iv. p. 511.]
28 THE LAW OF THE PRESSURE OF GASES [275
by the use of manometric gauges combined in a special manner. The object
is to test whether when the volume of a gas is halved its pressure is doubled,
and its attainment requires two gauges indicating pressures which are in the
ratio of 2 : 1. To this end we may employ a pair of independent gauges as
nearly as possible similar to one another, the similarity being tested by
combination in parallel, to borrow an electrical term. When connected
below with one reservoir of air and above with another reservoir, or with
a vacuum, the two gauges should reach their settings simultaneously, or at
least so nearly that a suitable correction may be readily applied. For brevity
we may for the present assume precise similarity. If now the two gauges be
combined in series, so that the lowpressure chamber of the first communicates
with the highpressure chamber of the second, the combination constitutes a
gauge suitable for measuring a doubled pressure.
The Manometers.
The construction of the gauges is modelled upon that used extensively in
my researches upon the density of gases, so far as the principle is concerned,
although of course the details are very different. In fig. 1 A and B represent
[about ^] size the lower and upper chambers. As regards the glasswork,
these communicate by a short neck at D as well as by the curved tube ACB.
Through the neck is carried the glass measuringrod FDE, terminating
downwards at both ends in carefully prepared points E, F. The rod is held,
at D only, with cement, which also completely blocks up the passage, so that
when mercury stands in the curved tube the upper and lower chambers are
isolated from one another. The use of the gauge is fairly obvious. Suppose
for example that it is desired to adjust the pressure of gas in a vessel com
municating with G to the standard of the gauge. Mercury standing in C,
H is connected to the pump until a vacuum is established in the upper
chamber. From a hose and reservoir attached below, mercury is supplied
through / until the point F and its image in the mercury surface nearly
coincide. If E coincides with its image, the pressure is that defined ; other
wise adjustment must be made until the points E, F both coincide with
their images, or as we shall say until both mercury surfaces are set. The
pressure then corresponds to the column of mercury whose height is the
length of the measuringrod between the points E, F. The vertically of
EF is tested with a plumbline.
The measuringrods appear somewhat slender; but it is to be remem
bered that the instruments are used under conditions that are almost constant.
So far as the comparison of one gas with another is concerned, the qualifica
tion "almost" may indeed be omitted. The coincidence of the points
and their images is observed with the aid of four magnifiers of 20 millims.
focus, fixed in the necessary positions.
1902]
BETWEEN 75 AND 150 MILLIMETRES OF MERCURY
2!)
General Arrangement of Apparatus.
In fig. 2 is represented the connection of the manometers with one
another and with the gas reservoirs. The lefthand manometer can be
connected above through F with the pump or with the gas supply. The lower
chamber A communicates with the upper chamber D of the righthand
manometer and with an intermediate reservoir E, to which, as to the mano
meters, mercury can be supplied from below. The lower chamber C of the
Fig. 1.
Fig. 2.
righthand manometer is connected with the principal gas reservoir. This
consists of two bulbs, each of about 129 cub. centims. capacity, connected
together by a neck of very narrow bore. Three marks are provided, one G
above the upper bulb, a second H on the neck, and a third / below the lower
bulb, so adjusted that the included volumes are nearly equal. The use of the
sidetube JK will be explained presently.
When, as shown, the mercury stands at the lower mark, the double
volume is in action and the pressure is such as will balance the mercury in
30 THE LAW OF THE PRESSURE OF GASES [275
one (the righthand) manometer. A vacuum is established in the upper
chamber D from which a way is open through AB to the pump. When
the mercury is raised to the middle mark H, the volume is halved, and the
pressure to be dealt with is doubled. Gas sufficient to exert the single
pressure (75 millims.) must be supplied to the intermediate chamber,
which is now isolated from the pump by the mercury standing up in
AB. Both manometers can now be set, and the doubling of the pressure
verified.
The communication through F with the pump is free from obstruction,
but on a sidetube a threeway tap is provided communicating on the one
hand with the gas supply and on the other with a vertical tube, more than a
barometerheight long and terminating below under mercury, by means of
which a washout of the generating vessels can be effected when it is not
desired to evacuate them. The five tubes leading downwards from A, E,
C, I, K are all over a barometerheight in length and are terminated
by suitable hoses and reservoirs for the supply of mercury. When settings
are actually in progress, the mercury in the hoses is isolated from the
reservoirs by pinchcocks and the adjustment of the supply is effected by
squeezing the hoses. As explained in my former paper, the final adjust
ment must be made by squeezers which operate upon parts of the hoses which
lie flat upon the large wooden mercury tray underlying the whole. The
adjustment being somewhat complicated, a convenient arrangement is almost
a necessity.
The Side Apparatus.
By the aid of these manometers the determination of pressure is far more
accurate than with the ordinary mercury column and cathetometer, but since
the pressures are defined beforehand, the adjustment is thrown upon the
volume. The variable volume is introduced in the sidetube JK. This was
graduated to \ cub. centim., in the first instance by mercury from a burette.
Subsequently the narrow parts above and below the bulb (which as will
presently be seen are alone of importance) were calibrated with a weighed
column of mercury of volume equal to cub. centim. and occupying about
80 millims. of the length of the tube. The whole capacity of the tube
between the lowest and highest marks was 20 cub. centims. The object of
this addition is to meet a difficulty which inevitably presents itself in
apparatus of this sort. The volume occupied by the gas cannot be limited
to the capacities susceptible of being accurately gauged. Between the upper
mark and the mercury surface in C when set, a volume is necessarily
included which cannot be gauged with the same accuracy as the volumes
between G and H and between H and 7. The simplest view of the side
apparatus is that it is designed to measure this volume. In the notation
1902] BETWEEN 75 AND 150 MILLIMETRES OF MERCURY 31
subsequently used F 3 is the volume included when the mercury stands at C,
at G, and at the top mark /. Let us suppose that with a certain quantity
of gas imprisoned it is necessary in order to set the manometer CD, the
upper chamber being vacuous, to add to F 3 a further volume F B , amounting
to the greater part of the capacity of the sidetube, so that the whole
volume is V 3 + V 5 . When the second manometer is brought into use, the
volume must be halved, for which purpose the mercury is raised through the
bulb until it stands somewhere in the upper tube. The whole volume is now
V + V. And since
we see that V 3 = F 5  2 F 4 ,
which may be regarded as determining F 3 , F 4 and F 5 being known. A some
what close accommodation is required between V s , about 19 cub. centims.
in my apparatus, and the whole contents of the sidetube.
General Sketch of Theory.
As the complete calculation is rather complicated on account of the
numerous temperature corrections, it may be convenient to give a sketch of
the theory upon the assumption that the temperature is constant, not only
throughout the whole apparatus at one time, but also at the four different
times concerned. We shall see that it is not necessary to assume Boyle's law,
even for the subsidiary operations in the sidetube.
F! = volume of two large bulbs together between / and G (about 258 cub.
centims.),
F 2 = volume of upper bulb between G and H,
F 3 = volume between C, G and highest mark J on sidetube,
F 4 = measured volume on upper part of J from highest mark downwards,
F 8 = measured volume, including bulb, of side apparatus from highest
mark downwards,
Pj = small pressure (height of mercury in righthand manometer),
P 2 = large pressure (sum of heights of mercury in two manometers).
In the first pair of operations when the large bulbs are in use, the
pressure P 1 corresponds to the volume ( Fj f F 8 4 F 5 ) and the pressure P 2
corresponds to ( F 2 + F 3 + F 4 ), the quantity of gas being the same. Hence the
equation
P 1 (F 1 + F 3 + F 5 ) = J BP 2 (F 2 +r 3 + F 4 ), .................. (1)
B being a numerical quantity which would be unity according to Boyle's
law. In the second pair of operations with a different quantity of gas but with
the same pressures, the mercury stands at G throughout, and we have
P 1 (F 3 + F/) = 5P 2 (F 3 +F/); ........................ (2)
whence by subtraction
P 1 (F 1 + F 5 F 5 ') = J5P 2 (F 2 + F 4 F/). .................. (3)
32 THE LAW OF THE PRESSURE OF GASES [275
From this equation V 3 has been eliminated and B is expressed by means
of PI/P,, and the actually gauged volumes
v lt v,, r.v.', v 4 v 4 '.
It is important to remark that only the differences
(V.JY), (V<Vt)
are involved. The first is measured on the lower part of the sideapparatus
and the second on the upper part, while the capacity of the intervening bulb
does not appear.
If the principal volumes V l and V t are nearly in the right proportion,
there is nothing to prevent both V & V s ' and V 4 V 4 from being very small.
When the temperature changes are taken into account, V a , V 4 , V s are not
fully eliminated, but they appear with coefficients which are very small if the
temperature conditions are good.
Thermometers.
As so often happens, much of the practical difficulty of the experiment
turned upon temperature. The principal bulbs were drowned in a water
bath which could be effectively stirred, and so far there was no particular
impediment to accuracy. But the other volumes could not so well be
drowned, and it needed considerable precaution to ensure that the associated
thermometers would give the temperatures concerned with sufficient accuracy.
As regards the sidetube, a thermometer associated with its bulb and
wrapped well round with cottonwool was adequate. A third thermometer
was devoted to the space occupied by the manometers and the tube leading
from C to J. It was here that the difficulty was greatest on account of the
proximity of the observer. Three large panes of glass with enclosed air
spaces were introduced as screens, and although the temperature necessarily
rose during the observations, it is believed that the rise was adequately
represented in the thermometer readings. A single small gas flame, not
allowed to shine directly upon the apparatus, supplied the necessary illumi
nation, being suitably reflected from four small pieces of lookingglass fixed
to a wall behind the glass points of the manometers.
As regards the success of the arrangement for its purpose, it is to be
remembered that by far the larger part of any error that might arise is
eliminated in the final result, since it is only a question of a comparison of
observations with and without the large bulbs. Any systematic error made
in the first case as regards the temperature of the undrowned capacities will
be repeated in the second, and so lose its importance. A similar remark
applies to any deficiency in the comparison of the three thermometers with
one another.
1902] BETWEEN 75 AND 150 MILLIMETRES OF MERCURY
Comparison of Large Bulbs.
This comparison needs to be carried out with something like the full
precision aimed at in the final result, although it is to be noted that an error
enters to only the half of its proportional amount, since we have to do not
with the ratio of the capacities of the two bulbs, but with the ratio of the
capacity of the upper bulb to the capacity of the two bulbs together. Thus if
the volume of the upper bulb be unity and that of the lower (1 + a), the ratio
with which we are concerned is 2 + a : 1, differing from 2 : 1 by the propor
tional error a.
To adjust the capacities to approximate equality and to determine the
outstanding difference, the double bulb was mounted vertically, in connection
above with a Topler pump and below with a stopcock, such as is used with
a mercury burette. The " marks " were provided by small collars of metal
embracing tubing of 3 millims. bore and securely cemented, to the lower edge
of which the mercury could be set as in reading barometers. A measuring
flask, with a prolonged neck consisting of uniform tubing of 6 millims.
diameter, was prepared having nearly the same capacity as the bulbs. The
mercury required at a known temperature to fill the upper bulb between the
marks was run from the tap into this flask. Air specks being removed,
the flask was placed in a waterbath and the temperature varied until the
mercury stood at a fixed mark upon the neck of the flask. Subsequently
the mercury required at the same temperature to fill the lower bulb between
the middle and the lower marks was measured in the same way. On a mean
of two trials it was found that the flask needed to be 2'4 C. warmer in the
second case than in the first, showing that the capacity of the lower bulb was
a little the smaller. Taking the relative expansions of mercury and glass
for one degree to be '00016, we get as the proportional difference '00038.
Thus in the notation already employed,
Fi : F 2 = 2 '00038 = 199962 (4)
It appeared that so far as the measurements were concerned this ratio should
be correct to at least ^^5 but disturbances due to pressure introduce
uncertainty of about the same order.
Comparison of Gauges.
A simple method of comparing the gauges is to combine them in parallel
so that the pressures in the lower chambers are the same, and also the
pressures in the upper chambers, and then to find what slope must be given
to the longer measuringrod in order that its effective length may be equal
to that of the shorter rod maintained vertical. The mercury can then be set
B. V. 3
34 THE LAW OF THE PRESSURE OF GASES [275
to coincidence with all four points, and the equality of the gauges so arranged
actually tested. It is afterwards an easy matter to calculate back so as to
find the proportional difference of heights when both measuringrods are
vertical. Preliminary experiments of this kind upon the gauges, mounted on
separate levelling stands and connected by indiarubber tubing, had shown
that the difference was about ^ part.
It would be possible, having found by the combination in parallel an
adjustment to equality, to maintain the same sloped position during the
subsequent use when the gauges must be combined in series. But in this
case it would hardly be advisable to trust to woodwork in the mounting. At
any rate in my experiments the gauges were erected with measuringrods
vertical, an arrangement which has at least the advantage that a displace
ment is of less importance as well as more easily detected. At the close of
the observations upon the various gases it became necessary to compare
the gauges with full precision.
For this purpose, they were connected (without indiarubber) in parallel,
the upper chambers of both being in communication with the pump, and
the lower chambers of both in communication with the gas reservoirs 01.
Had the lengths of the measuringrods been absolutely equal, this equality
would be very simply proved by the possibility of so adjusting the pressure
of the gas and the supply of mercury to the two manometers that all four
mercury surfaces could be set simultaneously. It was very evident that no
such simultaneous setting was possible, and the problem remained to evaluate
the small outstanding difference. To pass from one manometer to the other,
either the volume or the temperature [of the gas] had to be varied.
In principle it would perhaps be simplest to keep the volume constant
and determine what difference of temperature (about half a degree) would be
required to make the transition. But the temperature of the undrowned
parts (now increased in volume) could not be ascertained with great precision,
so that I preferred to vary the volume and to trust to alternations backwards
and forwards for securing that the mean temperature in the two cases to be
compared should not be different. Thus in one set, including seven observa
tions following continuously, four alternate observations were settings with
one manometer and three were settings with the other. According to the
thermometers, the mean temperature in the first case was for the drowned
volume 11'38 and for the (much smaller) undrowned volume 12 0 76. In the
second case the corresponding temperatures were 11'39 and 12'80, so that
the differences could be neglected. The volume changes were effected in the
sidetube JK, and the mean difference in the two cases was '411 cub. centim.
It will be understood that in order to define the volume both manometers
were always set below. The whole volume was reckoned at 294 cub. centims.,
of which about 258 cub. centims. represents the capacity of the bulbs GI
1902] BETWEEN 75 AND 150 MILLIMETRES OF MERCURY 35
drowned in the waterbath. According to these data the proportional
difference in the lengths of the measuringrods, equal to the proportional
difference of the above determined volumes, is '00140. Two other similar
sets of observations gave '00136, "00137 ; so that the mean adopted value is
00138. The measuringrod of the manometer on the right, fig. 2, is the
longer.
As in the case of the volumes, any error in the above comparison is halved
in the actual application. If H z be the length of the rod in the righthand
manometer, H l the length in the left, we are concerned only with the ratio
H l + 7/2 : H z . And from the value above determined we get
Hl + H * = 199862 (5)
The* Observations.
In commencing a set of observations the first step is to clear away any
residue of gas by making a high vacuum throughout the apparatus, the
mercury being lowered below the manometers and bulbs. The mercury
having been allowed to rise into the pump head of the Topler, the gas to be
experimented on is next admitted to a pressure of about 75 millims. This
occupies the manometers, the bulbs, and part of the capacity of the inter
mediate chamber E. The passage through the righthand manometer is
then closed by bringing up the mercury to the neighbourhood of G, and by
rise of mercury from I to H the pressure is doubled in the upper bulb. The
next step is to cut off the communication between A and B, and to renew
the vacuum in B. If the right amount of gas has been imprisoned, it is
now possible to make a setting, the mercury standing at A, C, H, and in the
side apparatus somewhere in the upper tube below J. If, as is almost certain
to be the case in view of the narrowness of the margin, a suitable setting
cannot be made, it becomes necessary to alter the amount of gas. This can
usually be effected, without disturbing the vacuum, by lowering the mercury
at C and allowing gas to pass in pistons in the curved tube CD either
from the intermediate chamber to the bulbs, or preferably in the reverse
direction.
When the right amount of gas has been obtained, the observations are
straightforward. On such occasion six readings were usually taken, extend
ing over about an hour, during which time the temperature always rose,
and the means were combined into what was considered to be one ob
servation.
A complete set included four observations with the large bulbs at
150 millimg. pressure and four at 75 millims: To. pass to the latter the
32
THE LAW OF THE PRESSURE OF OASES
[275
mercury must be lowered from H to / and in the lefthand manometer, and
the pump worked until a vacuum is established in D. It was considered
advisable to break up one of the sets of four ; for example, after two observa
tions at 150 millims. to take four at 75 millims., and afterwards the remaining
pair at 150 millims. In this way a check could be obtained upon the
quantity of gas, of which some might accidentally escape, and there were
also advantages in respect of temperature changes. These eight observations
with the large bulbs were combined with four in which the side apparatus
was alone in use, the mercury standing all the while at G. Of these, two
related to the 75 millims. pressure and two to the 150 millims. Finally, the
means were taken of all the corresponding observations.
The following table shows in the notation employed the correspondence of
volumes and temperatures :
I.
V,
01
V 3
TI V 6
tl
II.
III.
V 2
6*
v
rl %
1g
h
IV.
V 3
V 4 '
T<
In the first observation V^ is the volume of the two large bulbs and 0^ the
temperature of the waterbath, reckoned from some convenient neighbouring
temperature as a standard. F 3 is the ungauged volume already discussed
whose temperature T^ is given by the upper thermometer. F 5 is the (larger)
volume in the side apparatus whose temperature ^ is that of the lower
thermometer. In the second observation F 2 is the volume of the upper bulb
and #2 its temperature. F 4 is the volume in the side apparatus whose
temperature, as well as that of F,, is taken to be T 2 , the mean reading of the
upper thermometer. III. and IV. represent the corresponding observations
in which the large bulbs are not filled. The reading of the waterbath
thermometer is in every case denoted by 0, that of the upper thermometer by
T, and that of the lower thermometer by t. The temperature of the columns
of mercury in the manometers is also represented by r.
As an example of the actual quantities, the observations on air between
October 28 and November 5 may be cited. The values of F, and F, are
approximate. As appears from the formulae, F, occurs with a small co
efficient, as does also V lt except in the ratio Fj : F s otherwise provided for.
We have
F, = 2584, F, = 1905;
F 4 = 810, F 8 = 20493;
F 4 F/= 0841, F 5 F 5 '= 0266;
#, = 077, 2 = 059; ,= 257, * 3 = '141;
T,= 092, T 2 = 186, T,= 033, r 4 = '100.
1902] BETWEEN 75 AND 150 MILLIMETRES OF MERCURY 37
The volumes are in cubic centimetres and the temperatures are in
Centigrade degrees reckoned from 14.
An effort was made, and usually with success, to keep all the temperature
differences small, and especially the difference between 0, and 2 . It is
desirable also so to adjust the quantities of gas in the two cases that F 4  F 4 ',
F 6 F/ shall be small
The Reductions.
The simple theory has already been stated, but the actual reductions are
rather troublesome on account of the numerous temperature corrections.
These, however, are but small.
We have first to deal with the expansion of mercury in the manometers.
If, as in (5), the actual heights of the mercury (at the same temperature) be
H l} HZ, we have for the corresponding pressures H/(l + mr), where m='00017.
Thus in the notation already employed
p _ HI H 2
l = l + mr^ r l+mT 3 '
TT i 17" TT t TT
and P 2 =:
The quantity of gas at a given pressure occupying a known volume is to
be found by dividing the volume by the absolute temperature. Hence each
volume is to be divided by 1 + $0, 1 f ftr, 1 + fit, as the case may be, where ft
is the reciprocal of the absolute temperature taken as a standard. Thus in
the above example for air (p. 36),
8= L_ J_
p 273 + 14 287 '
Our equations, expressing that the quantities of gas are the same at the
single and at the double pressure, accordingly take the form
g 2 f F, F 3 F B ] _B(H l + HJ) f F 2 F 8 +F 4 ]
1 + mr, \l+ft0,l+ftr,l + ftt,} 1 +mr 2 (1 + ftff, 1 + frrj '
H z ( F 3 F/ ](#, + #,) F 3 +F/
l + mr 3 \l+ftr 8 '
where B is the numerical quantity to be determined according to Boyle's
law identical with unity.
By subtraction we deduce
_ _
(1 + mr,) (1 + 00,) V.H. (1 + mr 2 ) (1 +
38 THE LAW OF THE PRESSURE OF GASES [275
7, {tf a (1 + rnr,) (1 + ftr t ) #, (I + mr 4 ) (1 + ftr t )
1 1 I
a )(l+/?T a ) (l+mT 4 )(l+/3T 4 )[
_ __ i _)
V, 1(1 + 7^0(1 +#,) (1 + mr,)(l + W)
F,
.(6)
The first three terms on the right, viz., those in F s , F 4 , F 5 , vanish if
T, = T,, T S = T 4 , <! = <,. In the small terms we expand in powers of the small
temperatures (T, t), and further identify B (H^ + H 2 )IH a with 2. The five
terms on the right then assume the form
s
f(m + ft) ( Tl  T S  2r 2 + 2r 4 ) 4 P (2r 2 2  2r 4 2  r, 2 + r 8 2 )}
\
2F
y* {(ill + ft) (T,  T 4 )  & (T 2 2  T 4 2 )}
[m (T,  T,) + (t,  O + ft 2 (,  3 2 )]
in which m/8 and m 2 are neglected, while yS 2 is detained. In point of fact, the
terms of the second degree were seldom sensible.
Taking the data above given for the observations on air October 28
November 5, we find
Term in F 8 ........................... ='000012
F 4 ........................... =000002
F s ........................... = + 000034
(F 4 F/) .................. = + 000652
(F.F/) .................. =000103
+ 000569
1902] BETWEEN 75 AND 150 MILLIMETRES OF MERCURY 39
For the first term on the left of (6), we find
(l + mrjVw"" 00856 ''
so that B =  lH * ^ + r *) (* + ^ x '999687,
or when the numerical values are introduced from (4), (5),
B = 100002.
The deviation from Boyle's law is quite imperceptible.
It may be noted that a value of B exceeding unity indicates an excessive
compressibility, such as is manifested by carbonic acid under a pressure of a
few atmospheres.
Tlte Results.
Little now remains but to record the actual results. All the gases were,
it is needless to say, thoroughly dried.
Air.
Date. B.
April 1529, 1901 99986
May 2228, 1901  100003
October 28November 5, 1901 ... T0002
Mean '99997
Hydrogen.
Date. B.
July 613, 1901 '99999
July 1623, 1901 '99996
Mean '99997
The hydrogen was first absorbed in palladium, from which it was driven
off by heat as required.
Oxygen.
Date. B.
June 717, 1901 100022
July 21July 1, 1901 100044
September 1830, 1901 100005
October 1018, 1901 100027
Mean 100024
* [1901. A misprint of sign is here corrected.]
40 THE LAW OF THE PRESSURE OF GASES [275
The two first fillings of oxygen were with gas prepared by heating
permanganate of potash contained in a glass tube and sealed to the re
mainder of the apparatus. The desiccation was, as usual, by phosphoric
anhydride. In the third and fourth fillings the gas was from chlorate of
potash and had been stored over water.
Nitrous Oxide.
Date. B.
July 31August 5, 1901 100059
August 824, 1901 100074
Mean 100066
Argon.
Date. B.
December 28 January 1, 1902 ... 100024
January 29, 1902 100019
Mean r00021
The argon was from a stock which had been carefully purified some
years ago and has since stood over mercury. In this case the two sets of
observations recorded related to the same sample of gas imprisoned in the
apparatus. In all other cases the gas was renewed for a new set of
observations.
With regard to the accuracy of the results it was considered that
systematic errors should not exceed I0 fl 00 . In the comparison of one gas
with another most of the systematic errors are eliminated, and the mean of
two or three sets should be accurate according to the standard above stated.
That nitrous oxide should show itself more compressible than according to
Boyle's law is not surprising, but there appear to be deviations also in the
cases of oxygen and argon. Whether these deviations are to be regarded as
real departures from Boyle's law, or are to be attributed to some complication
relating to the glass or the mercury cannot be decided. At any rate they
are very minute. It will be noted that the oxygen numbers are not so
concordant as they ought to be. I am not in a position to suggest an ex
planation, and the discrepancies were hardly large enough to afford a handle
for further investigation.
If we are content with a standard of 7JT fo T7 , we may say that air, hydrogen,
oxygen, and argon obey Boyle's law at the pressures concerned and at the
ordinary temperatures (10 to 15)*.
Throughout the investigation I have been efficiently assisted by Mr Gordon,
to whom I desire to record my obligations.
[1010. For carbonic oxide B = 100005 ; see Phil. Tram. A, 204, p. 351.]
276.
ON THE PRESSURE OF VIBRATIONS.
[Philosophical Magazine, m. pp. 338346, 1902.]
THE importance of the consequences deduced by Boltzmann and W. Wien
from the doctrine of the pressure of radiation has naturally drawn increased
attention to this subject. That sethereal vibrations must exercise a pressure
upon a perfectly conducting, and therefore perfectly reflecting, boundary was
Maxwell's deduction from his general equations of the electromagnetic field ;
and the existence of the pressure of light has lately been confirmed experi
mentally by Lebedew. It seemed to me that it would be of interest to
inquire whether other kinds of vibration exercise a pressure, and if possible
to frame a general theory of the action.
We are at once confronted with a difference between the conditions to be
dealt with in the case of aethereal vibrations and, for example, the vibrations
of air. When a plate of polished silver advances against waves of light, the
waves indeed are reflected, but the medium itself must be supposed capable
of penetrating the plate; whereas in the corresponding case of aerial vibrations
the air as well as the vibrations are compressed by the advancing wall. In
other cases, however, a closer parallelism may be established. Thus the
transverse vibrations of a stretched string, or wire, may be supposed to
be limited by a small ring constrained to remain upon the equilibrium
line of the string, but capable of sliding freely upon it. In this arrangement
the string passes but the vibrations are compressed, when the ring moves
inwards.
We will commence with the very simple problem of a pendulum in which
a mass C is suspended by a string. B is a ring [which Fig. l.
embraces the string] constrained to the vertical line AD and
capable of moving along it ; BC = I, and 6 denotes the angle
between BC and A D at any time t. If B is held at rest, BC
is an ordinary pendulum, and it is supposed to be executing
small vibrations ; so that 6 = @ cos nt, where n 2 = g/l. The
tension of the string is approximately W, the weight of the
bob ; and the force tending to push B upwards is at time t
W (1 cos 6). Now this expression is closely related to the
potential energy of the pendulum, for which
V= Wl(Icos0).
42 ON THE PRESSURE OF VIBRATIONS [276
The mean upward force upon B is accordingly equal to the mean value
of Fr 1; or since the mean value of V is half the constant total energy E of
the system, we conclude that the mean force (L), driving B upwards, is
measured by %E/L
From the equation
LlE/l .................................... (1)
it is easy to deduce the effect of a slow motion upwards of the ring. The
work obtained at B must be at the expense of the energy of the system,
so that
By integration
E^EJi, .................................... (2)
where E l denotes the energy corresponding to 1=1. From (2) we see that
by withdrawing the ring B until I is infinitely great, the whole of the energy
of vibration may be abstracted in the form of work done by B, and this by
a uniform motion in which no regard is paid to the momentary phase of the
vibration.
The argument is nearly the same for the case of a stretched string
vibrating transversely in one plane. The string itself may be supposed to
be unlimited, while the vibrations are confined by two rings of which one
may be fixed and one movable.
If the origin of # be at one end of a string of length I, the transverse
displacement [and velocity] may be expressed by
simnr . 27r#
 + ..., ..................... (3)
TX
 + ..., ...... .............. (4)
where <,, < 2 , ... are coefficients depending upon the time. For the kinetic
and potential energies we have respectively (Theory of Sound, 128)
(5)
in which W represents the constant tension and p the longitudinal density of
the string. For each kind of </> the sums of T and V remain constant during
the vibration ; and the same is of course true of the totals given in (5).
From (3)
dy IT ( . TTX _ , 2Tnc
3*.
so that when x = I
1902] ON THE PRESSURE OF VIBRATIONS 43
Accordingly the force tending to drive out the ring at x = I is at time t
or in the mean taken over a long interval,
 W . Mean 2 ^ <j> g 2 .
Comparing with (5), we see that the mean force L has the value
21 x mean V; or since mean F= mean T= \E, E denoting the constant
total energy,
L = E/l (6)
The force driving out the ring is thus numerically equal to the longitudinal
density of the energy.
This result may readily be extended to cases where the vibrations are not
limited to one plane ; and indeed the case in which the plane of the string
uniformly revolves is especially simple in that T and V are then constant
with respect to time.
If the ring is allowed to move out slowly, we have
dE = Ldl=Edlfl,
or on integration
E = E l l~\ (7)
analogous to (5), though different from it in the power of I involved. If
I increase without limit, the whole energy of the vibrations may be abstracted
in the form of work done on the ring.
We will now pass on to consider the case of air in a cylinder, vibrating in
one dimension and supposed to obey Boyle's law according to which p = a?p.
By the general hydrodynamical equation (Theory of Sound, 253 a),
(8)
where < denotes the velocitypotential and U the resultant velocity at
any point ; so that in the present case, if we integrate over a long interval
of time,
a?f\ogpdt + $JU*dt ........................... (9)
retains a constant value over the length of the cylinder. If p denote the
pressure that would prevail throughout, had there been no vibrations, p  p
is small and we may replace (9) by
(10)
The expression (10) has accordingly the same value at the piston
44 ON THE PRESSURE OF VIBRATIONS [276
for the mean of the whole column of length I. Now for the mean of the
whole column
and thus if/), denote the value of p at the piston where x = I,
......... (11)
It is not difficult to prove that the righthand member of (11) vanishes.
Thus, expressing the motion in terms of <j>, suppose that
STTX sirat
9 = cos j cos j ......................... (12)
Then
p p ~ PO d<j>/dt, U =
and since p 9 = a a p , we get
and this vanishes by (12). Accordingly
Again by (12)
so that
Now p /JU*dxdt represents twice the mean total kinetic energy of the
vibrations or, what is the same, the constant total energy E. Thus if L
denote the mean additional force due to the vibrations and tending to push
the piston out,
L = El~ l ........................... . ...... (14)
As in the case of the string, the total force is measured by the longitudinal
density of the total energy ; or, if we prefer so to express it, the additional
pressure is measured by the volumedensity of the energy.
In the last problem, as well as in that of the string, the vibrations are in
one dimension. In the case of air there is no difficulty in the extension
to two or three dimensions. Thus, if aerial vibrations be distributed equally
in all directions, the pressure due to them coincides with onethird of the
volumedensity of the energy. In the case of the string, where the vibrations
are transverse, we cannot find an analogue in three dimensions ; but a
membrane with a flexible and extensive boundary capable of slipping along
the surface, provides for two dimensions. If the vibrations be equally
1902] ON THE PRESSURE OF VIBRATIONS 45
distributed in the plane, the force outwards per unit length of contour will
be measured by onehalf of the superficial density of the total energy.
A more general treatment of the question may be effected by means of
Lagrange's theory. If I be one of the coordinates fixing the configuration
of a system, the corresponding equation is
d fdT\ dT dV
where T and V denote as usual the expressions for the kinetic and potential
energies. On integration over a time ti
[Ldt_l[dT] + l((dV
J t, t.idi 1 y tj^di
If dTjdV remain finite throughout, and if the range of integration be
sufficiently extended, the integrated term disappears, and we get
(Ldt 1 ffdV dT^
On the right hand of (16) the differentiations are partial, the coordinates
other than / and all the velocities being supposed constant.
We will apply our equation (16) in the first place to the simple pendulum
of fig. 1, I denoting the length of the vibrating portion of the string BC. If
x, y be the horizontal and vertical coordinates of C,
x = I sin 6, y = I  I cos 6 ;
and accordingly if the mass of G be taken to be unity,
r=r 2 (22cos0) + n9'.Jsin<9 + ^ 2 , ............ (17)
I', & denoting dl/dt, dO/dt. Also
V=gl(lcos0) ............................ (18)
From (17), (18)
(19)
These expressions are general ; but for our present purpose it will suffice if
we suppose that V is zero, that is that the ring is held at rest. Accordingly
dV_V dT_2T
dl ~~ I' dl~ I '
and (16) gives
On the right hand of (20) we find the mean values of V and of T. But these
mean values are equal. In fact
(21)
46 ON THE PRESSURE OF VIBRATIONS [27 G
if E denote the total energy. Hence, if L now denote the mean value,
.............................. (22)
the negative sign denoting that the mean force necessary to hold the ring at
rest must be applied in the direction which tends to diminish /, i.e. downwards.
In former equations (1), (6), (14), L had the reverse sign.
We will now consider more generally the case of one dimension, using
a method that will apply equally whether for example the vibrating body be
\ a stretched string, or a rod vibrating flexurally. All that we postulate
is homogeneity of constitution, so that what can be said about any part of
the length can be said equally about any other part. In applying Lagrange's
method the coordinates are I the length of the vibrating portion, and <j, < 2 , &c.
defining, as in (3), the displacement from equilibrium during the vibrations.
As functions of I, we suppose that
Foe l m , Tx l n ............................ (23)
Thus, if L be the force corresponding to I, we get by (16)
Ldt I/mV nT
]
in which
E representing as before the constant total energy. Accordingly, L now
representing the mean value,
In the case of a medium, like a stretched string, propagating waves of all
lengths with the same velocity, m = 1, n = 1, and L = E/l, as was found
before.
In the application to a rod vibrating flexurally, m =  3, n = 1, so that
L = 2E/l ............................... (25)
If m = n, L vanishes. This occurs in the case of the line of disconnected
pendulums considered by Reynolds in illustration of the theory of the group
velocity*, and the circumstance suggests that L represents the tendency of
a group of waves to spread. This conjecture is easily verified. If in con
formity with (13) we suppose that
and also that
. 27T* 27T 27T
<>! = sin , <J>j = cos >
See Proc. Math. Soc. ix. p. 21 (1877) ; this collection, i. p. 322. Also Theory of Sound,
Vol. i. Appendix.
1902] ON THE PRESSURE OF VIBRATIONS 47
r being the period of the vibration represented by the coordinate fa, we
obtain, remembering that the sum of T and V must remain constant,
V l m = T l n .47T/T 2 .
This gives the relation between T and I. Now v, the wavevelocity, is pro
portional to //T; so that
vcc l l $ n +l m ............................... (26)
Thus, if u denote the group velocity, we have by the general theory
u/v = $n$m; .............................. (27)
and in terms of u and v by (24)
J =f ............................... (28)
vl
Boltzmann's theory is founded upon the application of Carnot's cycle to
the radiation inclosed within movable reflecting walls. If the pressure
(p) of a body be regarded as a function of the volume v*, and the absolute
temperature 6, the general equation deduced from the second law of thermo
dynamics is
* ............... .............. <29)
where M dv represents the heat that must be communicated while the volume
alters by dv and dd = 0. In the application of (29) to radiation we have
evidently
M=U+p, .............................. (30)
where U denotes the density of the energy a function of 6 only. Hence f
If further, as for radiation and for aerial vibrations,
P = $U, ................................. (32)
it follows at once that
dlogU=4,dlogO,
whence
UK e\ ................................. (33)
the wellknown law of Stefan. It may be observed that the existence of
a pressure is demanded by (31), independently of (32).
If we generalize (32) by taking
P = lu .................................. (34)
where n is some numerical quantity, we obtain as the generalization of (33)
UK B n + 1 ............................... (35)
* Now with an altered meaning.
t Compare Lorentz, Amsterdam Proceedings, Ap. 1901.
48 ON THE PRESSURE OF VIBRATIONS [276
It is an interesting question whether any analogue of the second law
of thermodynamics can be found in the general theory of the pressure of
vibrations, whether for example the energy of the vibrations of a stretched
string is partially unavailable in the absence of appliances for distinguishing
phases. It might appear at first sight that the conclusion alread} 7 given,
as to the possibility of recovering the whole energy by mere retreat of the
inclosing ring, was a proof to the contrary. This argument, however, will
not appear conclusive, if we remember that a like proposition is true for the
energy of a gas confined adiabatically under a piston. The residual energy
of the molecules may be made as small as we please, but the completion of
the cycle by pushing the piston back will restore the molecular energy unless
we can first abolish the infinitesimal residue remaining after expansion,
and this can only be done with the aid of a laody at the absolute zero
of temperature. It would appear that we may find an analogue for
temperature, so far as the vibrations of one system are concerned; but,
so far as I can see, the analogy breaks down when we attempt a general
theory.
[1910. See further Phil Mag. Sept. 1905 "On the Momentum and
Pressure of Gaseous Vibrations, and on the Connection with the Virial
Theorem."]
277.
ON THE QUESTION OF HYDROGEN IN THE ATMOSPHERE.
[Philosophical Magazine, III. pp. 416422, April 1902.]
IT will be remembered that M. Armand Gautier, as the result of very
elaborate investigations, was led to the conclusion that air, even from the
Atlantic, contains by volume nearly two parts in 10,000 of free hydrogen.
The presence of so much hydrogen, nearly twothirds of the carbonic acid
which plays such an important part, is of interest in connexion with theories
pointing to the escape of light constituents from the planetary atmospheres.
Besides the free hydrogen, M. Gautier found in the air of woods and towns
considerable quantities of hydrocarbons yielding CO 2 when led over hot
copper oxide.
Spectroscopic Evidence.
In the Philosophical Magazine for Jan. 1901*, I described some observa
tions upon the spectrum of sparks taken in dried air at atmospheric pressure,
which seemed "to leave a minimum of room for the hydrogen found by
M. Gautier." Subsequently (April 1901), these experiments were repeated
with confirmatory results. The spectra, taken from platinum points, of pure
country air and the same to which 1QO()0 of hydrogen had been added were
certainly and easily distinguished by the visibility of the Cline. An improve
ment was afterwards effected by the substitution of aluminium points for
platinum. A strong preliminary heating reduced the (7line with a stream
of pure dried air to the least yet seen, only just continuously visible, and
contrasting strongly with the result of substituting the air to which the two
parts in 10,000 of hydrogen had been added.
To air from outside one thousandth part of hydrogen was introduced and
allowed time to mix thoroughly. Excess of chlorine was then added, and
* [This Collection, Vol. iv. p. 496.]
50 ON THE QUESTION OF HYDROGEN IN THE ATMOSPHERE [277
after a while the whole was exposed to strong sunshine, after which the
superfluous chlorine was removed by alkali. Tested in the spectroscope, this
sample showed only about the same signs of hydrogen as the pure air, indi
cating that the added hydrogen had effectively been removed a result which
somewhat surprised me.
As there now appeared to be a margin for further discrimination, three
samples were prepared, the first pure air, the second air to which was
added joTJoo ^ hydrogen, and the third air with addition of JQQQQ of
hydrogen. In the spectroscope the three were just certainly distinguishable,
showing C in the right order. The chlorinetreated mixture showed about
the same as the pure air. On repetition with fresh samples these results
were confirmed.
In my former note I mentioned that nitrous oxide and oxygen showed
the (7line as much as, if not more than, air. I cannot say whether this
result is inevitable, but the gases were prepared with ordinary care. In the
more recent repetition N 2 O showed the (7line about the same as the air to
which Y00QQ H 2 had been added. Oxygen, prepared from permanganate,
showed C much more than does pure air; and there was not much change
when oxygen from mixed chlorates of potash and soda was substituted.
The impurity in the oxygen, if it be an impurity*, does not appear
to be easily removed. The visibility of C was not perceptibly diminished
by passage of either kind of oxygen over hot copper oxide. On the other
hand the air containing JQQQQ of added hydrogen was reduced by the same
treatment to equality with pure air. Possibly the impurity is a hydrocarbon
not readily burnt.
Neither by treatment with chlorine could oxygen from either source be
freed from the property of exhibiting the (7line.
The spectroscopic evidence here set forth is certainly far from suggesting
that air, previously to any addition, already contains two parts in 10,000 of
free hydrogen. The passage from two to three parts in 10,000 might possibly
produce the observed change of visibility which followed the introduction
of one tenthousandth of hydrogen ; and the behaviour with chlorine and
hot copper oxide is not absolutely inconsistent with the initial hydrogen.
But the reconciliation seems to involve coincidences of little a priori
probability.
*
* It is possible that traces of hydrogen, derived from the electrodes or from the glass, show
more in oxygen than in air.
1902] ON THE QUESTION OF HYDROGEN IN THE ATMOSPHERE 51
Determinations by Combustion.
In M. Gautier's experiments large volumes of dried air were passed
through tubes containing copper oxide heated in a specially constructed
furnace, the water formed being collected in suitable phosphoric tubes and
accurately weighed. In unsystematic experiments the source of the water
so collected might be doubtful, but it is explained that the apparatus was
tested with pure dry oxygen, and that under these conditions the phosphoric
tube showed no increase of weight exceeding '1 mg. The work was evidently
very careful and thorough ; and the impression left upon the mind of the
reader is that the case is completely made out. Indeed, had I been acquainted
with the details, as set forth in Ann. d. Chimie, t. xxii., Jan. 1901, at an
earlier stage, I should probably have attempted no experiments of my own.
It so happens, however, that I had already begun some work, which has since
been further extended, and which has yielded results that I find rather
embarrassing, and am even tempted to suppress. For the conclusion to which
these determinations would lead me is that the hydrogen in countiy air is
but a small fraction, perhaps not more than one eighth part, of that given by
M. Gautier. Although I am well aware that my experience in these
matters is much inferior to his, and that I may be in error, I think it
proper that some record should be made of the experiments, which were
carefully conducted with the assistance of Mr Gordon and many times
repeated.
The quantity of air upon which I operated was almost uniformly 10 litres
much less than was used by M. Gautier. A glass aspirating bottle, originally
filled with water, was discharged upon the lawn, so that the water was
replaced by fresh country air. During an experiment the air was driven
forward, at the rate of about 1 litres per hour, by water entering below.
After traversing a bubbler charged with alkali, it was desiccated first by
passing over a surface of sulphuric acid, and subsequently by phosphoric
anhydride. Next followed the hot copper oxide, contained in a hard glass
tube and heated by an ordinary combustionfurnace. Next in order followed
the Utube charged with phosphoric anhydride whose increase of weight was
to indicate the absorption of water, formed in or derived from the furnace
tube. The Utube was protected upon the further (downstream) side by
other phosphoric tubes. It was provided with glass taps and was connected
on either side by short pieces of thick rubber of which but little was exposed
to the passing air. The counterpoise in the balance was a similar closed
phosphoric tube of very nearly the same volume, and allowance was made for
the pressure and temperature of the air included in the workingtube at the
moment when the taps were closed.
42
52 ON THE QUESTION OF HYDROGEN IN THE ATMOSPHERE [277
Two parts in 10,000 of free hydrogen, i.e. 2 c.c., yield on combustion the
same volume of water vapour, and of this the weight would be 1'5 mg. to be
collected in the phosphoric tube. Any water, due to hydrocarbons originally
present in the air and oxidized in the furnacetube, would be additional to
the above 1*5 mg.
The earlier experiments, executed at the end of 1900 and beginning of
1901, gave results which I found it difficult to interpret. The gain of weight
from the passage of 10 litres of fresh air was about '4 mg., that is, far too
little; and, what was even more surprising, this gain was not diminished
when the air after passage was collected and used over and over again. The
gain appeared to have nothing to do with hydrogen originally present in the
air, being maintained, for example, when a single litre of air was passed round
and round eight or nine times. Neither did the substitution of oxygen for
air make any important difference. Subsequently it was found that the
gain was scarcely diminished when the furnace remained cold during the
passage of the air.
Warned by M. Gautier, I was prepared for a possible gain of weight due
to retention of oxygen ; but this gain ought to be additional, and should
not mask the difference between air containing and not containing free
hydrogen. Faulty manipulation might be expected to entail an excessive
rather than a defective gain ; and the only cause to which I could attribute
the nonappearance of the hydrogen was a failure of the copper oxide to do
its work. The sample which I had employed was of the kind sold as
granulated. M. Gautier himself found a considerable length of copper oxide
necessary to complete the action. The question is, of course, not one of
length merely, but rather of the time during which the travelling gas remains
in close proximity to the oxidizing agent. Taking into account the slower
rate of passage in litres per hour, it would seem that my arrangement had
the advantage in this respect.
An attempt was made to improve the phosphoric anhydride by a
preliminary heating for many hours to 260 in a current of dry air, somewhat
as recommended by M. Gautier, but the results were not appreciably altered.
When one remembers the experiments of Baker, from which it appears that,
if all is thoroughly dry, heated phosphorus does not combine with oxygen, it
is difficult to feel confidence in this process.
It was certain that the sample of phosphoric anhydride hitherto employed
in this work was inferior. When a tube of it which had been used for some
time (in other work) and had become gummy at the ends, was strongly
heated over a spirit flame, occasional flashes could be seen in the dark.
Treatment in the cold with ozonized air seemed to effect an improvement ;
but experiments in this direction were not pursued to a definite conclusion
1902] ON THE QUESTION OF HYDROGEN IN THE ATMOSPHERE 53
in consequence of the discovery that when another sample of phosphoric
anhydride was substituted for that hitherto in use the anomaly disappeared.
Thus in four trials where 10 litres of air were passed without a furnace, the
gains were :
Nov. 26, 1901
Dec. 2,
00016
00003
+ 00014
+ 00006
Mean 00000
thus on the whole no gain of weight. The errors would appear somewhat
to exceed '1 mg., but it may be noted that in this and following tables the
real error is liable to appear exaggerated. If in consequence of an error
of weighing, or of allowance for weight of included air, or of a varied condition
of the outer surface of the tube, a recorded gain is too high, the next is likely
to appear too low.
In the operations which followed, the furnacetube was charged with
copper oxide prepared in situ by oxidation of small pieces of thin copper foil
with which the tube was packed. Examination once or twice after a breakage
showed that the oxidation was not complete, but no measurements were
taken until there was no appreciable further absorption of oxygen. After
the copper oxide has been exposed to the air of the room, many hours'
heating to redness in a current of dry air are required to remove the adherent
moisture.
The results, referring in each case to 10 litres of fresh country air, are as
follows, the weights being in gms. :
eo. 6,
1901
+ 00023
9,
, }
+ 00042
11,
,,
+ 00010
13,
,,
+ 00014
14,
H '00025
15,
;= ooooi
17,
+ 00031
18,
+ 00028
20,
+ 00025
24,
tj
+ 00016
Mean +00021
It will be seen that the mean water collected is only about one seventh of
that corresponding to the complete combustion of the hydrogen, according to
M. Gautier's estimate of the amount.
As has already been suggested, a defective gain of weight can hardly be
explained by faulty manipulation. The important question is as to the
efficiency of the copper oxide. Did my furnacetube allow the main part of
the free hydrogen to pass unburnt ? The question is one that can hardly be
54 ON THE QUESTION OF HYDROGEN IN THE ATMOSPHERE [277
answered directly, but I may say that variations of temperature (within
moderate limits) did not influence the result.
What it is possible to examine satisfactorily is the effect of small additions
of hydrogen to the air as collected. In my later experiments the added
hydrogen was only 1 c.c., that is, JQOOO ^y volume, or half the quantity
originally present according to M. Gautier. The hydrogen was first diluted
in a gas pipette with about 100 c.c. of air and allowed time to diffuse. The
10 litre aspi ratingbottle being initially full of water, the diluted hydrogen
was introduced at the top, and was followed by 10 litres of air from the open,
after which the mixture stood over night, precautions which had been found
sufficient to ensure a complete mixture in the spectroscopic work. The
results were:
Dec. 19, 1901
+ 00091
+ 00084
+ O0103
Mean +'00093
The additional gain is thus '00072, very nearly the full amount (O007. r ))
corresponding to the 1 c.c. of added hydrogen. We may say then that
the copper oxide was competent to account for a small addition of hydrogen
to air.
Following a suggestion from the spectroscopic experiments, I have
examined the effect of treatment with chlorine. To 10 litres of air 10 c.c.
(in one case 5 c.c.) of hydrogen were added, followed by excess of chlorine
and exposure to sunshine. The excess of chlorine having been removed by
alkali, the air was desiccated and passed through the furnace as usual. The
resulting gains of weight in the phosphoric tube were :
Dec. 31, 1901
Jan. 2, 1902
6,
8,
+ 00011
+ 00050
+ 00044
+ 00030
Mean + O0034
According to these figures, the treatment with chlorine after hydrogen
left the gains somewhat larger than in the case of pure air ; but the sunshine,
especially on Jan. 2, was feeble, and the difference of '13 mg. may be the
consequence of incomplete insolation. In any case the added hydrogen was
very nearly removed by the chlorine.
In two experiments 10 litres of pure air were subjected to the chlorine
treatment, and after removal of excess were mixed with 1 c.c. of hydrogen.
The gains of weight recorded in the phosphoric tubes were :
Jan. 1, 1902
7,
+ 00082
+ 00069
Mean +O0075
1902] ON THE QUESTION OF HYDROGEN IN THE ATMOSPHERE 55
The difference between this and 00093 would suggest that perhaps
some original hydrogen had been removed by chlorine, but the amount
is very small.
I shall best fulfil my intention if I refrain from attempting to sum up the
whole of the evidence. So far as my own work is concerned, the natural
inference from it would be that the free hydrogen in country air does not
exceed gQ^oo ^ tne vo ^ ume  If I mav make a suggestion, it would be in
favour of working with some such quantity as 10 litres, collected in glass
bottles, and of comparisons between fresh air and air already passed once or
twice through the furnacetube.
.
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.
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  :
.
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.
b.'rr'C"' " 'xTr:
v <& i," V ii' vir.'
i &iihu0W ttb'tt Jnfl .anhewo! ban fc:pu*
' 9^7001 8iW>i.1fi
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278.
DOES CHEMICAL TRANSFORMATION INFLUENCE WEIGHT?
[Nature, LXVI. pp. 58, 59, 1902.]
IN Nature (Vol. LXIV. p. 181, 1901)* I directed attention to experiments
by Heydweiller (Drude Ann. Vol. v. p. 394) from which he inferred that
some chemical transformations, such as the solution of copper sulphate in
water, were attended by real, though minute, changes of weight, and I pointed
out certain difficulties involved in the acceptance of this statement. In
connexion with another subject, it has lately occurred to me that such changes
of weight would really be in opposition to the laws of thermodynamics, and
I propose now briefly to sketch the argument from which this opposition
appears.
It is known f that by suitable arrangements the dissolution of salt may
be effected reversibly at a given temperature. During the process, a certain
amount of work is gained and a certain amount of heat at the given
temperature has to be supplied. In the reverse process, of course, an equal
amount of work has to be performed and an equal amount of heat is recovered.
The temperature being given, these operations are not affected (it is assumed)
by the height above the earth's surface at which they may be supposed to
take place.
Conceive now that the temperature is uniform throughout and that the
materials are initially at a low level and in one state (.4). Let them be
raised to a high level and there be transformed into the other state (B).
Subsequently let them be brought down to the low level and transformed
back into state A. The reverse transformations above and below compensate
one another thermodynamically, and if the weights are the same in the two
states, so do the operations of raising and lowering. But if the weights in
states A and B are different, the cycle of operations may be so executed that
work is gained. Such a difference of weight is therefore excluded, unless,
[Vol. iv. p. 549.]
t "On the Dissipation of Energy," Nature, . p. 454, 1875 ; Scientific Papert, Vol. i. p. 238.
1902] DOES CHEMICAL TRANSFORMATION INFLUENCE WEIGHT? 57
indeed, hitherto unsuspected thermal effects accompany a rising or falling
against or with gravity. It is scarcely necessary to say that we are not here
concerned with the differences of temperature and pressure which may
actually be met with at different levels over the earth's surface.
There are many chemical transformations which cannot easily be supposed
to take place reversibly. But this, though it might complicate the statement,
does not affect the essence of the argument ; and the conclusion appears to
be general.
If the reasoning here put forward be accepted, it increases the difficulty
of admitting the reality of such changes of weight as have been suspected,
and it justifies a severe criticism of experimental arrangements. In my
former letter I pointed out a possible source of error.
It is to be hoped that the matter may soon be cleared up, for it is scarcely
creditable to science that doubt should hang over such a fundamental question.
But for my own part I would wish to say that I fully recognize how much
easier it is to criticize than to experiment.
RXAttT
279.
IS ROTATORY POLARIZATION INFLUENCED BY THE
EARTH'S MOTION?
[Philosophical Magazine, iv. pp. 215220, 1902.]
THE question whether the rotation of the plane of polarization of light
propagated along the axis of a quartz crystal is affected by the direction
of this axis relatively to that of the earth's orbital motion, is of considerable
theoretical importance. According to an investigation of Lorentz, an effect
of the first order might be looked for. Such an effect would be rendered
apparent by comparing the rotations when the direction of propagation of
the light is parallel to that of the earth's motion and in the reverse direction,
and it might amount to 10000 of the whole rotation*. According to Larmor's
theoryf there should be no effect of the first order.
The question was examined experimentally many years ago by Mascart J,
who came to the conclusion that the reversal of the ray left the rotation
unchanged to 2 oooo part. In most of the experiments, however, the accuracy
was insufficient to lend support to the above conclusion.
Dr Larmor (/. c. p. 220) having expressed the opinion that it might be
desirable to reexamine the question, I have made some observations which
carry the test as far as can readily be done. It appears that the rotation is
certainly not altered by  100000 part, and probably not by the half of this,
when the direction of propagation of the light is altered from that of the
earth's motion to the opposite direction.
I should scarcely have been able to carry the test to so satisfactory a
point, had it not been for the kindness of Prof. MacGregor, who allowed
me the use of certain valuable quartz crystals belonging to the Edinburgh
collection of apparatus. These crystals, five in number, are all righthanded,
and measure about 50 mm. each in the direction of the optical axis, to which
the polished faces are approximately perpendicular. They were prepared for
* This fraction representing approximately the ratio of the velocity of the earth in its orbit
to the velocity of light.
t Jtther and Matter, Cambridge, 1900.
+ Annalet de VEcole Normal*, Vol. i. p. 157 (1872).
1902] IS ROTATORY POLARIZATION INFLUENCED BY THE EARTH'S MOTION ? 59
Prof. Tait, and were employed by him for his "rotatory polarization spectro
scope of great dispersion*." For the most part they are nearly free from
blemish, and well adapted to the purpose in view.
In principle the experiment is very simple, scarcely differing from ordinary
polarimetry, as, for example, in determining the rotation due to sugar and
other active bodies. But the apparatus needs to be specially mounted upon
a long stiff board, itself supported upon a point, so that the absolute direction
of the light may be reversed without danger of even the slightest relative
displacement of the parts. The board swings round in the horizontal plane ;
and if its length is directed from east to west, or from west to east, observa
tions taken at noon (in June) correspond pretty accurately to propagation
of the light with or against the earth's motion in its orbit. Similar com
parisons at 6 o'clock are nearly independent of the earth's motion.
In another respect the experiment is peculiar on account of the enormous
amount of the rotation to be dealt with. For sodium light the rotation
is 22 per millimetre of quartz, so that the whole rotation is 5500, or more
than 15 complete revolutions. In the preliminary experiments, with one
of the crystals only, sodium light was employed ; but the observations were
unsatisfactory, even although the light was resolved into a spectrum. If the
flame was well supplied with salt, the extinction of the Dline by suitable
adjustment of the nicol still left the neighbouring region of the spectrum
so bright as to prejudice the observation by lessening the sensitiveness of the
eye. This effect, which is quite distinct from what is ordinarily called .the
broadening of the Dlines and can be made still more pronounced by
stimulating the flame with oxygen, does not appear to present itself in any
other method of observation, and is of interest in connexion with the theory
of luminous emission. A very moderate rotation of the nicol revives the
Dlines sufficiently to extinguish the neighbouring spectrum, just as the
first glimpse of the limb of the sun after a total eclipse extinguishes
the corona f.
When all five quartzes were brought into use it was hopeless to expect
good results from a sodaflame. From the fact that the rotation is as X~ s we
see that there must be 11 difference of rotation for the two Dlines, so that
a satisfactory extinction is out of the question. For the observations about
to be recorded a socalled vacuumtube, charged with helium, was employed,
the yellow line (situated close to the Dlines) being chosen. It was actuated
by a Ruhmkorff coil and four Grove cells, situated at some distance away.
* Nature, Vol. XXH. 1880 ; Tait's Scientific Papers, Vol. i. p. 423.
t July 6. A doubt having suggested itself as to whether this effect might not be due to an
actual whitening of the Bunsen flame, such as sometimes occurs rather unexpectedly, the
experiment was repeated with a flame of pure hydrogen. The region of the spectrum in the
neighbourhood of D was even brighter than before. An attempt to produce an analogous effect
with lithium was a failure, apparently in consequence of insufficient brightness of the flame.
60 IS ROTATORY POLARIZATION [279
The various parts, all mounted upon the pivoted board, will now be
specified in order. First came the helium tube with capillary vertical, then
at a distance of 25 cm. a collimating spectaclelens, followed by the polarizing
nicol. The field of view presented by this nicol was contracted to a circular
aperture 7 mm. in diameter, and was further divided into two parts by
a " sugarcell." This cell was the same as that formerly used in a cognate
research on the rotation of the plane of polarization in bisulphide of carbon
under magnetic force*. "The polarimeter employed is on the principle
of Laurent, but according to a suggestion of Poynting (Phil. Mag. July 1880)
the halfwave plate of quartz is replaced by a cell containing syrup, so
arranged that the two halves of the field of view are subjected to small
rotations differing by about 2. The difference of thickness 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 then
rotated by a thickness of syrup equal to the entire width of the cell (say
inch), but in the lower half of the field part of the thickness of syrup is
replaced by glass, and the rotation is correspondingly less. With a pretty
strong syrup a difference of 2 may be obtained with a glass T 8 ff inch
[inch = 2*54 cm.] thick. For the best results the operating boundary should
be a true plane 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 sufficiently
flat for the purpose, i.e. capable of exhibiting a field of view sharply
divided into two parts.... By this use of sugar, halfshade polarimeters may
be made of large dimensions at short notice and at very little cost. The
syrup should be filtered (hot) through paper, and the cell must be closed
to prevent evaporation."
The light next traversed the quartz crystals, each mounted upon a small
stand admitting of adjustment in azimuth and level so as to bring the optical
axis into parallelism with the line of vision. The analysing nicol, mounted
near the end of the board, was distant 102 cm. from the polarizer. After
passing the nicol the light traversed in succession a directvision prism of
sufficient aperture and a small operaglass focussed upon the sugarcell. The
aperture limiting the field had been so chosen that, as seen through the
spectroscope, the yellow image under observation was sufficiently separated
from the neighbouring red and green images corresponding to other spectral
lines of helium. The position of the analysing nicol was read with a vernier
to tenths of a degree an accuracy which just sufficed, and the setting could
be made by causing the two halves of the field of view afforded by the sugar
cell to appear equally dark.
A good deal of time was spent in preliminary experiment before the best
* Phil. Trans. CLXXVI. p. 343 (1885); Scientific Paper*, Vol. n. p. 363.
1902] INFLUENCED BY THE EARTH'S MOTION? 61
procedure was hit upon. It is necessary that the optic axes of the crystals
be adjusted pretty accurately to the line of vision, and this in several cases
involved considerable obliquity of the terminal faces. In these adjustments
the sugarcell and its diaphragm are best dispensed with, the crystals being
turned until the rotation required to darken the field is a minimum and the
darkness itself satisfactory. When the first crystal has been adjusted, a
second is introduced and adjusted in its turn, and so on. In some cases
a further shift of the crystal parallel to itself was required in order to remove
an imperfection from the part of the field to be utilized. In the end a fairly
satisfactory darkness was attained, but decidedly inferior to that obtainable
when the quartzes were removed. Part of the residual light may have been
due to want of adjustment ; but more seemed to originate in imperfections in
the quartzes themselves.
In my former experiments upon bisulphide of carbon advantage was
found from a device for rocking the plane of polarization through a small
constant angle*. During the observations now under discussion this effect
was obtained by the introduction of a second sugarcell, not divided into two
parts or seen in focus, just in front of the analysing nicol. The cell was
mounted so that it could slide horizontally in and out up to fixed stops. The
thickness of the cell being sufficient, the strength of the syrup was adjusted
to the desired point. Thus when the nicol was correctly set, the upper half
of the field was just distinctly the brighter when the cell was in, and the
lower half with equal distinctness the brighter when the cell was out, the
object to be aimed at in the setting of the nicol being the equality of these
small differences. For the results now to be given the setting of the nicol
was by myself and the reading of the vernier was by Mr Gordon. A second
observer is a distinct advantage.
As a specimen, chosen at random, I will give in full all the readings
made in the neighbourhood of noon on June 19. Five readings were taken
in each position and then the board was reversed. The headings "East" and
"West" indicate the end at which the observer was sitting; "East" therefore
meaning that the course of the light was from West to East.
The mean of the three " Easts " is 45'75, and of the two " Wests " is
4571; so that
All these numbers are in decimals of a degree. The progressive
alteration in the readings corresponds to the rise of temperature. It would
appear that, as was natural, the quartzes lagged somewhat behind the
thermometer.
* Loc. cit. ; Scientific Papers, Vol. n. p. 366.
THE EARTH'S MOTION
TABLE I.
[279
Timell"30
Temp. 17 4
East
Timell h 50 m
Temp. 177
West
Time 12 h 5"
Temp. 17 '9
East
Time 12 h 16 1 Time 12 h 25 ro
Temp. 17 9 Temp. 179
West East
457
455
46fi
456
456
454
4:. !)
454
457
457
456
458
455
456
457
459
457
459
457
458
460
461
461
460
460
4558
4562
4564
4580
4604
TABLE II. Noon.
Date
E. W.
June 17
18
19....
Main....
+ 03
05
+ 04
+ 007
Three sets of observations were taken at noon, and the results are
recorded in Table II. In two other sets taken about 6 h the differences
E W were even less. The comparison of the two hours serves to check
possible errors, e.g. of a magnetic character, such as might be caused by the
magnetism of the Ruhmkorff coil, if insufficiently distant.
It seems certain that at neither hour does the difference E W actually
amount to ^Q of a degree, i.e. to 1QOO()( j of the whole rotation. In all
probability the influence of the reversal is much less, if indeed it exists
at all.
P.S. Since the above observations were made, I see from the Amsterdam
Proceedings (May 28, 1902) that Lorentz maintains his opinion against the
criticism of Larmor. Lorentz's theoretical result contains an unknown
quantity which might be adjusted so as to make the influence of the earth's
motion evanescent ; but for this special adjustment there appears to be
no theoretical reason. I hope that the above experimental demonstration
of the absence of effect, to a high order of accuracy, will be found all the
more interesting.
280.
DOES MOTION THROUGH THE AETHER CAUSE DOUBLE
REFRACTION ?
[Philosophical Magazine, iv. pp. 678683, 1902*.]
THE wellknown negative result of the MichelsonMorley experiment in
which interference takes place between two rays, one travelling to and fro
in the direction of the earth's motion, and the other to and fro in a per
pendicular direction, is most naturally interpreted as proving that the aether
in the laboratory shares the earth's motion. But other phenomena, especially
stellar aberration, favour the opposite theory of a stationary aether. The
difficulty thus arising has been met by the at first sight startling hypothesis
of FitzGerald and Lorentz that solid bodies, such as the stone platform of
Michelson's apparatus, alter their relative dimensions, when rotated, in such
a way as to compensate the optical change that might naturally be looked
for. Larmor (JEther and Matter, Cambridge, 1900) has shown that a good
case may be made out for this view.
It occurred to me that such a deformation of matter when moving
through the aether might be accompanied by a sensible double refraction ;
and as the beginning of double refraction can be tested with extraordinary
delicacy, I thought that even a small chance of arriving at a positive result
justified a careful experiment. Whether the result were positive or negative,
it might at least afford further guidance for speculation upon this important
and delicate subject.
So far as liquids are concerned, the experiment is of no great difficulty,
and the conclusion may be stated that there is no double refraction of the
order to be expected, that is comparable with 10~ 8 of the single refraction f.
But the question arises whether experiments upon liquids really settle the
matter. Probably no complete answer can be given, unless in the light of
* Read before Section A of the British Association at Belfaot.
t 10~ 8 = (10 4 ) 2 , where 10* is the ratio of the velocity of the earth in its orbit to the velocity
of light.
64 DOES MOTION THROUGH THE .ETHER CAUSE DOUBLE REFRACTION ? [280
some particular theory of these relations. But it may be remarked that the
liquid condition is no obstacle to the development of double refraction under
electric stress, as is shown in Dr Kerr's experiments.
The apparatus was mounted upon the same revolving board as was
employed for somewhat analogous experiments upon the rotation in quartz
(Phil. Mag. Vol. iv. p. 215, 1902)*. Light, at first from the electric arc but
later and preferably from lime heated by an oxyhydrogen jet, after passing
a spectaclelens so held as to form an image of the source upon the analysing
nicol, was polarized by the first nicol in a plane inclined to the horizontal
at 45. The liquid, held in a horizontal tube closed at the ends by plates of
thin glass, was placed, of course, between the nicols. When at 12 o'clock the
board stands north and south, the earth's motion is transverse and the
situation is such as to exhibit any double refraction which may ensue. It
might be supposed, for instance, that luminous vibrations parallel to the
earth's motion, i.e. east and west, are propagated a little differently from
those whose direction is transverse to the earth's motion, i.e. vertical. But if
the board be turned through a right angle so as to point east and west, both
directions of vibration for light passing the tube are transverse to the earth's
motion, and therefore no double refraction could manifest itself. The question
is whether turning the board from the north and south position to the east
and west position makes any difference. In no case is any effect to be
expected from a rotation through 180, and such effect as a rotation through
90 may entail must be of the second order in the ratio which expresses the
velocity of the earth relatively to that of light.
It should not be overlooked that according to the theory of a stationary
aether, we have to do not only with the motion of the earth in its orbit, but
also with that of the sun in space. The latter is supposed to be much the
smaller, and to be directed towards the constellation Hercules. In the month
of April, when successful experiments were first made, the two motions would
approximately conspire.
If the suggested double refraction, due to the earth's motion, were large
enough, it would suffice to set the analysing nicol to extinction in one
position of the board, and to observe the revival of light consequent on
a rotation of the latter through 90. But a more delicate method is possible
and necessary. Between the polarizing nicol and the liquid column we
introduce a strip of glass whose length is horizontal and transverse to the
board. This strip, being supported (at two points) near the middle of its
length, and being somewhat loaded at its ends, is in a condition of strain,
and causes a revival of light except in the neighbourhood of a horizontal
band along the "neutral axis." Above and below this band the strained
[Supra p. 58.]
1902] DOES MOTION THROUGH THE .OTHER CAUSE DOUBLE REFRACTION? 65
condition of the glass produces just such a double refraction as might be
caused by the motion of the liquid through the aether, so that the existence
of the latter would be evidenced by a displacement of the dark band upwards
or downwards. In order the better to observe a displacement, two horizontal
wires are disposed close to the bent glass so as just to inclose the band, and
a small operaglass focussed upon these is introduced beyond the analysing
nicol. The slightest motion of the band is rendered evident by changes in
the feeble illumination just inside the wires.
The board is mounted upon a point so as to revolve with the utmost
freedom. The point is carried on the table and faces upwards. The bearing
is a small depression in an iron strap, rigidly attached to the board, and
raised sufficiently to give stability. The gasleading tubes are connected
in such a manner as to give rise to no forces which could appreciably vary as
the board turns.
Observations were made upon bisulphide of carbon in a tube 76 cms. long,
and upon water in a tube 73 cms. long. In neither case could the slightest
shift of the band be seen on rotation of the board from the northsouth
position to the eastwest position, whether at noon or at 6 P.M. The time
required to pass from one observation to the other did not exceed 15 seconds,
and the alternate observations were repeated until it was quite certain that
nothing could be detected.
Of course the significance of this result depends entirely upon the delicacy
of the apparatus, and it is worth little without an estimate of the smallest
double refraction that would have been detected. It may even be objected
that the investigation stands selfcondemned. In consequence of the earth's
magnetism there must be a rotation of the plane of polarization when the
light traverses the bisulphide of carbon in the north and south position;
and this effect, it may be argued, ought to manifest itself upon rotation
of the board.
To take the objection first, it is easy to calculate the rotation of the
plane of polarization. For one c.G.S. unit of magnetic potential the rotation
in CS 2 at 18 is '042 minute of angle*. In the present case the length is
76 cms. and the earth's horizontal force is *18; so that the whole rotation
to be expected f is
76 x 18 x 042 = 58'.
So small a rotation of the plane, which would show itself, if at all, by & fading
and not by a displacement of the band, is below the limit of observation.
The delicacy of the apparatus for its purpose may, indeed, be inferred
indirectly from the rotation of the nicol found necessary to engender a
* Phil. Trans. CLXXVI. p. 343 (1885) ; Scientific Papers, Vol. n. p. 377.
t The difference between astronomical and magnetic north is here neglected.
R. V. 5
66 DOES MOTION THROUGH THE <BTHER CAUSE DOUBLE REFRACTION ? [280
t
marked revival of light at the darkest part of the band. If #be this aiigle,>
the revived light is sin* 0, expressed as a fraction of the maximum obtainable
with parallel nicols. In the actual observation the nicols remain accurately
crossed, and the question is as to the effect of a double refraction causing
e.g. a retardation of vertical vibrations relatively to horizontal ones. If this
retardation amounted to X, X being the wavelength, the effect would be
the same as of a rotation of the nicol through 90. In general, a retardation
of phase e, in place of TT, gives a revival of light measured by sin 1 (e). If
the revivals of light in the two cases be the same, we may equate 6 to ^e.
Hence if we find that rotation 6 produces a sensible effect in lessening the
darkness at the darkest place, we may infer that there is delicacy sufficient to
detect a relative retardation of 20 due to double refraction. This comparison
would apply if the test for double refraction were made by simple observation
of the revival of light. As actually carried out by location of the band, the
test must be many times more delicate.
It was found that a marked fading of the band attended a rotation of the
nicol through 4'. According to this e would be ^ ; or since a retardation
of $\ corresponds to = TT, a retardation amounting to T ^y x \ should be
perceptible many times over, regard being paid to the superior delicacy of the
method in which a band is displaced relatively to fixed marks.
Another and perhaps more satisfactory method of determining the
sensitiveness was by introducing a thin upright strip of glass which could
be compressed in the direction of its length by small loads. These loads
were applied symmetrically in such a manner as to cause no flexure. The
double refraction due to the loads is of exactly the character to be tested for,
and accordingly this method affords a very direct check. If the load be given,
the effect is independent of the length of the strip and of its thickness along
the line of vision, but is inversely as the width. The strip actually employed
had a width of 15 mm.; and the application (or removal) of a total of 50 gms.
caused a marked shifting of the band, while 25 gms. was just perceptible
with certainty.
To interpret this we may employ some results of Wertheim (Mascart's
Traitd d'Optique, t. II. p. 232), who found that it requires a load of 10 kilo
grams per millimetre of width to give a relative retardation of \, so that
with the actual strip the load would need to be 150 kilograms. The
retardation just perceptible is accordingly \ : 6000. This may be
considered to agree well with what was expected from the effect of rotating
the nicol.
We have now only to compare the relative retardation which would
be detected with the whole retardation incurred in traversing the 76 cm.
of bisulphide of carbon. In this length there are contained 1,200,000 wave
1902] DOES MOTION THROUGH THE .ETHER CAUSE DOUBLE REFRACTION ? 67
lengths of yellow light, or 2,400,000 half wavelengths. The retardation due
to the refraction may be reckoned at '6 of this, or 1,440,000 half wavelengths.
Thus the double refraction that might be detected, estimated as a fraction of
the whole refraction, is 12 x 10~ 10 . The effect to be expected is of the
order 10~ 8 , so that there is nearly 100 times to spare. The above relates
to the bisulphide of carbon. With the water the delicacy of the test was
somewhat less.
When it is attempted to replace the liquid by solid matter, the difficulties
of experiment are greatly increased. The best results that I have been able
to obtain were with built up thicknesses of plateglass. A sufficient thickness
in one piece is liable to exhibit too much double refraction from the effect of
internal strains. A number of triangular pieces of plateglass, no larger than
necessary, and about 6 mm. thick, were put together in a trough to a total
thickness of about 110 mm. The interstices between the faces being filled
up with bisulphide of carbon, the internal reflexions were sufficiently reduced.
One difficulty is to get quit of motes and threads which adhere to the glass
and become extraordinarily conspicuous. Advantage was thought to be
derived from shaking up the bisulphide of carbon with strong sulphuric
acid. At the best the residual motes and specks in the glass interfere very
seriously with the observation, and the loss of light due to imperfect
transparency operates in the same direction. The least load upon the
upright strip that could be detected with certainty was now 100 grms.,
so that as compared with the observations upon liquid there was a loss
of delicacy of four times. In addition to this, the effect to be expected is
reduced in the proportion of 7 : 1, that being the ratio of lengths traversed by
the light. Thus in all we lose 28 times as compared with the liquid. In the
latter case we calculated a margin of 100 times, so that here there would
remain a margin of about 3 times.
A subsequent attempt was made to increase the total thickness of the
combined glasses to about 220 mm., but no real advantage was gained. The
loss of light and increase of disturbance from motes and residual double
refraction prejudiced the delicacy in about the same proportion as the length
of path was increased.
But although the results of the observations upon solids are very much
less satisfactory than in the case of liquids, enough remains to justify us in
concluding that even here there is no double refraction (of the order to
be expected) due to motion through the aether.
52
281.
ON THE DISTILLATION OF BINARY MIXTURES.
'
[Philosophical Magazine, iv. pp. 521537, 1902.]
AT various times during the past twenty years I have turned my
attention to the theory of distillation, and have made experiments upon
a question, as to which information seemed to be almost entirely lacking,
viz., the relation between the strengths of liquid and vapour which are in
equilibrium with one another when a binary mixture is subjected to distilla
tion. In order to be intelligible I must set forth a little in detail some
matters which are now fairly well known and understood, although they were
not so at the time when my notes were written.
Distillation of a Pure Liquid.
The temperature of the saturated vapour just over the liquid depends
upon the pressure. If the end of the condensertube, e.g., of the Liebig type,
be open, the pressure is of necessity nearly atmospheric. Suppose that in
this tube a piston, moving freely, separates pure vapour from pure air.
Then the whole wall of the condenser on the vapour side is almost at boiling
point. If we imagine the piston removed, the air and vapour may mix, and
it is now the total pressure which is atmospheric. Wherever the temperature
is below boiling there must be admixture of air sufficient to bring up the
pressure.
Two or more Liquids which press independently.
This is the case of liquids like water and bisulphide of carbon whose
vapourpressures are simply added So long as the number of ingredients
remains unchanged, the composition of the vapour rising from the boiling
mixture is a function of the temperature (or total pressure) only. Hence
in simple distillation the composition of the distillate remains constant until
perhaps one constituent of the liquid (not necessarily the most volatile) is
exhausted. At this point the distillate, as well as the boilingtemperature,
changes discontinuously and the altered values are preserved until a second
constituent is exhausted, and so on. None of the separate distillates thus
obtained would be altered by repetition of the process at the same pressure.
1902] ON THE DISTILLATION OF BINARY MIXTURES 69
Liquids which form true Mixtures.
The above is as far as possible from what happens in the case of miscible
liquids, e.g., water and common alcohol. Here the composition of the vapour,
as well as the boilingpoint under given pressure, depends upon the com
position of the liquid, and all three will in general change continuously
as the distillation proceeds. But, so long as the total pressure is fixed, to
a given composition of the liquid corresponds a definite composition of the
vapour; and it is the function of experiment to determine the relation
between the two. The results of such experiments may be exhibited
graphically upon a square diagram (e.g. figs. 3 and 4, pp. 76 and 79) in the
form of a curve stretching between opposite corners of the square, the abscissa
of any point upon the curve representing the composition of the liquid and
the ordinate representing the composition of the vapour in equilibrium with
it. For the pure substances at the ends of the scale, represented by opposite
corners of the square, the "compositions of liquid and vapour are necessarily
the same.
The character of the separation capable of being effected by distillation
depends in great measure upon whether or not the curve meets the diagonal
at any intermediate point, as well as at the extremities. If there be no
such intersection, the curve lies entirely in the upper (or in the lower)
triangular half of the square, so that for all mixtures the distillate is richer
(or poorer) than the liquid. As the distillation of a limited quantity of
mixed liquid proceeds, the composition of the residue moves always in one
direction and must finally approach one or other condition of purity.
If on the other hand the curve crosses the diagonal, the point of
intersection represents a state of things in which the liquid and vapour
have the same composition, so that distillation ceases to produce any effect.
This happens for example with a solution of hydrochloric acid at a strength
of 20 per cent. (fig. 3) and with aqueous alcohol at a strength of 96 per cent.
By no process of distillation can originally weak alcohol be strengthened
beyond the point named, and if (Le Bel) we start with still stronger alcohol
(prepared by chemical desiccation) the effect of distillation is reversed. The
vapour being now weaker (in alcohol) than the liquid, the residue in the
retort strengthens until it reaches purity.
In the case of substances which have no tendency to mix, e.g., water and
bisulphide of carbon, the composition of the vapour is, as we have seen,
always the same. The representative curve, reducing to a straight line
parallel to the axis of abscissae, or rather to the broken line AEFD (fig. 1),
necessarily crosses the diagonal. The point of intersection (H) represents
a condition of things in which the compositions of the liquid and vapour are
70
ON THE DISTILLATION OF BINARY MIXTURES
[281
the same. As distillation proceeds, the residue retains its composition, and
both ingredients are exhausted together.
If we commence with a liquid containing CS a in excess of the above
proportion, the excess gradually increases until nothing but CS 2 remains
c
100
60
A
S 8 D
HX'
E
j
/ 
H 2 50 100
behind. In the same way, if the water be originally in excess, the excess
accentuates itself until the (finite) residue is pure water. The critical
condition is thus in a sense unstable, and can only be realized by adjustment
beforehand.
The conclusions drawn above may be generalized. Whatever may be the
ingredients of a binary mixture, in the upper triangular half of the square
the vapour is stronger (we will say) than the liquid, in the lower half weaker.
Hence, as the liquid distils away, progress from a point in the upper half is
towards diminishing abscissae, and in the lower half towards increasing
abscissae. When, as in fig. 1, the curve in its course from A to D crosses AD
from left to right, the condition represented by the point of intersection H is
unstable. When, as in the case of hydrochloric acid (fig. 3), the crossing
takes place from right to left, i.e. from the lower half to the upper half of the
square, the progress from points in the neighbourhood is always towards
the point of intersection, so that the state represented thereby is stable. We
may sum up by saying that if, as the liquid strengthens, the vapour having
been weaker than the liquid becomes the stronger, the point of transition,
representing constant distillation, is stable ; but if the vapour having been
at first the stronger becomes the weaker, then the point of transition is
unstable.
The question presents itself, whether as the liquid strengthens (in a
particular ingredient) the vapour necessarily strengthens with it. Does the
curve on our diagram slope everywhere upwards on its course from A to D ?
Although a formal proof may be lacking, it would seem probable that this
must be so when the ingredients mix in all proportions. A limiting case
1902] ON THE DISTILLATION OF BINARY MIXTURES 71
is when two ingredients tfo not mix at all, e.g., water and bisulphide of
carbon, or when the mixture divides itself into two parts of constant
composition as when ether and water are associated in certain proportions.
In these cases the composition of the vapour is constant for the whole or
for a part of the range (Konowalow), and the representative curve is without
slope.
Konowaloiu's Theorem.
An important connexion has been formulated by Konowalow* between
the vapourpressure, regarded as a function of the composition of the liquid
with which it is in equilibrium, and the existence of a point of constant
distillation. "The pressure of the vapour from a fluid consisting of two
different substances is in general a function of the composition of th
mixture.... Let such a mixture, confined in a closed space, be maintained
at a constant temperature. We may conceive this space bounded by fixed
walls and by a movable piston. The conditions of stable equilibrium are
then (1) that the external pressure operative upon the piston should be
equal to the pressure of the saturated vapour at the given temperature ;
(2) that by increase, or diminution, of the vapour space the pressure should
become respectively not greater, or not less, than the external pressure. In
expansion the vapourpressure can thus either remain constant, or become
smaller. On the basis of this law we can establish a relation between the
composition of the liquid and that of the vapour."
Before proceeding further I must remark that the principle, as stated,
appears to need elucidation. Why should the equilibrium of the piston
under a constant load be stable ? There must of course be some position of
stable equilibrium for a given load and temperature ; but this might, for all
that appears, correspond to complete evaporation of the liquid or to complete
condensation of the vapour.
The following argument, however, suffices to show that Konowalow's
principle is a necessary consequence of the second law of Thermodynamics.
Suppose that the cylinder in which are contained the given liquid and
vapour communicates by a lateral channel (fig. 2) with a large reservoir
filled with liquid of similar composition, and that all are maintained at the
prescribed temperature. As a first operation close the tap between the
vessels, and then let the piston rise a little. The motion is supposed to
be so slow that equilibrium prevails throughout. The result of the expansion
may be that the compositions of the liquid and of the vapour undergo a
change. Now open the tap, and allow diffusion to take place, if necessary,
until equilibrium is again established. On account of the large quantity of
liquid in the reservoir the pressure is sensibly restored to its original value
Wied. Ann. xiv. p. 48 (1881).
72
ON THE DISTILLATION OF BINARY MIXTURES
[281
and remains undisturbed as the piston is slowly pushed back to its first
position. During this cycle of operations work cannot be gained ; and thus
is excluded the possibility of a rise of pressure during the expansion. It
follows that a fall of pressure cannot accompany compression.
Upon the basis of this principle Konowalow proceeds as follows : Suppose
that at a particular compositionratio the pressure of vapour increases as the
liquid becomes richer in a specified component. In this case the expansion
of the mass cannot enrich the liquid ; for if this result occurred the pressure
would rise, which we have proved it cannot do. During the expansion fresh
vapour is formed ; and if the composition of the vapour were poorer than that
of the liquid, the latter would inevitably be enriched by the operation. We
conclude that at the point in question the vapour cannot be poorer than the
liquid. In like manner if the vapourpressure falls with increasing richness
Fig. 2.
of liquid, compression of a given mass cannot enrich the liquid, and this
requires that the vapour be not richer than the corresponding liquid. If we
suppose the vapourpressure to be plotted as a function of richness of corre
sponding liquid, we may express these results by saying that rising parts of
the pressurecurve can have no representation in the lower triangle of our
former diagrams (where vapour is poorer than liquid), and that falling parts
cannot be represented in the upper triangle.
It is now evident that the passage from a rising to a falling part of
the pressurecurve can only occur when the vapour is neither richer nor
poorer than the liquid, and we arrive at Konowalow's important theorem
that any mixture, which corresponds to a maximum or minimum of vapour
pressure, has (at the temperature in question) the same composition as its
vapour.
The particular case in which one ingredient is wholly involatile is worth
a moment's notice. The vapour over a solution of salt in water can never
have the same composition as the liquid; and from this we may conclude
1902] ON THE DISTILLATION OF BINARY MIXTURES 73
that the vapourpressure has no maximum or minimum, or rather that there
is no transition anywhere between rising and falling.
The converse of Konowalow's theorem is also not without importance.
Consider two mixtures of slightly differing composition, one of which is
richer than its vapour and the other poorer. Expansion of the first entails
an enrichment of the liquid, and during the operation the pressure cannot
rise. Expansion of the second impoverishes the liquid, and again the
pressure cannot rise. The curve exhibiting pressure as a function of
composition (of liquid), if it slopes at all at the two points, must slope in
opposite directions. Hence by approaching nearer and nearer to the point
where the compositions of vapour and liquid are the same, we see that the
vapourpressure must there be stationary in value.
An example of the use of the converse theorem is afforded by the
consideration of mixtures of water and common alcohol. The question of
the existence of a mixture having the same composition as its vapour is not
easily settled directly, but the recent observations of Noyes and Warfel*
show conclusively that the mixture containing 96 per cent, of alcohol by
weight has a minimum boilingpoint, and accordingly distils without change.
It may be noted that the curve given by Konowalow himself would point to
the contrary conclusion.
In the practical conduct of distillation it is the pressure that is constant
rather than the temperature. Inasmuch, however, as pressure always rises
with temperature, a maximum or minimum pressure when temperature is
given necessarily corresponds with a minimum or maximum temperature
when pressure is given. In the case of a solution of hydrochloric acid., for
example, the thermometer marks a maximum temperature at the point where
the solution distils without change.
Calculation of Residue.
Before proceeding to the experimental part of this paper it may be well
to explain further the significance of the curves exhibiting the relative
compositions of liquid and vapour. If w represent the whole quantity
(weight) of liquid, say alcohol and water, remaining in the retort at any time,
y the quantity of one ingredient (alcohol), the abscissa of the curve is y/w.
As the distillation proceeds for a short time w becomes w + dw, and y becomes
y + dy f; and the composition of the vapour, that is the ordinate 17 of the
diagram, is dy/dw. Thus
= y/w, 77 = dy/dw,
* Am. Chem. Soc. xxm. p. 463 (1901).
t dw and dy being negative.
74 ON THE DISTILLATION OF BINARY MIXTURES  {281
while the functional relation between and ij is given by the curve, and may
be analytically expressed by rj =/(). Thus
whence "
.
w , being corresponding values of w and f.
When is small the curve is often approximately straight. If we
set/(f) = ,f wefmd
f/f. = (/)".
For example, in the case of alcohol and water, we have for very weak
mixtures 77 = 12f approximately, so that * = 12. As the distillation proceeds,
w diminishes and j? soon becomes exceedingly small. The halving of w
implies a diminution of in the ratio of 2" : 1. The residue in the retort
thus approximates rapidly to pure water.
On the other hand, in the case of acetic acid and water K is about f .
When weak acetic acid is distilled the residue strengthens, but the earlier
stages of the process are covered by the formula given, which now assumes
the form
/&(>* .
In order to double the strength of the liquid remaining in the retort,
15/16 of it would have to be distilled away, or again, in order to increase the
strength in the ratio of 3 : 2, the distillation must proceed until the liquid is
reduced in the ratio of 16 : 81, or nearly of 1 : 5. An experiment of this sort
upon acetic acid is recorded below.
Observations.
The experimental results about to be given were obtained by simple
distillation of mixtures of known composition. In order to avoid too rapid
a change of composition, somewhat large quantities were charged into a retort
and were kept in vigorous ebullition. By special jacketing arrangements
security was taken that the upper part of the retort should be maintained
at a distinctly higher temperature than the liquid, so that there could be no
premature condensation which would vitiate the result. All the vapour
rising from the liquid must be condensed in the specially provided Liebig
condenser and be collected as distillate. Subject to this condition, and in
view of the rapid stirring effected by the rising vapour, it would seem safe
to assume that the distillate really represents the vapour which is in
1902] ON THE DISTILLATION OF BINARY MIXTURES 75
equilibrium with the liquid at the time in question. The compositions of
the liquid and vapour are of course continually changing as the distillation
proceeds.
The distillates (including the first drop) were collected in 50 c.c. measuring
flasks. It will save circumlocution to speak of a particular case, and I will
take that of alcohol and water, for which the analyses were made by specific
gravity. The successive collections of 50 c.c. show an increasing specific
gravity corresponding to a diminishing strength. The specific gravity of
each gives the total weight, and the strength, deduced from tables, allows us
to calculate the alcohol and water in each collection. The total alcohol and
water originally present in the retort being known in the same way, we are
able to deduce by subtraction the quantities remaining in the retort at each
stage, and thus to compare the strengths of corresponding liquid and vapour.
In the reduction any particular distillate is considered to correspond with the
mean condition of the liquid before and after its separation therefrom.
If the process above sketched could be absolutely relied on, it would
be possible, starting with a strong spirit in the retort, to obtain from one
distillation data relating to a great variety of strengths. But this method is
not to be recommended, as the errors would tend to accumulate. The first
50 c.c., condensed under somewhat abnormal conditions, was not used directly,
but only to allow for the change going on in the retort. The 2nd, 3rd, and
4th collections were usually calculated so as to show the strengths of these
distillates in comparison with that of the liquid, but they were regarded
rather as checks upon one another than as independent results relating to an
altered state of affairs.
Alcohol and Water.
Observations upon mixtures of water and ethyl alcohol, sufficient to give
a nearly complete curve, were made in 1891 and again in 1898 with good
general agreement. The specific gravities were found in the balance with
a bottle of 20 c.c. capacity, and the calculations of strength were by Mendeleef's
tables with appropriate temperature correction. The results of the second
series are given in the accompanying table and are exhibited as a curve, A, in
fig. 3. The strengths are throughout reckoned by weight.
The observation of May 4 thus signifies that to a liquid containing by
weight 1'97 per cent, of alcohol there corresponds a vapour containing by
weight 17'5 per cent, of alcohol. From the results of May 24 we see that
when the liquid reaches 92 per cent, the vapour is but little the stronger,
and the difference practically disappears at 95 per cent. Indeed according
to May 25 the vapour is a little the weaker at this point. The difference,
76
ON THE DISTILLATION OF BINARY MIXTURES
[281
however, is not to be trusted, since the difficulties of manipulation, depending
partly upon the attraction of strong alcohol for aqueous vapour, are much
Date
1898
Strength
of Liquid
Strength
of Vapour
Mav 4
01970
1750
y \
,. >
, 5.
03982
0601
3159
3979
M 9
O988
5145
10
2586
6803
13
4562
7412
16:
, 17
6606
7739
7976
8414
20
8221
8622
23
8594
8849
24
9241
9284
25
9555
9545
Fig. 3.
10 20 50 40 50 60 70 80 90 100
A. Alcohol and Water. B. Hydrochloric Acid and Water. C. Acetic Acid and Water.
1902]
ON THE DISTILLATION OF BINARY MIXTURES
77
increased at this stage. It was these difficulties and the uncertainty as to
what exactly happened with spirit stronger than 95 per cent, that retarded
the publication of the work. I had intended to make further experiments
upon this point, but the matter was postponed from time to time. The
observations of Noyes and Warfel (I. c.) seem now to remove all doubt.
The existence of a minimum boilingtemperature for a strength of 96 per
cent, shows that the curve there crosses the diagonal. Between this point
and 100 per cent, the vapour is the weaker, and the curve lies in the lower
triangular half of the square. But the deviation from the diagonal in this
region is probably extremely small.
The following from Noyes and Warfel's table may be useful :
Strength
Boilingpoint
Strength
Boilingpoint
100
78300
85
78645
99
78243
80
79050
98
78205
75
79505
97
78181
65
80438
96
78174
55
8177
95
78177
48
8243
94
78195
35
8387
93
78:227
26
8541
92
78259
20
8732
91
78270
10
9180
90
78323
10000
Hydrochloric Acid and Water.
One of the ingredients of the mixture being gaseous under ordinary
conditions, the observations are limited to that portion of the curve for which
the strength of the liquid does not exceed 35 per cent., unless freezing
appliances are called into play. No attempt was made in the present
experiments to pass the above limit, the object being merely to determine
with moderate accuracy that part of the curve with which we are usually
concerned in the laboratory. It was known from the experiments of Roscoe
and others that the curve would cross the diagonal at the strength of about
20 per cent.
The general plan of the work was the same as in the case of alcohol and
water, but the strengths were usually determined chemically. In the case of
the stronger acids it was not possible to condense the vapour at atmospheric
temperature ; and I contented myself with a calculation in which the strength
of the vapour was inferred from observations of the quantity and strength of
the liquid in the retort before and after the operation. Results obtained in
this way are doubtless of minor accuracy.
ON THE DISTILLATION OF BINARY MIXTURES
[281
It may be worth while to reproduce in tabular form the data relating to
the weakest acid.
Distillation of Hydrochloric Acid. Sept 13, 1898.
Per
No.
Volume
in c.c.
Specific
gravity
Total
weight
Per
centage
ofHCl
Weight
of
HC1
Weight
of
H 2
HC1 re
maining
in retort
H a O re
rosining
in retort
Total
remain
ing
centage
ofHCl
remain
ing
1800
1031
18558
60
1114
17444
...
...
...
60
1
250
10013
25032
026
065
2497
1107
14947
16054
69
2
250
10023
25058
045
112
2494
1096
12453
13549
81
3
250
10038
25095
076
190
2490
1077
9963
11039
98
4
250
10076
25190
153
386
2480
1040
7483
8523
122
The first column contains the numbers of the successive distillates from
I to 4, the entry referring to the mixture with which the retort was
originally charged. The volume of this mixture was 1800 c.c. of specific
gravity 1'031 and of 6*0 per cent, strength. Of the total weight 1855'8 gms.,
111*4 gms. is hydrochloric acid and 1744*4 gms. water. In like manner the
volume of the first distillate is 250 c.c., the specific gravity 1*0013, the total
weight 25032 gms., of which 0*65 gms. is HC1 and 249*7 gms. H 2 O. The
residue in the retort after the first 250 c.c. has been distilled over is
accordingly composed of 111*40*65 or 110*7 gms. HC1 and 1744*4249*7
or 1494*7 gms. H 2 0, making 1605'4 gms. in all. At this stage the percentage
strength of the liquid remaining in the retort is 6*9. The strengths of the
liquid in the retort after the 1st, 2nd, ... 4th distillates have been removed
are found in this way to be 6'9, 8*1, 9*8, and 12'2 per cent. The first distillate,
whose strength is 0'26 per cent., thus corresponds with a liquid whose strength
varied from 6*0 to 6*9 per cent., on an average 6'45 per cent. We thus obtain
the following corresponding strengths :
Percentage Strengths.
Liquid
6*45
750
895
11*00
Vapour
0*26
045
076
1*53
It is hardly worth while to record all the separate results. In addition
o the above the following will suffice for the construction of the curve.
1902]
ON THE DISTILLATION OF BINARY MIXTURES
79
In the three last the strengths of the distillates were not directly observed,
but were calculated from the condition of the liquid before and after as
already mentioned.
Percentage Strengths.
Date (1898)
Liquid
Vapour
Sept 21
145
446
28 . ...
18'0
128
Oct. 10
12
203
233
189
324 J
14
, 15....
252
27'4
442
566
18 .
290
688
;; 19 :...
328
 883
The results are plotted in Curve B, fig. 3.
"
Ammonia and Water.
In this case the analysis was by specific gravity and the results were
somewhat rough, the intention being merely to obtain an approximation to
the form of the curve. On this account they are plotted upon a smaller
Fig. 4.
100
.0. Ammonia and Water.
100
D. Sulphuric Acid and Water.
scale C, fig. 4i the dotted portion of the curve being conjecturally added to
indicate the progress towards the corner of the square.
ON THE DISTILLATION OF BINARY MIXTURES
[281
Sulphuric Acid and Water.
The distillates were here determined chemically. From acid in the
retort of less strength than 60 per cent, the distillate failed to redden litmus.
From 75 per cent, acid the distillate contained about onethousandth part of
H 2 S0 4 . From 81 per cent, the distillate contained 16 per cent. ; from
90 per cent, the distillate contained 7'1 per cent. ; and from 93 per cent,
liquid the distillate contained 12'8 per cent, of acid. The curve is given in D,
fig. 4, the dotted portion for strengths of liquid greater than 93 per cent,
being conjectural.
Acetic Acid and Water.
This case was examined as likely to exemplify a very different behaviour
from any of the others, since it was known that these substances are not
easily separated by distillation. The retort was charged with 1000 c.c. of
mixture and two distillates were collected of 150 c.c. each. The analyses
were conducted chemically and the results calculated as already explained.
Thus the first distillate was considered to correspond with the mean strength
in the retort before and after its separation. The following were the results
obtained :
Acetic Acid and Water.
Date
1902
Strength of
Liquid
Strength of
Vapour
Aug 18
0677
0510
11
1458
1136
6
2682
2035
18
3746
2810
5
4998
3849
23
6156
4907
22
7227
6045
21
8166
7306
20
9070
8f22
:
It appears that the vapour is always weaker than the liquid, but that
the difference is never great. A plot of the corresponding strengths is given
in C (fig. 3, p. 76).
In illustration of the preceding theory an experiment was tried in which
threequarters of the original volume of liquid was distilled over. The
original liquid consisted of 1000 c.c. of acid of density I'OIO, and of strength
(as determined chemically) '0757, representing as usual the proportion of the
1902] ON THE DISTILLATION OF BINARY MIXTURES 81
weight of acetic acid to the whole weight. The residue measuring 250 c.c.
was of density T016 and of strength '1100. From these data we find
log (f/fo) = '1624, log (wjw) = 5995,
whence
l* = 27, * = 73.
The number denoted by K represents the ratio of strengths of vapour and
liquid when weak mixtures are distilled.
A new Apparatus with uniform Regime.
In the theory and experiments so far considered the distillation has
always been supposed to be simple, that is, the vapour rising from the
boiling liquid is supposed to be removed and to be condensed as a whole,
so that the distillate has the same composition as the vapour leaving the
boiling liquid. In practice, as is well known, this condition is often and
advantageously violated. A preliminary partial condensation of the vapour
in the stillhead frees it from some of the less volatile ingredient ; and, when
the residue is condensed and collected, the more volatile ingredient is obtained
in a nearer approach to purity. Prof. S. Young has shown that the principle
is more effectively carried out if the stillhead be maintained at a suitable
temperature.
Even with a preliminary partial condensation in the stillhead, the
" fractionation " of a mixture is usually regarded as a very tedious operation.
The stock of mixture in the retort is constantly changing its composition
as the distillation and partial condensation proceed, and no uniform regime
can be established. Although theoretical simplicity and practical convenience
are not always conjoined, a uniform regime seems very desirable, and it
excludes the usual arrangement in which the whole supply of mixture is
charged into the retort. The return into the retort of the liquid first con
densed from the original vapour is also objectionable.
The problem of distillation may be stated to be the separation from a
binary mixture of the whole of the two components in, as nearly as may be,
a state of purity. There is no theoretical reason why this should not be
effected at one operation ; but for this purpose the mixture must be fed in
continuously and not at the place of highest or lowest temperature. A
description of the procedure followed in some illustrative experiments will
make the nature of the process plain.
The mixtures actually employed were of water and common alcohol. The
choice was perhaps not a happy one, as in consequence of the peculiar
properties of strong alcohol it was unlikely that a distillate could be obtained
stronger than about 90 per cent. As regards apparatus, the retort and still
R. v. 6
82 ON THE DISTILLATION OF BINARY MIXTURES [281
head are replaced by a long length (12 metres) of copper tubing, 15 mm. in
diameter. This is divided into two parts, arranged in spirals, like the worms
of common condensers, and mounted in separate iron pails. The lower and
longer spiral was surrounded with water which was kept boiling. The water
surrounding the upper spiral was maintained at a suitable temperature,
usually 77 C. The copper tubes forming the two spirals were connected
by a straight length of glass, or brass, tubing of somewhat greater bore, and
provided with a lateral junction through which the material could be supplied.
The connecting piece and the spirals were so arranged that the entire length
was on a slight and nearly uniform gradient, rising from near the bottom of
the lower pail to the top of the upper pail. On leaving the latter the tube
turned downwards and was connected with an ordinary Liebig's condenser
capable of condensing the whole of the vapour which entered it. At the
lower end of the system of tubing the watery constituent is collected. In
strictness the receiver should be connected airtight and be maintained at
100. In distilling the stronger mixtures (60 or 75 per cent, alcohol) this
precaution was found advisable or necessary ; but in the case of the weaker
ones the water could be allowed to discharge itself through a short length of
pipe whose end was either exposed to the atmosphere or slightly sealed by
the liquid in the receiver.
The feed of the mixture was arranged as a visible and rather rapid
succession of drops, and was maintained at a uniform rate. In the case of
the stronger mixtures the evaporating power of the lower coil was hardly
sufficient, and was assisted by applying heat to the feed, so that a good
proportion was evaporated before reaching the main tube. The weaker
mixtures on the other hand could be fed in without any preliminary heating.
The uniform rdgime should be maintained long enough to ensure that the
liquids collected at the two ends shall be fairly representative and not
complicated by anything special that may happen before the uniform regime
is established.
During the operation every part of the tube (not too near the ends) is
occupied by a double stream an ascending stream of vapour and a descending
stream of liquid. Between these streams an exchange of material is constantly
taking place, the liquid, as it descends, becoming more aqueous and the
vapour, as it rises, becoming more alcoholic. In view of the slowness of the
feed and the length of the tube, we may regard the liquid and the vapour as
being everywhere in approximate equilibrium. At the lower end, since the
pressure is atmospheric and the temperature scarcely below 100, there
can be little alcohol; for similar reasons at the upper end there cannot be
much water, although the exclusion is here less complete on account of the
peculiar character of the curve representing the relation of compositions at
this extreme.
1902] ON THE DISTILLATION OF BINARY MIXTURES 83
Experiments were tried with four different strengths of mixture 20, 40,
60, and 75 per cent, of alcohol. In all cases the water was collected nearly
pure, never containing more than per cent, of alcohol. The alcoholic part
condensed from the upper end varied but little. The weakest (from the
20 per cent, mixture) was of 89 per cent., and the strongest was of 90*3 per
cent. All strengths are given by weight, and were calculated by Mendeleef's
tables from the observed specific gravities with suitable temperature correction.
The watery constituent which, not having been evaporated, sometimes looked
a little dirty, was usually redistilled so as to obviate any risk of its purity
being overestimated. In some cases it could not be distinguished from pure
water.
The apparatus illustrates very well the principles of ideal distillation, and
its performance may be regarded as satisfactory. When once the conditions,
as to preliminary heating (if necessary) and as to rate of feed, have been
found for a particular mixture, the continued working is almost selfacting,
or at any rate could be made so without much difficulty ; and it is probable
that separations, otherwise very troublesome, could be easily effected by
use of it.
62
i
282.
NOTE ON THE THEORY OF THE FORTNIGHTLY TIDE.
[Philosophical Magazine, v. pp. 136141, 1903.]
AN adequate calculation d priori of the tide of fortnightly period that
which depends upon the moon's motion in declination would be of great
interest as affording the means, by comparison with observation, of deter
mining the extent to which the solid earth yields to the tidegenerating
force. On the assumption that the fortnightly tide over an absolutely rigid
earth would be sensibly equal to its "equilibrium value," Prof. G. Darwin*
has estimated that the actual rigidity must be at least as great as that
of steel, in accordance with the earlier surmises of Lord Kelvin.
But is an "equilibrium theory" adequate? The known properties of
a system vibrating about a configuration of thoroughly stable equilibrium
would certainly suggest an affirmative answer, when it is considered that
a fortnight is a long period in comparison with those of the more obvious free
oscillations. It is to be remembered, however, that a tidally undisturbed sea
is not in equilibrium, 'and that in virtue of the rotation of the earth the tides
are really oscillations about a condition of steady motion. In Laplace's theory
the rotation of the earth is taken fully into account, but the sea must be
supposed to cover the entire globe, or at any rate to be bounded only by
coasts running all round the globe along parallels of latitude. The resulting
differential equation was not solved by Laplace, who contented himself with
remarking that in virtue of friction the solution for the case of fortnightly and
(still more) semiannual tides could not differ much from the " equilibrium
values."
The sufficiency of Laplace's argument has been questioned, and apparently
with reason, by Darwin f, who accordingly resumed Laplace's differential
equation in which frictional forces are neglected. Taking the case of an
ocean of uniform depth completely covering the globe and following the
indications of Lord Kelvin +, he arrives at a complete evaluation of Laplace's
* Thomson and Tart's Natural Philosophy, 2nd ed. Vol. i. pt. n. p. 400 (1883).
t Proceedings of the Royal Society, Vol. XLI. p. 337 (1886).
J Phil. Mag. Vol. L. p. 280 (1875).
1903] NOTE ON THE THEORY OF THE FORTNIGHTLY TIDE 85
" Oscillation of the First Species." A summary of Darwin's work has been
given by Lamb * from'which the following extracts are taken. The equilibrium
value of the fortnightly tide being
the actual tide for a depth of 7260 feet is found to be
/#'= 1515  10000 ^ + 15153 fj. 4  12120 /*
 2076/i 10 + 0516/i 12  0097/* 14 + 0018^ M  0002 pP,
whence at the poles (jj, = + 1)
= ftf'x154,
and at the equator (//, = 0)
^=^x455.
Again, for a depth of 29040 feet, we get
C/J5T = 2359  TOGO/* 2 + 5898^
 1623/4 6 + 0258/* 8  0026/4' + 0002 /*",
making at the poles f =  IE' x '470,
and at the equator f = \H' x '708.
It appears that with such oceans as we have to deal with the tide thus
calculated is less than half its equilibrium amount.
The large discrepancy here exhibited leads Darwin to doubt whether
" it will ever be possible to evaluate the effective rigidity of the earth's mass
by means of tidal observations."
From the point of view of general mechanical theory, the question at once
arises as to what is the meaning of this considerable deviation of a longperiod
oscillation from its equilibrium value. A satisfactory answer has been
provided by Lambf; and I propose to consider the question further from this
point of view in order to estimate if possible how far an equilibrium theory
may apply to the fortnightly tides of the actual ocean.
The tidal oscillations are included in the general equations of small
vibrations, provided that we retain in the latter the socalled gyrostatic
terms. By a suitable choice of coordinates, as in the usual theory of normal
coordinates, these equations may be reduced to the form
+...=Q 3 ,
CD
in which &., = . (2)
* Hydrodynamics, 210, Cambridge, 1895.
t Hydrodynamics, 196, 198, 207.
86 NOTE ON THE THEORY OF THE FORTNIGHTLY TIDE [282
From these we may fall back upon the case of small oscillations about stable
equilibrium by omitting the terms in $; but in general tidal theory these
terms are to be retained. If the oscillations are free, the quantities Q, repre
senting impressed forces, are to be omitted.
If the coefficients y9 are small, an approximate theory of the free vibrations
may be developed on the lines of Theory of Sound, 102, where there are
supposed to be small dissipative (but no rotatory) terms. For example, the
frequencies are unaltered if we neglect the squares of the yS's. Further, the
next approximation shows that the frequency of the slowest vibration is
diminished by the operation of the 's; or more generally that the effect
of the $'s is to cause the values of the various frequencies to repel one
another*.
To investigate forced vibrations of given period we are to assume that all
the variables are proportional to e i<rt , where a is real. If the period is very
long, a is correspondingly small, and the terms in q and q diminish generally
in importance relatively to the terms in q. In the limit the latter terms
alone survive, and we get
?i = Qi/Ci, ? 2 = &/c 2 , &c., ..................... (3)
which are the " equilibrium values." But, as Prof. Lamb has shown, exceptions
may arise when one or more of the c's vanish. This state of things implies
the possibility of steady motions of disturbance in the absence of impressed
forces. For example, if C? = 0, we have as a solution q z = constant, with
qi =  & 2 &/Cl , q 3 =  &292/C3 , &C.
In illustration Prof. Lamb considers the case of two degrees of freedom, for
which the general equations are
chtfi + C! ?! + &=&, a 2 2 + c 2 ? 2 # = &; ......... (4)
supposing that c 2 = and also that Q 2 = 0, while Qi remaining finite is pro
portional to e i<rt , as usual. We find
, .
so that in the case of a disturbance of very long period, when a approaches
zero,
Since a 2 is positive, <?, is less than its equilibrium value ; and it is accompanied
by a motion of type q t , although there is no extraneous force of the latter
type.
It is clear then that in cases where a steady motion of disturbance is
possible the outcome of an extraneous force of long period may differ greatly
* See Art. 283 below.
1903] NOTE ON THE THEORY OF THE FORTNIGHTLY TIDE 87
from what the equilibrium theory would suggest. It may, however, be
remarked that the particular problem above investigated is rather special
in character. In illustration of this let us suppose that there are three
degrees of freedom, and that c 2 , c a , Q^, Q 3 are evanescent. The equations then
become
+ i<rj3 13 q 3 = Q l ,
+ i0 a q a = 0,
n q l +ip n qt =0;
whence, regard being paid to (2),
<)
When <r= 0, the value of q l reduces to Qj/c,, unless /? a = 0, so that in general
the equilibrium v^alue applies. But this is only so far as regards q t . The
corresponding values of q 2 , q 3 are
q* = qiPa/0*, g8 = ft&,/; .................. (8)
and thus the equilibrium solution, considered as a whole, is finitely departed
from. And a consideration of the general equations (1) shows that it is only
in very special cases that there can be any other outcome when the possibility
of steady motion of disturbance is admitted.
It thus becomes of great importance in tidal theory to ascertain what
steady motions are possible, and this question also has been treated by Lamb
( 207). It may be convenient to repeat his statement. In terms of the
usual polar coordinates Laplace's equations are
(9)
d%_ 1 (d (hu sin 6) d(hv))
di~~^sm0\ W~ r d }'
where u, v are the velocities along and perpendicular to the meridian, f is the
elevation at any point, f the equilibrium value of , a denotes the earth's
radius, n the angular velocity of rotation, and h the depth of the ocean at any
point. To determine the free steady motions, we are to put = as well as
du/dt, dv/dt, dtydt. Thus
_ __ _# _ 9 .
2nasi
and
If h sec 6 be constant, (13) is satisfied identically. In any other case a
restriction is imposed upon f. If h be constant or a function of the latitude
88 NOTE ON THE THEORY OF THE FORTNIGHTLY TIDE [282
only, must be independent of &>; in other words the elevation must be
sj'mmetrical about the polar axis. In correspondence therewith u must
be zero and v constant along each parallel of latitude.
In the application to an ocean completely covering the earth, such as
is considered in Darwin's solution, the above conditions are easily satisfied,
and the free steady motions, thus shown to be possible, explain the large
deviation of the calculated fortnightly tide from the equilibrium value. What
does not appear to have been sufficiently recognized is the extent to which this
state of things must be disturbed by the limitations of the actual ocean.
Since v must be constant along every parallel of latitude, it follows that
a single barrier extending from pole to pole would suffice to render impossible
all steady motion ; and when this condition is secured a tide of sufficiently
long period cannot deviate from its equilibrium value. Now the actual state
of things corresponds more nearly to the latter than to the former ideal.
From the north pole to Cape Horn the barriers exist ; and thus it is only
in the region south of Cape Horn that the circulating steady motion can
establish itself. It would seem that this restricted and not wholly un
obstructed area would fail to disturb greatly the state of things that would
prevail, were every parallel of latitude barred. If this conclusion be admitted,
the theoretical fortnightly tide will not differ materially from its equilibrium
value, and Darwin's former calculation as to the earth's rigidity will regain its
significance.
Some caution is required in estimating the weight of the argument above
adduced. Though there were no free disturbance possible of infinitely long
period, it would come to the same, or to a worse, thing if free periods existed
comparable with that of the forces, which is itself by hypothesis a long period.
On this account a blocking of every parallel of latitude by small detached
islands would not suffice, although meeting the theoretical requirement of the
limiting case.
It would serve as a check and be otherwise interesting if it were possible
to calculate the fortnightly tide for an ocean of uniform depth bounded by two
meridians. The solution must differ widely from that appropriate to an
unlimited ocean ; but, although the conditions are apparently simple, it does
not seem to be attainable by Laplace's methods. A similar solution for the
semidiurnal tide would be interesting for other reasons.
In any case I think that observations and reductions of the fortnightly
tide should be pursued. Observation is competent to determine not merely
the general magnitude of the tide but the law as dependent upon latitude and
longitude. Should the observed law conform to that of the equilibrium theory,
it would go a long way to verify d posteriori the applicability of this theory to
the circumstances of the case.
283.
ON THE FREE VIBRATIONS OF SYSTEMS AFFECTED WITH
SMALL ROTATORY TERMS.
[Philosophical Magazine, v. pp. 293297, 1903.]
BY a suitable choice of coordinates the expressions for the kinetic and
potential energies of the system may be reduced to the forms
..., ........................ (1)
............................ (2)
If there be no dissipative forces, the equations of free vibration are
... =
where /3 r8 = ftgr ', and under the restriction contemplated all the quantities
ft are small.
If in equations (3) we suppose that the whole motion is proportional
to e,
(c,  orX) fc + ur M & + ^&3& + =
and it is known that whatever may be the magnitudes of the yS's, the values
of the o's are real. The frequencies are equal to o/2?r.
If there were no rotatory terms, the above system of equations would
be satisfied by supposing one coordinate <j> r to vary suitably, while the
remaining coordinates vanish. In the actual case there will be in general a
corresponding solution in which the value of any other coordinate (j> t will be
small relatively to < r .
Hence if we omit the terms of the second order in /3, the rth equation
becomes
c r ov 2 a r < r = 0, (5)
90 ON THE FREE VIBRATIONS OF SYSTEMS [283
from which we see that o> is approximately the same as if there were no
rotatory terms.
From the sth equation we obtain
(c.  <r r 'a.) <f>, + io>& r < r = 0,
terms of the second order being omitted ; whence
A .A *>&r *>&,.
* : *^^~W^7)'
where on the right the values of o>, <r g from the first approximation (5) may
be used. This equation determines the altered type of vibration ; and we see
that the coordinates <f> g are in the same phase, but that this phase differs by a
quarter period from the phase of (f> r .
We have seen that when the rotatory terms are small, the value of o> may
be calculated approximately without allowance for the change of type ; but by
means of (6) we may obtain a still closer approximation, in which the squares
of the #'s are retained. The rth equation (4) gives
Since the squares of the as are positive, as well as a r , a,, c r , we recognize
that the effect of y8 rg is to increase <r r 2 if o> 2 be already greater than <r 2 , and to
diminish it if it be already the smaller. Under the influence of the 's the
o's may be considered to repel one another. If the smallest value of <r r
be finite, it will be lowered by the action of the rotatory terms*.
The vigour of the repulsion increases as the difference between <r r and <r g
diminishes. If <r r and <r s are equal, the formulae (6), (7) break down, unless
indeed = 0. It is clear that the original assumption that < is small
relatively to <f> r fails in this case, and the reason is not far to seek. When two
normal modes have exactly the same frequency, they may be combined in any
proportions without alteration of frequency, and the combination is as much
entitled to be considered normal as its constituents. But the smallest
alteration in the system will in general render the normal modes determinate ;
and there is no reason why the modes thus determined should not differ
finitely from those originally chosen.
A simple example is afforded by a circular membrane vibrating so that one
diameter is nodal. When all is symmetrical, any diameter may be chosen to
be nodal ; but if a small excentric load be attached, the nodal diameter must
either itself pass through the load or be perpendicular to the diameter that
This conclusion was given in Phil. Mag. v. p. 188 (1903) [p. 86 above], but without some
reservations presently to be discussed. Similar reservations are called for in Theory of Sound,
90, 102.
1903] AFFECTED WITH SMALL ROTATORY TERMS 91
does so (Theory of Sound, 208). Under the influence of the load the two
originally coincident frequencies separate.
In considering the modifications required when equal frequencies occur, it
may suffice to limit ourselves to the case where two normal modes only have
originally the same frequency, and we will suppose that these are the first and
second. Accordingly, the coincidence being supposed to be exact,
cr 2 ............................... (8)
The relation between fa and fa and the altered frequencies are to be
obtained from the first two equations of (3), in which the terms in fa, fa, &c.
are at first neglected as being of the second order of small quantities. Thus
(d  o 2 aO fa + ifffrtfa = 0, (c,  <7 2 a 2 ) fa  i<rft u fa = ....... (9)
in which the two admissible values of tr 2 are given by
(c 1 a 1 <r 2 )(c 2 o 20  2 )  2 A 2 2 = ................... (10)
If one of the factors of the first term, e.g. the second, be finite, $ 12 2 may
be neglected and a value of o 2 is found by equating the first factor to zero ;
but in the present case both factors are small together. On writing o for a
in the small term, (10) becomes
(o 2 oo 2 ) 2 =(r 2 ^ 12 2 /a 1 a 2 , ........................ (11)
so that
<J 2 o 2 =o /3 12 Maia 2 ), ..................... (12)
or
(13)
The disturbance of the frequency from its original value is now of the first
order in /3 12 , and one frequency is raised and the other depressed by the same
amount.
As regards the ratios in which fa, fa enter into the new normal modes, we
have from (9)
 1= ^ )= . ...................
fa
From (14) we see that in the new normal vibrations the two original
coordinates are combined so as to be in quadrature with one another, and in
such proportion that the energies of the constituent motions are equal.
The value of any other coordinate fa accompanying fa and fa in vibration
er is obtained from the sth equation (4). Thus, squares of /8*s being neglected,
(c t o*a l )fa + i<r0 n <l> l + iff/3* fa=0, ............... (15)
in which, if we please, we may substitute for fa in terms of fa from (14).
92 SYSTEMS AFFECTED WITH SMALL ROTATORY TERMS [283
For the second approximation to a we get from (15) and the two first
equations (4)
in which the summation extends to all values of * other than 1 and 2. In the
coefficients of the second terms it is to be observed that /3 12 = /3<u, and that
', so that the determinant of the equations becomes
terms of the fourth order in being omitted. In (16) Cj o 2 aj, Cjcr'aa
are each of the order ft. Correct to the third order we obtain with the
use of (12)
whence
c,<7 2 a,
In (18) y3 12 is supposed to be of not higher order of small quantities than
& &* For example, we are not at liberty to put $ 12 = 0.
In the above we have considered the modification introduced by the
y9's into a vibration which when undisturbed is one of two with equal
frequencies. If the type of vibration under consideration be one of those
whose frequency is not repeated, the original formulae (6), (7) undergo no
essential modification.
In the following paper some of the principles of the present are applied to
a hydrodynamical example.
284.
ON THE VIBRATIONS OF A RECTANGULAR SHEET
OF ROTATING LIQUID.
[Philosophical Magazine, v. pp. 297301, 1903.]
THE problem of the free vibrations of a rotating sheet of gravitating
liquid of small uniform depth has been solved in the case where the boundary
is circular*. When the boundary is rectangular, the difficulty of a complete
solution is much greater ; but I have thought that it would be of interest to
obtain a partial solution, applicable when the angular velocity of rotation is
small.
If be the elevation, u, v the component velocities of the relative motion
at any point, the equations of free vibration, when these quantities are pro
portional to e i<7t , are
i<ru 2nv = g d/dx, icrv + 2nu =  g d/dy, ......... (1)
and
in which n denotes the angular velocity of rotation, h the depth of the water
(as rotating), and g the acceleration of gravity. The boundary walls will be
supposed to be situated at x = + ^TT, y y^.
When n is evanescent, one of the principal vibrations is represented by
u=cosx, # = 0; ........................... (3)
and is proportional to sin x, so that
<r* = gh ..................................... (4)
This determines the frequency when n = 0. And since by symmetry a positive
and a negative n must influence the frequency alike, we conclude that (4) still
holds so long as n* is neglected. Thus to our order of approximation the
* Kelvin, Phil. Mag. Aug. 1880 ; Lamb, Hydrodynamics, 200, 202, 203.
94 ON THE VIBRATIONS OF A RECTANGULAR SHEET [284
frequency is uninfluenced by the rotation, and the problem is reduced to
finding the effect of the rotation upon that mode of vibration to which (3) is
assumed to be a first approximation. The equation for f is at the same time
reduced to
Since v is itself of the order n, the first of equations (1) shows that u, as well
as f, satisfies (5).
Taking u and v as given in (3) and the corresponding as the first
approximation, we add terms u', v, ', proportional to n, whose forms are to
be determined from the equations
i<ru'=gdldx, .................................... (6)
tW =  g d'/dy 2n cos x, ........................ (7)
(d*/dx* + d*/dy* + l)(Z',u',v') = Q ................... (8)
They represent in fact a motion which would be possible in the absence
of rotation under forces parallel to v and proportional to cos a;. This con
sideration shows that u' is an odd function of both x and y, and v' an even
function. If we assume
, ..................... (9)
the boundary condition to be satisfied at x= ^ir is provided for, whatever
functions of y A 2 , A t> &c. may be. If we eliminate " from (6), (7), we find
dv' du' 2n . dA z dA t 2n .
j = j + sin x = j sin 2# + == sin 4# + . . . + sin a; :
dx dy 10 dy dy 10
or, on integration,
dA /dy being the constant of integration. In (10) the A'a are to be so chosen
that v' = when y=y* for all values of x between  TT and + ^TT.
From (8) we see that A,, A t , &c. are to be taken so as to satisfy
^".O. ^'^. = 0, & o.,
or, since the A's are odd functions of y,
), A 4 = ^sinh (v!5. y), &c.
Also A = B sin y.
In these equations the B's are absolute constants.
1903] OF ROTATING LIQUID 95
The boundary conditions at y = y l now take the form
= cos x B cos y l
V3 . B z cosh (V3 . y x ) . cos 2#
\/15.B 4 cosh(\/15 . y^.cos 4# ... , (11)
which can be satisfied if cos x be expressed between the limits of x in the
series
cos#= (7 + (7 8 cos 2# + C 4 cos 4# + (12)
By Fourier's theorem we find that (12) holds between x = TT and
The 5's are thus determined by (11), and we get
_ 2m 2 siny . _ 2m _2_ _4_ sinh ( V3 . y)
a TT cos yx ' a \/3 3?r cosh (V3 . y^)
A = 2m 2m 4(l) TO+1 sinh (y V4m 2  1)
2m ~ a /s /(4m 2  1) (4m 2  1) TT cosh (y x V4m a  1) '
Hence, finally, for the complete values of u and v to this order of approxi
mation
sin
TT cos y x
......
The limiting values of x have been supposed, for the sake of brevity,
to be + \TT. If we denote them by sc l) we are to replace x, y, y^ in (14), (15)
by \irx\x^ \iry\sK^ \iry^x^ At the same time (4) becomes
As was to be expected, the small terms in (14), (15) are in quadrature with
the principal term. The success of the approximation requires that the
frequency of revolution be small in comparison with that of vibration.
If y l be such that cos ( TT y^) vanishes, or even becomes very small,
the solution expressed in (14), (15) fails. This happens, for example, when
the boundary is square, so that y l = x 1 . The inference is that the assumed
solution (3) does not, or rather does not continue to, represent the facts of
the case as a first approximation.
96 ON THE VIBRATIONS OF A RECTANGULAR SHEET [284
From the principles explained in the previous paper, or independently,
it is evident that in the case of the square (3) must be replaced by
u = cosx, v=icosy, (17)
corresponding to which
(18)
These values satisfy all the conditions when there is no rotation, and
(T O = ^(gh), as in (4). For the second approximation we retain these terms,
adding to them u', v, ', which are to be treated as small. So far, the
procedure is the same as in the formation of (6), (7) ; but now we must be
prepared for an alteration of a from its initial value o by a quantity of the
first order. Hence, with neglect of w a ,
i (a <r ) cos x + i<r u' + 2m cosy=g d^'jdx .......... (19)
T (<r <T O ) cos y + i<r v' + 2n cos x = g d^'/dy .......... (20)
These equations are the same as would apply in the absence of rotation if we
suppose impressed forces to act parallel to u and v proportional to
i(<r o )cos# + 2ni cosy, ........................ (21)
+ (o <r c ) cos y + In cos x, ........................ (22)
respectively.
The complete solution of (19), (20) to the first order of n would lead
to rather long expressions. The point of greatest interest is the alteration
of frequency, and this can perhaps be most easily treated by a simple
mechanical consideration. The forces given in (21), (22) must be such as
not wholly to disturb the initial motion (17) with which they synchronize.
Accordingly (21) must be free from a component capable of stimulating
a vibration similar to u = cos x, and in like manner (22) must be incapable of
stimulating a motion similar to v = cosy. The necessary conditions are
// cos x {(<r <T O ) cos x + 2w cos y}dxdy= 0,
//cos y {(<r <r ) cos y T 2n cos x} dxdy = 0,
the integration being taken over the whole area. On account of the symmetry
the two conditions coincide ; and it is sufficient to integrate for x and y
between the limits and \ir. Thus
(<r  o ) . TT . ITT = 2n . 1 . 1,
so that
(23)
Since n and <r are of the same dimensions, this result holds good, whatever
may be the side of the square.
1903] OF ROTATING LIQUID 97
It may be of interest, and serve as a confirmation of the above procedure,
to mention that when applied to the principal vibration in a rotating circular
trough it gives
'"i' ........................... (24)
where z t is the first root of Ji (z) = 0, equal to 1841, so that
< 25 >
An accordant result may be deduced from the analysis given by Lamb, 203,
by putting s=l, and taking account of the properties of the function Jj.
The corresponding value of is given by
infl} ...................... (26)
[1911. This subject is pursued in "Notes respecting Tidal Oscillations
upon a Rotating Globe," Proceedings Royal Society, A, Vol. LXXXII. p. 448,
1909.]
B. V,
285.
ON THE SPECTRUM OF AN IRREGULAR DISTURBANCE.
[Philosophical Magazine, v. pp. 238243, 1903.]
IN my paper " On the Character of the Complete Radiation at a given
Temperature"*, I have traced the consequences of supposing white light to
consist of a random aggregation of impulses of certain specified types, and
have shown how to calculate the distribution of energy in the resulting
spectrum. The argument applies, of course, to all vibrations capable of
propagation along a line, and it is convenient to fix the ideas upon the
transverse vibrations of a stretched string. Suppose that this is initially
at rest in its equilibrium position and that velocities represented by $(x)
are communicated to the various parts. The whole energy is proportional
/foo
to I {<f> (x)} z dx ; and it is desired to know how this energy is distributed
J oo
among the various components into which the disturbance may be analysed.
By Fourier's theorem,
(1)
where
/, (k) = [ + cos kv <f> (v) dv, / 2 (k) = f + * siu Jb < (v) dv ....... (2)
J oo J oo
It was shown that the desired information is contained in the formula
] dfc ............ (3)
As an example, we may take an impulse localized in the neighbourhood of a
point, and represented by
*(*)** .................................. (4)
Equation (1) becomes
e*** = J__ r gi* C o 8 kx dk, .. ... .(5)
CV/7T JO
* Phil. Mag. xxvii. p. 460 (1889); Scientific Paper*, in. p. 268.
1.903] ON THE SPECTRUM OF AN IRREGULAR DISTURBANCE 99
while for the distribution of energy in the spectrum by (3)
P" e *cw dx= ^ f%K*f<fib (6)
" If an infinite number of impulses, similar (but not necessarily equal) to
(4) and of arbitrary sign, be distributed at random over the whole range from
 oo to + oo , the intensity of the resultant for an absolutely definite value of k
would be indeterminate. Only the probabilities of various resultants could be
assigned. And if the value of k were changed, by however little, the resultant
would again be indeterminate. Within the smallest assignable range of k
there is room for an infinite number of independent combinations. We are
thus concerned only with an average, and the intensity of each component
may be taken to be proportional to the total number of impulses (if equal)
without regard to their phaserelations. In the aggregate vibration, the law
according to which the energy is distributed is still for all practical purposes
that expressed by (6)."
The factor e~ c2a;2 in the impulse was introduced in order to obviate discon
tinuity. The larger c is supposed to be, the more highly localized is the
impulse. If we suppose c to become infinite, the impulse is infinitely narrow,
and the disturbances at neighbouring points, however close, become inde
pendent of one another. It would seem therefore from (6) that in the
spectrum of an absolutely irregular disturbance (where the ordinates of the
representative curve are independent at all points) the energy between k and
k + dfc is proportional to dk simply, or that the energy curve is a straight line
when k is taken as abscissa. If we take the wavelength X, (to which k is
reciprocal) as abscissa, the ordinate of the energy curve would be as X" 2 .
The simple manner in which dk occurs in Fourier's theorem has always
led me to favour the choice of k, rather than of \, as independent variable.
This may be a matter of convenience or of individual preference ; but some
thing more important is involved in the alternative of whether the energy
of absolutely arbitrary disturbance is proportional to dk or to d\. In
Prof. Schuster's very important application of optical methods to the problems
of meteorology, which seems to promise a revolution in that and kindred
sciences, the latter is the conclusion arrived at. " Absolute irregularity would
show itself by an energycurve which is independent of the wavelength ;
i.e.. a straight line when the energy and wavelength or period are taken as
rectangular coordinates..."*. It is possible that the discrepancy may depend
upon some ambiguity ; but in any case I have thought that it would not be
amiss to reconsider the question, using a different and more elementary
method.
For this purpose we will regard the string as fixed at the two points
* "The Periodogram of Magnetic Declination, &c." Camb. Phil. Trans, xvm. p. 108 (1899).
72
100 ON THE SPECTRUM OF AN IRREGULAR DISTURBANCE [285
x = and x=L The possible vibrations are then confined to the wellknown
" harmonics," and k is limited to an infinite series of detached values forming
an arithmetical progression. The general value of the displacement y at
time / is
^ . STTX ( . STTdt n . STTat\
t/ = Ssmj 1 4, cos j + .#gSin j I, ............... (7)
in which a is the velocity of propagation and s is one of the series 1, 2, 3
From (7) the constant total energy (T+ V) is readily calculated. Thus
(Theory of Sound, 128) if M denote the whole mass, T, the period of
component s,
T+V=^M.^ As ^ 9 Bt3 , ........................ (8)
an equation which gives the distribution of energy among the various modes.
The initial values of y and y are
_, . . sirx . tra ^ n . STTX
2/o=24,sm
whence
2
(9)
If we suppose that y = Q throughout and that y is finite only in the
neighbourhood of x = f, we have A s = 0, and
* ~~ TTOS 8m ~T ' ' '
where Y=fy dx. The energy in the various modes being proportional to
BS/T,*, or to
in which s 2 T, a = Tj 2 , is thus independent of s except for the factor sin 2 (str^fl).
And even this limited dependence on s disappears if we take the mean with
respect to f. We may conclude that in the mean the energy of every mode is
the same ; and since the modes are uniformly spaced with respect to their
frequency (proportional to s) and not with respect to their period or wave
length, this result corresponds with a constant ordinate of the energy curve
when k is taken as abscissa.
It is to be noted that the above corresponds to an arbitrary localized
velocity. We shall obtain a higher and perhaps objectionable degree of
discontinuity, if we make a similar supposition with respect to the displace
ment. Setting in (9) y = throughout and y = except in the neighbourhood
of we get B. = and
1903] ON THE SPECTRUM OF AN IRREGULAR DISTURBANCE 101
where Y 1 =fy dx. By (8) the mean energy in the various modes is now
proportional to l/r g 2 or to s 2 . When I is made infinite, so that r g may be
treated as continuous, we have an energy curve in which the ordinate is
proportional to s 2 or k 2 , k being abscissa.
We may sum up by saying that if the velocity curve is arbitrary at every
point the energy between k and k + dk varies as dk, but if the displacement
be arbitrary the energy over the same range varies as k*dk.
In Schuster's Periodogram, as applied to meteorology, the conception of
energy does not necessarily enter, and the definitions may be made' at
pleasure. But unless some strong argument should appear to the contrary,
it would be well to follow optical (or rather mechanical) analogy, and this, if
I understand him, Schuster professes to do. If the energy associated with
the curve < (x) to be analysed is represented by /{< (a;)} 2 dx, <f> (x) must be
assimilated to the velocity and not to the displacement of a stretched string.
We have seen that when (j>(x) is arbitrary at all points the ordinate of
the energy curve is independent of k. In the curves with which we are
concerned in meteorology the values of $(x) at neighbouring points are
related, being influenced by the same accidental causes. But at sufficiently
distant points the values of $ (x) will be independent. Equation (6) suggests
that in such cases the ordinate of the energy curve (k being abscissa) will
tend to become constant when k is small enough.
Another illustration of the application of Fourier's theorem to the analysis
of irregular curves may be drawn from the optical theory of gratings. For
this purpose we imagine the aperture of a telescope to be reduced to a
horizontal strip bounded below by a straight edge and above by the curve
to be analysed, such as might be provided by a selfregistering tidegauge.
Any periodicities in the curve will then exhibit themselves by bright lines
in the image of a source of homogeneous light, corresponding to the usual
diffraction spectra of the various orders. An aperture of the kind required
may be obtained by holding the edge of a straight lath against the teeth of
a handsaw. When the combination is held square in front of the telescope,
we have spectra corresponding to the number of teeth. When the aperture is
inclined, not only do the previously existing spectra open out, but new spectra
appear in intermediate positions. These depend upon the fact that the period
now involves a sequence of two teeth inasmuch as alternate teeth are bent in
opposite directions out of the general plane.
The theory of diffraction* shows that the method is rigorous when the
source of light is a point and when we consider the illumination at those
points of the focal plane which lie upon the horizontal axis (parallel to the
straight edge of the aperture).
* See, for example, "Wave Theory of Light," Encyc. Brit., Scientific Papers, in. pp. 80, 87.
Make = 0.
102 ON THE SPECTRUM OP AN IRREGULAR DISTURBANCE [285
In order to illustrate the matter further, Mr Gordon constructed an
aperture (cut from writingpaper) in which the curved boundary* had the
equation
y = sin 2# + sin (3# + TT).
The complete period was about half an inch and the maximum ordinate about
one inch. The aperture was placed in front of a 3inch telescope provided
with a highpower eyepiece. When desired, the plane of the aperture could
be considerably sloped so as to bring more periods into action and increase
the dispersion.
The light employed was from a paraffin lampf, and it was convenient to
limit it by slits. Of these the first was vertical, as in ordinary spectrum work,
and it was crossed by another so that at pleasure a linear or a point source
could be used. In the latter case the spectrum observed agreed with
expectation. Subdued spectra of the first order (corresponding to the
complete period) and traces of the fourth and fifth orders were indeed present,
as well as the second and third orders alone represented in the aperture
curve. But along the horizontal axis of the diffraction pattern these subsidiary
spectra vanished ; so that the absence of all components, except the second
and third, from the aperturecurve could be inferred from the observation.
It will be evident from what has already been said that confusion arises
when the pointsource is replaced by a linear one ; and this is what theory
would lead us to expect. In a diffractiongrating, as usually constructed,
where all the lines are of equal length, the spectra are of the same character
whether the source be elongated, or not, in the vertical direction ; but it is
otherwise here. The inadmissibility of a linear source and the necessity for
limiting the observation to the axis seriously diminish the prospect of making
this method a practical one for the discovery of unknown periods in curves
registering a meteorological or similar phenomena; but the fact that the
analysis can be made at all in this way, without any calculation, is at least
curious and instructive.
It may be added that a similar method is applicable when the phenomena
to be analysed occur discontinuously. Thus if the occurrence of earthquakes
be recorded by ruling fine vertical lines of given length with abscissa? propor
tional to time, so as to constitute a grating, the positions of the bright places
in the resulting spectrum will represent the periodicities that may be present
in the time distribution of the earthquakes. And in this case the use of a
linear source of light, from which to form the spectrum, is admissible.
[1910. Compare Schuster, Phil. Mag. v. p. 344, 1903.]
* Figured in Thomson and Tait's Natural Philosophy, 62.
f Doubtless a more powerful source would be better.
286.
CONSIDERATIONS RESPECTING THE COMBUSTION OF
MODERN PROPELLANTS IN CLOSED VESSELS AND IN
GUNS.
[Minutes of Explosives Committee, 1903.]
Closed Vessels.
THE rate of combustion (if the term may be allowed) per unit of surface
is assumed to be some function f(p) of the pressure at the moment (t) under
consideration. This assumption does not imply that the pressure rather, for
instance, than the temperature, is the governing circumstance, but merely
that the pressure sufficiently determines the state of affairs. To a high pressure
will, of necessity, correspond a high temperature.
The case of a tubular Cordite of annular section is the simplest. If r
denote the external and r the internal radius at time t,
dr/dt = dr'/dt=f(p) (1)
Thus, r + r remains constant, and with it the surface at any time exposed.
Accordingly the rate of total combustion depends only upon the pressure, and
this simplifies the question considerably.
In addition to (1) another relation is required. It is usual to assume that
the pressure is proportional to the quantity of propellant already burned.
This supposition will be made in some of the calculations which follow;
but it may be remarked that it cannot be accurately true. In ordinary
practice, the pressure is atmospheric (and not evanescent) when the combustion
begins ; and, as the combustion progresses, the temperature rises. Even if
it were otherwise, Boyle's law would not apply strictly to the high pressures
here involved.
On these accounts the pressure would be greater than it is estimated to
be. On the other hand, the increase in space available for the gases as the
104 CONSIDERATIONS RESPECTING THE COMBUSTION OP [286
solid propellant disappears tells in the other direction, so that there is here
a tendency to compensation. Upon the whole, the simple law may not be far
from the mark, so long as the temperature does not greatly vary.
In the present case it suffices to assume proportionality between the
increment of pressure and the rate of burning, or
dp/dtAf(p), (2)
where A is constant in each experiment, but may vary in different experi
ments, if the weight of the charge, or the aggregate surface, or the space, be
altered.
For (2) the relation between pressure and time is determined by a simple
integration. Or conversely, if this relation be known from experiment,
(2) determines the form of /(/>). It is evident that for this purpose the use
of tubular Cordite offers advantages.
If f(p) = p n , where n is positive, (2) gives
p=A(ln)t + C, (3)
C being a constant of integration.
If n be less than unity, (3) may be written in the form
^ = ^(ln)(<< ), (4)
t being the time at which the pressure is zero. This law obtains so long as
the Cordite burns with constant surface. In the ideal case the walls of the
tubes become everywhere infinitely thin and disappear simultaneously, the
surface exposed falling discontinuously from a constant finite value to zero.
From (4), if the time occupied in the rise of pressure from zero to the
maximum P be t ly
P>" = A(ln)t,;
so that if the time be measured from the moment of zero pressure
(p/py^ (5)
For example, if, as has sometimes been supposed, n = ,
P/P = (t/W (6)
the pressure being proportional to the square of the time which has elapsed
from the commencement of the burning.
If in (3), n be greater than unity, the pressure cannot be supposed to be
zero. The meaning of this is that under such a law the burning could
not commence from a zero of pressure. In order to obtain a practical result,
we should have to take into account an initial pressure, whether atmospheric
or (in virtue of a primer) exceeding atmosphere. These initial pressures
being relatively very small, we may expect the commencement of the burning
to be slow and uncertain.
1903] MODERN PROPELLANTS IN CLOSED VESSELS AND IN GUNS 105
The case n = 1, probably somewhat closely approached in practice, is
critical. Equation (3) is replaced by
logp = At+C, ................................. (7)
or, p = De At ..................................... (8)
signifying that the pressure rises according to the law of compound interest.
Here again the pressure cannot rise at all from zero, and from an initial
atmospheric pressure the rise would be comparatively slow.
If the Cordite be solid instead of tubular, the surface continually diminishes
as the combustion proceeds. The conclusion that if n be not less than unity,
the pressure is incapable of rising from zero still holds good. If n= , the
calculation is easily made ; and it appears that if fc, be the total time of
burning, (6) is replaced by
(9)
The problem of finding the relation between R (the initial radius of solid
Cordite) and 1^, R 2 (the inner and outer radii of tubular Cordite) in order
that the times of burning of equal weights in the same vessel may be equal,
has been solved by Lieutenant Wright, R.N. (O.C. Minute 46,198/19. 10. 98).
The conclusion is
R 2 R 1 .................................. (10)
But, as has already been remarked, the law of the rise of pressure is quite
different in the two cases. Thus, even though (10) be satisfied, a pressure
nearly equal to the common maximum is reached earlier in the case of the
solid than in that of the tubular Cordite, notwithstanding that equal times
are needed for the final maximum. This is a natural consequence of the
circumstances that the surface of the solid Cordite diminishes gradually to
zero, while that of the tubular remains constant until the last moment.
Combustion in Guns.
When we consider the case of a gun in place of a closed vessel, the
question is further complicated by the increasing space becoming available
for the gases of combustion as the shot advances. Of special importance is
the relation between the pressure and the travel (x) of the shot. The total
work done is represented by j pdx, so that if the muzzlevelocity of a given
shot from a given gun is prescribed, the mean pressure (estimated with regard
to distance} is thereby determined. A lowered chamber, or maximum, pressure
of necessity involves a raised pressure at some other part of the course of
the shot.
106 CONSIDERATIONS RESPECTING THE COMBUSTION OF [286
The muzzle velocity, giving the mean pressure, and the maximum pressure
(as determined by crusher gauges) are two very important data, but they by
no means exhaust all that it is desirable to know. More telling are such
investigations as those of Sir A. Noble where, by delicate chronoscopes, the
shot is timed as it passes various points of the bore. From these times the
accelerations may be deduced, to which approximately the pressures are
proportional. In this way we may find the strength necessary in the various
parts of the gun when a given shot is fired with a given propellant.
But, for the purposes of the Explosives Committee, the converse problem
seems the more important. Indeed, I am strongly impressed with the
feeling that the natural order of procedure is first to determine, either from
the strength of the gun, or for other reasons, what pressures it is desired to
have in other words, to define p as a function of x and then to investigate,
by theory and experiment, how nearly the desired law of pressure can be
obtained by varying the nature and form of the propellant. Of course,
experience may show that an improvement is attainable by an altered design
of gun and an altered law of pressure, but at a given time the practical
problem is to fit the propellant to the gun.
The question as to what law of pressure is desirable must be decided by
experts. Economy of propellant suggests that the combustion should be
finished at a comparatively early stage, say before the shot has travelled
more than one quarter of its course. But at the present time the object
appears rather to be to get the most out of the gun. If the maximum
pressure is laid down, the aim would then be to reach this pressure early,
and to maintain it without much drop over the strong part of the gun.
Subsequently the pressure should fall somewhat rapidly to that considered
desirable at the muzzle. From the curves that I have seen, I should
suppose that there is not much to complain of in the manner of reaching the
maximum, but that the maximum itself is insufficiently maintained. It
cannot be right that the high pressure should operate over a very small part
only of the travel of the shot.
These considerations suggest the problem of finding whether it is
possible to maintain the highest pressure over a finite travel of the shot, and
[ believe that an answer can be given sufficient to afford practical guidance,
although no doubt leaving much to be desired from a theoretical point of
view.
Although some deduction for friction and perhaps for other complications
would be proper, it may suffice to assume that the momentary acceleration is
proportional to the pressure operative behind the shot, so that
(11)
In (11) the changes of x represent the travel of the shot, but the origin of x
may be chosen at convenience.
1903J MODERN PROPELLANTS IN CLOSED VESSELS AND IN GUNS 107
The rate of total combustion at time t may be equated to Sf(p), when
8 is the (momentary) surface of the propellant, and f(p) denotes, as in (1),
a suitable function of the pressure. The pressure itself depends upon the
total amount of the gases, i.e., upon the quantity of propellant already burned,
upon the volume, and upon the temperature. The changes of temperature
during the course of the combustion are certainly not insignificant, and they
would have to be allowed for if the object were a precise quantitative estimate.
But so long as the pressure is nearly constant, they must be unimportant,
and this suffices for the immediate purpose. In the general problem,
especially during the final expansion towards the muzzle of the gun, a
different treatment might be necessary.
Neglecting then the change of temperature, we have to consider the
effect of volume. To the original volume in the chamber we have, of course,
to add that provided by the forward displacement of the shot from its seat.
But a further question arises as to what is to be regarded as the chamber
volume. As the propellant burns away, the actual gasvolume is increased
independently of the motion of the shot. But in consequence of the deviation
from Boyle's law at high pressures, we shall obtain the closest approach to
the facts if we reckon the chamber volume throughout as that part originally
unoccupied by propellant. On the whole the volume may be considered to
be represented by ac, where x is measured, not from the initial position of the
base of the shot, but from a point further behind. The initial value of x may
be denoted by # .
In accordance with the suppositions already detailed, the product (px)
of the pressure and volume represents the total quantity of gas, and its
differential coefficient with respect to time is the rate of total combustion.
Hence as our second equation we may take
~ ........................... (12)
when B is some constant; or in a form more telling for our immediate
purpose
In (13) dxjdt represents the instantaneous velocity of the shot.
We are now in a position to examine effectively what is implied in a
pressurecurve such that over a finite range p is independent of x, so that
this part of the curve BG reduces to a straight line parallel to the axis of x
(fig. 1). If in (13) p is constant, dpjdx is zero, and we see that S must be
proportional to dx/dt, independently altogether of the form of f. Hence, in
order that the maximum pressure may be maintained, the operative surface
of the propellant must increase, and that somewhat rapidly, so as to remain
108 CONSIDERATIONS RESPECTING THE COMBUSTION OF [286
proportional to the increasing velocity of the shot. This conclusion appears
to be of some importance, and it shows how hopeless must be the attempt to
obtain the desired feature in the pressurecurve so long as we limit our
selves to solid Cordite, of which the surface is all the time diminishing.
Even if we substitute simple tubular Cordite we merely maintain the surface,
and it still remains impossible, according to our equation, to keep the
pressure constant.
Fig. l.
If it be desired to approximate at all closely to the condition of a main
tained maximum pressure, other means must be resorted to. We might
imagine fresh surfaces brought into play by the removal of an inactive
coating; but the most natural device would seem to be the adoption of
;i multitubular form of propellant. When the section is simply annular, the
total surtiuv is maintained constant because the internal surface increases
at the same rate as the external surface decreases. It is only necessary to
provide two or more perforations in order that the gain of internal surface
may exceed the loss of external surface.
The theory of the gain or loss of surface during combustion of a pro
pellant burning practically in two dimensions, is more simple than might
at first be supposed, at any rate so far as the earlier stages are concerned.
If the boundary be circular, the rate of change of surface is the same what
ever be the radius of the circle. It is on this principle that an inner circular
surface always balances an external circular surface. Hence if there be two
circular perforations through a round stick, the net gain of surface is at the
same rate as the loss of surface which ensues when there are no perforations.
And by increasing the number of perforations the gain may be made aa
rapid as we please.
But it is not necessary to limit these statements to circular boundaries.
So long as the boundary remains oval, i.e., of one curvature throughout, the
result is the same, whether it be circular or not. For example, the gain at an
internal circular perforation is the same as the loss at an external elliptical
1903]
MODERN PROPELLANTS IN CLOSED VESSELS AND IN GUNS
109
boundary, and this form of external boundary might, perhaps, be recom
mended when it is desired to work with two perforations. It must not be
overlooked, however, that, in the final stage, a distinction will arise between
the behaviour of a simple annular and a multitubular form. In the former
case the surface disappears suddenly, while the latter must involve the
separation of portions whose burning will be more like that of threads of
unperforated material.
It may be argued that we do not desire an absolute maintenance of the
highest pressure over any finite portion of the curve. Perhaps the require
ments of the case may be met by a propellant of which the surface remains
constant over the space in question. The pressure must indeed fall, but
possibly not faster than is admissible. This state of things may be sym
bolized by the curve (fig. 2), in which BO is shown straight, though no
longer horizontal. In this case, as in the former, we have still to consider
what is implied in the corners, or places of strong curvature, desirable at
B and G, and the required information can be obtained from (13).
Fig. 2.
In passing through B the value of dp/dx drops suddenly from a large
positive value to zero (fig. 1), or to a negative value (fig. 2), while there is
no sudden change in the values of p or x or dx/dt. Hence at the point B
there must be a sudden drop in the value of S; and again, for the same
reason, there must be another sudden drop in S at the point C. The
requirements for S are thus somewhat peculiar. At B there is to be a
sudden drop. From B to G, S should increase, or at any rate not diminish,
and again at G there is to be a sudden drop. The last drop is naturally
obtained by the burning out of a simple tubular propellant, so that $ falls to
zero and remains at zero. But how is the sudden drop in the value of S,
required at B, to be secured ?
It would appear that the desired features in the curve of pressure can
only be obtained by combining in the charge two forms of propellant. If
we are limited to simple annular sections, there must be two of different
times of burning. Of these, one, the thinner in the walls, burns out at B,
110 CONSIDERATIONS RESPECTING THE COMBUSTION OF [286
giving the first drop of operative surface. From B to G the second portion
continues to burn with constant surface, until in its turn it disappears at C.
From C to D the gases expand without addition to their mass.
It will be understood that sharp corners are spoken of merely for the
sake of brevity. In practice it would not be possible, or perhaps desirable,
to have them quite sharp. The whole surface of a quantity of annular
propellant can never disappear with absolute simultaneity, and the corners
would inevitably be rounded off.
If the propellant is entirely burned away at C, the remaining part of the
curve CD is out of control, but it can be calculated upon known principles,
and is (I believe) well understood. In designing a propellant to suit a given
gun, attention would first be given to defining the positions of B and G.
The next step would be to find, by calculation, and, if necessary, by special
experiment, the forms and proportional quantities of the two kinds of annular
propellant necessary. It would seem that in the present problem, i.e., to get
the most out of the gun, the solid form of propellant has no place.
It may be of interest to notice that over any straight portion, such as BC,
of the pressurecurve, the relation between x and t can readily be calculated.
For by (11) we have
d*x/dt* = a + bx, .............................. (14)
where a, b are constants, of which a is positive and b positive or negative
jiccording as the pressure rises or falls with increasing x. Equation (14)
may also be written,
where be = a ; and the solution is
c + x^Qe^' + He^'*, ..................... (16)
or, c + x = G' cos (V6' . t) + H' sin (*JV . t) ................ (17)
The first form obtains when b is positive, the second when b is negative,
6' being equal to  6. G, H, G', H' are constants of integration, to be deter
mined by the initial circumstances at B, the commencement of the straight
portion of the pressurecurve.
In the case of fig. 1, where b = 0, (14) becomes
d*x/dP = a, .............................. (18)
whence x = L + Mt + %aP, ........................... (19)
the motion being that of a falling body.
Another problem, whose solution would be of interest from the theoretical
point of view, is that corresponding to S = constant, as in the combustion of
1903] MODERN PROPELLANTS IN CLOSED VESSELS AND IN GUNS 111
a propellant of tubular form. The case where f(p) is taken proportional to
p could be dealt with graphically. For from (11), (12), with omission of
constant multipliers, we get
d
or on integration p = jr+pii, ........................... (20)
Pi, x 1 being simultaneous values of p and x when dx/dt = 0.
In (20) we may again replace p by dPac/dt*, so that
dte _Xip! + dx/dt
dt 2 x
an equation by means of which the curve showing the relation between x
and t could be constructed. But it may be questioned whether the actual
solution would convey anything of practical value.
Although at some points confirmation may be needed, the results of this
discussion seem to me to show that the proper course is first to define the
pressurecurve to be aimed at. Until this has been done, at least in some
degree, it is impossible to say whether one propellant is better or worse than
another. When the pressurecurve has been laid down, it will be possible
to approximate to it by a suitable choice of propellant. I am disposed to
think that the requirements of the case cannot be met by a single form of
propellant, but that a combination of two forms of tubular character may
suffice.
In conclusion it may be observed that when the power of the gun is
pushed by the employment of a propellant capable of giving rather high
forward pressures, we can hardly expect to retain the highest standard in
respect of uniformity of ballistics. The latter requirement, as well as
economy of material, is promoted by an early burning out of the propellant,
which may then assume the solid or unperforated form. When nearly the
maximum work is obtained from the propellant, a slight irregularity in the
manner of burning has but little effect upon the ballistics.
287.
ON THE BENDING OF WAVES ROUND A SPHERICAL OBSTACLE.
[Proceedings of the Royal Society, LXXII. pp. 4041, 1903.]
IN the Proceedings for January 21, 1903, Mr H. M. Macdonald discusses
the effect of a reflecting spherical obstacle upon electrical and aerial waves
for the case where the radius of the sphere is large compared with the wave
length (\) of the vibrations. The remarkable success of Marconi in signalling
across the Atlantic suggests a more decided bending or diffraction of the
waves round the protuberant earth than had been expected, and it imparts
a great interest to the theoretical problem. Mr Macdonald's results, if they
can be accepted, certainly explain Marconi's success ; but they appear to me
to be open to objection.
If C be the source of sound, P a point upon the sphere whose centre is
at 0, <i the velocitypotential at P due to the source (in the absence of the
sphere), ^ the angle subtended by OC, Mr Macdonald finds for the actual
potential at P,
so that there is no true shadow near the surface of the sphere. If C be
infinitely distant, and p denote (as usual) the cosine of the angle between OP
and OC,
That the sound should vanish at the point opposite, and be quadrupled at the
point immediately under the source is what would be expected ; but that
(however large the sphere) the shadow should be so imperfect at, for example,
p=  $, is indeed startling.
The first objection that I have to offer is that nothing of this sort is
observed in the case of light. The relation of wavelength to diameter of
1903] ON THE BENDING OF WAVES ROUND A SPHERICAL OBSTACLE 113
obstacle is about the same in Marconi's phenomenon as when visible light
impinges upon a sphere 1 inch (2'54 cm.) in diameter. So far as I am aware
no creeping of light into the dark hemisphere through any sensible angle is
observed under these conditions even though the sphere is highly polished*.
But I shall doubtless be asked whether I have any complaint against the
mathematical argument which leads up to (2).
As in Theory of Sound, 334, the question relates to the ratio between a
certain function of c (the radius) and its differential coefficient with respect
to c. The function is that which occurs in the representation of a disturbance
which travels outwards, and ( 323) may be denoted by
(3)
where k = 2Tr/X,
 . *W>+ ! *S?
The differential coefficient of (3) is
pikc
^r{(l+ikc)f n (ikc)ikcf n '(ikc)}, .................. (5)
so that the ratio in question takes the form
Cf n (ikc)
(l + ikc)f n (ikc)ikcf n '(ikcY
In these expressions n is the order of the Legendre's function P n (p) which
occurs in the series representative of the velocitypotential.
When kc is very great, the ratio expressed in (6) may assume a simplified
form. From (4) we see that, if n be finite,
f n (ikc) = l, ikcf n ' (ike) = 0,
ultimately, so that
i "
independent of n.
This is the foundation of the simple result reached by Mr Macdonald. Its
validity depends, therefore, upon the applicability of (7) to all values of n
that need to be regarded. If when kc is infinite, only finite values of n are
* It may be remarked that at the centre of the shadow thrown at some distance (say 1 metre)
behind, there is a bright spot similar to that seen when a disc is substituted for the sphere. This
effect is observed with a magnifying lens. If the eye, situated at the centre of the shadow, be
focused upon the sphere, the edge of the obstacle is seen bounded by a very narrow ring of light.
R.V. 8
114 ON THE BENDING OF WAVES ROUND A SPHERICAL OBSTACLE [287
important, (7) is sufficiently established ; but ( 328) it appears that under
these conditions the most important terms are of infinite order. I think it
will be found that for the most important terms n is approximately equal
to kc, and that accordingly (7) is not available. In any case it could not
be relied upon without a further examination.
In Theory of Sound, 328, the problem is treated for the case where kc is
small, and the calculation is pushed as far as kc = 2. The results indicate no
definite shadow. I have commenced a calculation for kc=lO, about the
highest value for which the method is practicable. But it is doubtful
whether even this value is high enough to throw light upon what happens
when kc is really large*.
* [See Art. 292 below.]
288.
ON THE PROPORTION OF ARGON IN THE VAPOUR
RISING FROM LIQUID AIR.
[Philosophical Magazine, v. pp. 677680, 1903.]
THE boilingpoint of argon being intermediate between those of nitrogen
and oxygen, it may be expected that any operations of evaporation and
condensation which increase the oxygen relatively to the nitrogen will at
the same time increase the argon relatively to the nitrogen and diminish it
relatively to the oxygen. In the experiments about to be detailed the gas
analysed was that given off from liquid air, either freshly collected, or after
standing (with evaporation) for some time from a day to a week. The
analyses were for oxygen and for argon, and were made upon different, though
similar, samples. Thus after an analysis of a sample for oxygen by Hempel's
method with copper and ammonia, 4 or 5 litres would be collected in a
graduated holder, and then the first analysis confirmed on a third sample.
In no case, except one to be specified later, was the quantity of gas withdrawn
sufficient to disturb sensibly the composition. The liquid was held in Dewar's
vessels, but the evolution of gas from below was always sufficient to keep the
mass well mixed.
The examination for argon was made in a large testtube inverted over
alkali, into which the gas was fed intermittently from the holder. The
nitrogen was gradually oxidized by the electric discharge from a Ruhmkorff
coil in connexion with the public supply of alternating current, the proportion
of oxygen being maintained suitably by additions of oxygen or hydrogen as
might be required. In the latter case the feed should be very slow, and the
electric discharge should be near the top of the testtube. Great care is
required to prevent the hydrogen getting into excess ; for if this should
occur, the recovery of the normal condition by addition of oxygen is a very
risky process. After sufficient gas from the holder, usually about 2 litres, had
been introduced, the discharge was continued until no more nitrogen remained,
as was evidenced by the cessation of contraction and by the disappearance of
the nitrogen line from the spectrum of the discharge when the terminals were
connected with a leydenjar. When it was certain that all nitrogen had been
removed, the residual oxygen was taken up by ignition of a piece of phosphorus.
On cooling, the residue of argon was measured, and its amount expressed as
a percentage of the total gas taken from the holder,
82
116
ON THE PROPORTION OF ARGON IN THE VAPOUR
[288
The results are shown in the following table. The oxygen, expressed as a
percentage of the whole, varied from 30 to about 98. From 43 to 90 per cent.
of oxygen, the argon, as a percentage of the whole, scarcely varied from 2'0.
Percentage of
Oxygen
Percentage of
Argon
Argon as a percentage of
the Nitrogen and Argon
90
43
64
75
90
13
20
20
21
2O
19
35
56
84
200
98
100
076
oo
330
1000
The experiment entered under the head of 98 per cent, oxygen is not
comparable with the others. In this case 5 litres of gas were collected as
the last portion coming away from a stock of liquid as it dried up. Nor was
the subsequent treatment quite parallel, for the whole of the oxygen was first
removed with copper and ammonia leaving 125 c.c. of mixed nitrogen and
argon, of which again by subsequent analysis 42 c.c. was found to be argon.
The last entry corresponding to 100 per cent, of oxygen is theoretical and
does not represent any actual experiment.
It must be clearly understood that these results relate to the vapour
rising from the liquid, and not to the composition of the liquid itself. So
far as the oxygen content is concerned, the comparison may be made by
means of Mr Baly's observations (Phil. Mag. XLIX. p. 517, 1900). It will
appear, for example, that when the vapour contains 30 per cent, of oxygen,
the liquid will contain about 60 per cent., and that when the vapour contains
90 per cent, the liquid will be of 95 or 96 per cent. At every stage the liquid
will be the stronger in the less volatile constituents ; so that the proportion
of argon to nitrogen, or to nitrogen + argon, will be higher in the liquid than
in the vapour.
The constancy of the proportion of argon to the whole over a considerable
range may be explained to a certain extent, for it will appear that the
proportion must rise to a maximum and thence decrease to zero. To
understand this, we must remember that "liquid air" is something of a
misnomer. In the usual process the whole of the air concerned is not
condensed, but only a part ; and the part that is condensed is of course
not a sample of the whole. As compared with the atmosphere the liquid
contains the less volatile ingredients in increased proportion, and the part
not condensed and rejected contains the more volatile ingredients in increased
1903] RISING FROM LIQUID AIR 117
proportion. The vapour coming away from the liquid as first collected has
the same composition as the gas rejected in the process of condensation. At
the beginning of our table, a point, however, which it would be difficult to
reach in actual experiment, we should have an oxygen content much below
20 per cent., a ratio of argon to nitrogen + argon below 1 per cent., and in all
probability a ratio of argon to the whole also below 1 per cent.
The object which I had in view was principally to obtain information as
to the most advantageous procedure for the preparation of argon. So many
laboratories are now provided with apparatus for liquifying air, that it will
usually be convenient to start in this way if a sufficient advantage can be
gained. The above results show clearly that the advantage that may be
gained is great. Something depends upon the procedure to be adopted for
eliminating the nitrogen. Upon a moderate scale and where there is a
supply of alternating current, the method of oxidation, as in the analyses,
is probably the most convenient. In this case it may be an advantage to
retain the oxygen. If the oxygen content be about 60 per cent., as in the
third experiment, the proportion is about sufficient to oxidize the nitrogen.
We may compare this with the mixture of atmospheric air and oxygen which
behaves in the same manner. In the latter case the proportion of argon
would be reduced from 2'0 per cent, to about '4 per cent., so that the
advantage of using the liquid air amounts to about five times. In the
arrangement that I described for oxidizing nitrogen upon a large scale*
the mixed gases were absorbed at the rate of 20 litres per hour.
In the alternative method the nitrogen is absorbed by magnesium or
preferably by calcium formed in situ by heating a mixture of lime and
magnesium as proposed by Maquennef. In this case it is necessary first to
remove the oxygen; but oxygen is so much more easily dealt with than
nitrogen that its presence, even in large proportion, is scarcely an objection.
On this view, and on the supposition that liquid air is available in large
quantities, it is advantageous to allow the evaporation to proceed to great
lengths. A 20 per cent, mixture of argon and nitrogen (experiment 5) is
easily obtained. Prof. Dewar has shown me a note of an experiment executed
in 1899, in which a mixture of argon and nitrogen was obtained containing
25 per cent, of the former. In the 6th experiment 33 per cent, was reached,
and there is no theoretical limit.
P.S. I see that Sir W. Ramsay (Proc. Roy. Soc., March 1903) alludes to
an experiment in which the argon content was doubled by starting from
liquid air.
* Chem. Soc. Journ. LXXI. p. 181 (1897) ; Scientific Papers, Vol. iv. p. 270.
t I employed this method successfully in a lecture before the Royal Institution in April 1895
(Scientific Papers, Vol. iv. p. 188). In a subsequent use of it I experienced a disagreeable
explosion, presumably on account of the lime being insufficiently freed from combined water.
1>S9.
ON THE THEORY OF OPTICAL IMAGES, WITH SPECIAL
REFERENCE TO THE MICROSCOPE.
(SUPPLEMENTARY PAPER.)
[Journ. R. Micr. Soc. pp. 474482, 1903.]
IN the memoir, above* reprinted from the Philosophical Magazine, I dis
cussed the theories of Abbe and Helmholtz, and endeavoured to show their
correlation. It appeared that the method of the former, while ingenious
and capable of giving interesting results in certain directions, was inap
plicable to many of the problems which it is necessary to attack. As an
example of this, it may suffice to mention the case of a selfluminous
object.
The work of Helmholtz, to which attention has recently been recalled
by Mr J. W. Gordon in a lively criticism (p. 381), was founded upon the
processes already developed by Airy, Verdet, and others for the performance
of the telescope. The theories both of Abbe and Helmholtz pointed to a
tolerably definite limit to the powers of the Microscope, dependent, however,
upon the wavelength of the light employed and upon the medium in which
the object is imbedded. It appeared that two neighbours, whether consti
tuting a single pair of points or forming part of an extended series of equi
distant points, could not be properly distinguished if the distance were less
than half the wavelength of the light employed. The importance of this
conclusion, as imposing a limit upon our powers of direct observation, can
hardly be overestimated ; but there has been in some quarters a tendency to
ascribe to it a more precise character than it can bear, or even to mistake
its meaning altogether. A few words upon this subject may not be out
of place.
The first point to be emphasised is that nothing whatever is said as to
the smallness of a single object that may be made visible. The eye, whether
' [In Journ. n. Micr. Soc. See this collection, iv. p. 235.]
1903] THE THEORY OF OPTICAL IMAGES 119
unaided or armed with a telescope, is able to see as points of light stars
subtending no sensible angle. The visibility of the star is a question of
brightness simply, and has nothing to do with resolving power. The latter
element enters only when it is a question of recognising the duplicity of a
double star, or of distinguishing detail upon the surface of a planet. So in
the Microscope there is nothing except lack of light to hinder the visibility
of an object however small. But if its dimensions be much less than the
half wavelength, it can only be seen as a whole, and its parts cannot be
distinctly separated, although in cases near the border line some inference
may be possible founded upon experience of what appearances are presented
in various cases. Thus a practised astronomer may conclude with certainty
that a star is double, although its components cannot be properly seen. He
knows that a single star would present a round (though false) disc, and any
departure from this condition of things he attributes to a complication. A
slightly oval disc may suffice not only to prove that the star is double but
even to fix the line upon which the components lie, and their probable
distance apart.
What has been said about a luminous point applies equally to a luminous
line. If bright enough, it will be visible, however narrow ; but if the real
width be much less than the half wavelength, the apparent width will be
illusory. The luminous line may be regarded as dividing the otherwise dark
field into two portions; and we see that this separation does not require a
luminous interval of finite width, but may occur, however narrow the interval,
provided that its intrinsic brightness be proportionally increased.
The consideration of a luminous line upon a dark ground is introduced
here for comparison with the case, suggested by Mr Gordon, of a dark line
upon a (uniformly) bright ground. Calculations to be given later confirm
Mr Gordon's conclusion that the line may be visible (but not in its true
width), although the actual width fall considerably short of the half wave
length. Although in both these cases there is something that may be
described as resolution, what is seen as distinct from the ground is really
but a single object. So far as I see, there is no escape from the general con
clusion, as to the microscopic, limit, glimpsed originally by Fraunhofer and
afterwards formulated by Abbe and Helmholtz ; but it must be remembered
that near the limit the question is one of degree, and that the degree may
vary with the character of the detail whose visibility is under consideration.
Mr Gordon comments upon the fact that Helmholtz gave no direct proof
of his pronouncement that a grating composed of parallel, equidistant, in
finitely narrow, luminous lines shows no structure at a certain degree of
closeness, and he appears to regard the question as still open. This matter
was, however, fully discussed in my paper of 1896, where it is proved that as
120 THE THEORY OF OPTICAL IMAGES [289
the gratinginterval diminishes, structure finally disappears when the distance
between the geometrical images of neighbouring lines falls to equality with
half the width of the diffraction pattern due to a single line, reckoned from
the first blackness on one side to the first blackness on the other. It is easy
to see that the same limit obtains when the lines have a finite width,
provided, of course, that the widths and intrinsic luminosities of the lines
are equal. If the gratingtnterra/, that is the distance between centres or
corresponding edges of neighbouring lines, be less than the amount above
mentioned, no structure can be seen. The microscopic limit occurs when the
gratinginterval is equal to half the wavelength of the light in operation.
The method employed in 1896 depends upon the use of Fourier's theorem.
The critical case, where the structure has just disappeared, may be treated
in a somewhat more elementary manner as follows. It is required to prove
that
sin'M sin^M + Tr) sin 2 (u TT) ^ sin 2 (u + 27r) n .
~ ' " 2
obtained by writing TT for v in (22) [Art. 222], is the same for all values of u.
In (76) the (sine) 2 have all the same value, so that what has to be proved
may be written
This follows readily from the expression for the sine in factors. If we write
sin u = Cu (u + TT) (u ir)(u + 2?r)
or log sin u = log C + log u + log (u + TT) + . . . ,
we get on differentiation
d log sin wll 1
du U M + 7T U IT
dMogsinw 11 1
andagam ^L._ + ( __ + ___ + ....
In these equations
d log sin {u = u _ d 3 log sin u _ 1
du du 3 ~ sin'M '
from which (77) follows.
We infer that a grating of the degree of closeness in question presents
to the eye a uniform field of light and no structure, but it is not proved by
this method that structure might not reappear at a greater degree of
closeness. If however we take v = ?r, that is, suppose the lines to be exactly
1903] WITH SPECIAL REFERENCE TO THE MICROSCOPE 121
twice as close as above, a similar method applies. The illumination at the
point is now expressed by
in 2 u sin 2 (u + ^ IT) sin 2 (u
orby
The value of the first series has above been shown to be unity, and by a
like method the same may be proved of the second. The illumination for all
values of u is thus equal to 2. That it should be twice as great as before
might have been expected.
But my principal object at present is to consider the problem, suggested
by Mr Gordon, of a dark line of finite width upon a uniformly bright
ground. The problem assumes two forms according as the various parts of
the ground are supposed to be selfluminous or to give rise to waves which
are all in one phase. The latter is the case of an opaque wire or other
linear obstacle upon which impinge plane waves of light in a direction
parallel to the axis of the instrument (telescope or microscope), and as it is
somewhat the simpler we may consider it first*.
In (28) [Art. 222] we have the expression for the resultant amplitude at
any point u due to a series of points or lines, whose geometrical images are
situated at u = 0, u = + v, u = 2v, &c. If all values of u are equally
geometrical images of a uniformly bright ground of light, we have to
consider
sum , ,,_ ft .
du = ir (78)
At present we suppose that the bright ground is interrupted at points corre
sponding to u = a, u = a, so that 2a represents the width of the geometrical
image of the dark obstacle. The amplitude at u is the same for a given
numerical value of u, whether u be positive or negative. It will suffice there
fore to suppose u positive. If u < a, we have
. , . [ +a sinu, [ a smu ., [ a+u smu, /( _ m
A(u)=l du\ du\ du ...... (79)
./ u Jo u Jo u
* It should be remarked that in point of fact the field is limited through the operation of a
cause not taken into account in the formation of (28) [Art. 222], It is there assumed that equality
of phase in the light emitted from the various points of the object carries with it a like equality
of phase at the geometrical images of these points. This will hold good only near the centre of
the field. At a moderate distance out the illumination is destroyed by the phasedifferences here
neglected.
122 THE THEORY OF OPTICAL IMAGES
which gives the resultant amplitude at any point u as a function of u and a.
If it > a, we have
u +*smu ,
du ....... (80)
*
By (78) the first term is equal to TT.
The integral in (79), (80) is known as the sineintegral. In the usual
notation
f* sin u . ...
I du=si(x) ........................... (81)
so that (79) may be written
A(u) = 7rsi(au)si(a + u), .................. (82)
and (80) may be written
A (u) = TT + si (u  a)  si (u + a) ................ (83)
The function si has been tabulated by Dr Glaisher*.
At the centre of the geometrical image of the bar, u = 0, and (82)
becomes
4 (0) = IT  2 si (a) ............................ (84)
If x is small, (81) gives
9i <*> = *870T3 + 5.1.2*3.4.5"' ; ............ (85)
so that in (82) if be small,
^W'^TTTi^F ................... < 86 >
From this we see that over the whole geometrical image of the bar the
amplitude of vibration is nearly the same. If we write / for the intensity,
where / (u) = [A (u)} 9 , and denote by / the value of / corresponding to a
uniform ground (a = 0), then
(87)
This gives the proportional loss of illumination over the image of the bar,
and it suffices for the information required near the limit of visibility. For
example, if the loss of light over the image be oneeighth of the maximum,
2 = jfo tr ; so that a single bar upon a bright ground might well remain
apparent when its width is reduced to ^ of the minimum gratinginterval
(2?r) necessary for visibility.
The above gives the loss of brightness over the region occupied by
Phil. Tram., 1870.
1903]
WITH SPECIAL REFERENCE TO THE MICROSCOPE
123
the geometrical image. Outside this region we have from (80), when 2a
is small,
whence
ru+a. s j n u s [ n u
A (u) = TT  I du = TT 2ot ,
7 / (u) 4a sin u
.(88)
.(89)
Here (89) identifies itself with (87) when u is small, and it does not alter
greatly until u = \ir. The slightly darkened image of the bar has thus a
width corresponding to the interval u = IT, exceeding to a great extent
the width of the geometrical image when the latter is very small. The con
clusion is that, although a very narrow dark bar on a bright ground may
make itself visible, the apparent width is quite illusory.
A(u).
M
a = l
a = 2
a = 3
+ 1520
 068
 556
1
1807
+ 347
 221
2
2509
1384
+ 646
3
3259
2538
1717
4
3711
3322
2633
5
3745
3536
3173
6
3507
3326
3326
7
3263
3027
3242
8
2932
2909
3114
The annexed table gives the values of A (u) for a = 1, 2, 3 for u = 0, 1 ... 8.
Corresponding to any value of a,
u (oo )= TT = 3142.
It will be remembered that 2a is the width of the geometrical image of the
bar, so that when a = 3 the width is about the same as the minimum re
solvable grating interval (2ir).
We now pass to the case of a selfluminous ground interrupted by a dark
bar. As in (22) [Art. 222], we have for the illumination at any point u
within the geometrical image
a+w sin 2 M ,
* 1 ...... (90)
and for any point on the positive side beyond the geometrical image
124
THE THEORY OF OPTICAL IMAGES
[289
2a denoting as before the width of the geometrical image of the bar, while u
is reckoned from the centre of symmetry. If o = 0,
(92)
The integrals in (90), (91) may be reduced to dependence upon the sine
integral. It may be proved* that
ri!*) ....... (93)
Thus, inside the geometrical image,
/ ( ,,, = T si(2
and beyond it,
/() = w + 8i(2 u 
At the centre (u = 0)
Ct
; ...(94)
a  ) . ...(95)
= 7r2si(2a)
2 sin 2 a
..(96)
As in the former case an approximate expression (85) for si (#) gives the
desired information near the limit of visibility. If a be small, we have for
the illumination within the geometrical image from (90)
/( M ) = 7r2a, (97)
so that / "7 / =?5.. (98)
1 H 7T
The visibility of a bar of width 2a is thus only half as great as before.
Outside the geometrical image we have approximately, when u considerably
exceeds a,
[ u+a sm*u , _ sin'tt
/(U) = TT  du = TT2a , (99)
J  a U U 9
whence I*  1 (u) = 2a tin* u }
/ 7T M 2
The following table gives some values of 7 (u) calculated from (94), (95).
u
a = i
a = l
a = 2
2170
1349
453
1
2442
1797
827
2
2921
2621
1711
3
...
3056
2565
* E.g. by writing ru for u in the integral to be examined and differentiating with respect to r.
Or (93) may be verified by differentiating with respect to x.
1903] WITH SPECIAL REFERENCE TO THE MICROSCOPE 125
The complete value of / (u), when u is great, is TT. The width of the
geometrical image of the bar is 2a, and the smallest resolvable grating
interval is TT. The dark bar should be easily recognisable in the first case
when its width is but onethird of the minimum grating interval.
In conclusion I may mention the results of a simple experiment con
ducted almost entirely without apparatus. In front of the naked eye was
held a piece of copper foil perforated by a fine needlehole. Observed
through this, the structure of some gauze just disappeared at a distance
from the eye equal to 17 in. (inch = 2'54 cm.), the gauze containing 46 meshes
to the inch. On the other hand, a single wire '034 in. in diameter remained
fairly visible up to a distance of 20 ft. or 240 in. The ratio between the
angles subtended by the periodic structure of the gauze and the diameter of
the wire was thus
022 240
034 X 17 "
Using this in (98), we find for the proportional loss of illumination at the
centre of the wire
about what might have been expected.
290.
ON THE PRODUCTION AND DISTRIBUTION OF SOUND.
[Philosophical Magazine, Vol. vi. pp. 289305, 1903.]
Theory of Conical Trumpet.
THE theory of small periodic vibrations having their origin at a single
point of a gas and thence spreading symmetrically has long been known.
The following statement is from Theory of Sound*, 280. In it a denotes
the velocity of sound, and k = 2Tr/X, X being the wavelength.
" If the velocitypotential be
<}> = ~ r cosk(atr) (1)
\\T have for the total current crossing a sphere of radius r,
47T; 2 ~ = A {cos k (at  ?*)  kr sin k (at r)} = A cos kat,
when r is small enough. If the maximum rate of introduction of fluid be
denoted by A, the corresponding potential is given by (1).
"It will be observed that when the source, as measured by A, is finite,
the potential and the pressurevariation (proportional to d<f>/dt) are infinite
at the pole. But this does not, as might for a moment be supposed, imply
an infinite emission of energy. If the pressure be divided into two parts,
one of which has the same phase as the velocity, and the other the same
phase as the acceleration, it will be found that the former part, on which the
work depends, is finite. The infinite part of the pressure does no work on
the whole, but merely keeps up the vibration of the air immediately round
the source, whose effective inertia is indefinitely great.
" We will now investigate the energy emitted from a simple source of
given magnitude, supposing for the sake of greater generality that the source
* Macmillan & Co., first edition 1878, second edition 1896.
1903] ON THE PRODUCTION AND DISTRIBUTION OF SOUND 127
is situated at the vertex of a rigid cone of solid angle o>. If the rate of intro
duction of fluid at the source be A cos kat, we have
wr^d^jdr = A cos kat
ultimately, corresponding to
</> =  cos k(atr\ ..... ................... (2)
dd> kaA . .,
whence  =  smK(at r). ........................ (3)
dt <ar
and a)r*d<j>/dr = A (cos k (at  r)  kr sin k (at  r)} ............. (4)
Thus if dW be the work transmitted in time dt, we get, since Bp =  p d<j>/dt,
dW
. . . x .
sm fc (at r) cos K (at r) + p sm 2 k (at r).
Of the righthand member the first term is entirely periodic, and in the
second the mean value of sm 2 k(at r) is ^. Thus in the long run
<>
" It will be remarked that when the source is given, the amplitude varies
inversely as to, and therefore the intensity inversely as o> 2 . For an acute
cone the intensity is greater, not only on account of the diminution in the
solid angle through which the sound is distributed, but also because the total
energy emitted from the source is itself increased.
" When the source is in the open, we have only to put o> = 4>7r, and when
it is close to a rigid plane, o> = 27r.
" These results find an interesting application in the theory of the
speakingtrumpet, or (by the law of reciprocity, 109, 294) hearingtrumpet.
If the diameter of the large open end be small in comparison with the wave
length (27T/A;), the waves on arrival suffer copious reflexion, and the ultimate
result, which must depend largely on the precise relative lengths of the tube
and of the wave, requires to be determined by a different process. But by
sufficiently prolonging the cone, this reflexion may be diminished, and it will
tend to cease when the diameter of the open end includes a large number
of wavelengths. Apart from friction it would therefore be possible by
diminishing o> to obtain from a given source any desired amount of energy,
and at the same time by lengthening the cone to secure the unimpeded
transference of this energy from the tube to the surrounding air.
" From the theory of diffraction it appears that the sound will not fall
off to any great extent in a lateral direction, unless the diameter at the large
end exceed half a wavelength. The ordinary explanation of the effect of a
common trumpet, depending upon a supposed concentration of rays in the
axial direction, is thus untenable."
128 ON THE PRODUCTION AND DISTRIBUTION OF SOUND [290
Data respecting FogSignals.
The above theory should throw light upon the production of sound in
" fogsignals," where sirens, or vibrating reeds, are associated with long conical
trumpets. In the practice of the Trinity House these are actuated by air
compressed to a pressure (above atmosphere) of 25 Ibs. per square inch, or
1760 gms. per sq. cm., a pressure which appears rather high. According to
Stone the highest pressure used in orchestral windinstruments is 40 inches
(102 cm.) of water.
As might be expected from the high pressure, the energy consumed during
the sounding of the signal is very considerable. The high note of the
St Catherine's Service signal takes 130 horsepower, and the corresponding
note of a Scottish signal (tested at St Catherine's in 1901) requires as much
as 600 horsepower. The question obtrudes itself whether these enormous
powers are really utilized for the production of sound, or whether from some
cause, possibly unavoidable, a large proportion may not be wasted.
Comparison with Musical Instruments, dec.
These statements as to horsepower may be better appreciated if I record
for comparison the results of some rough measurements, made in 1901, upon
the power absorbed by smaller instruments. In the calculation it will suffice
to regard the compressions and rarefactions as isothermal. Thus if v , p
represent the volume and pressure of air in its natural (atmospheric) condition,
v, p the corresponding quantities under compression, so that according to
Boyle's law pv = p v , then the work (If) of compression is given by
W = Po v \og(p/p ), ........................... (6)
or, if the compression be small,
In C.G.S. measure p (the atmospheric pressure) will be 10 8 , and if v be
measured in cubic centimetres, W will be expressed in ergs. If in (7) v
be understood to mean the volume compressed per second of time, W will be
given in ergs per second, of which 7'46 x 10 9 go to the horsepower.
The first example is that of a small horn (without valves) blown by the
lips. It resonates to e of my harmonium, and the pitch when sounded is
about e' flat. From one inspiration I can blow it for about 30 seconds with
a pressure (in the mouth) of 1 inch (3'8 cm.) of mercury. The contents of
the lungs may be taken at 1200 c.c., giving 40 c.c. per second as the wind
consumption. This is the value of v , and (pp )/p is fa. Hence W in
ergs per second will be 2 x 10 8 , or in horsepower
W= 00027 H.P.
1903] ON THE PRODUCTION AND DISTRIBUTION OF SOUND 129
The sound from this very small horsepower is unpleasantly loud when given
in a room of moderate dimensions.
In the case of the harmonium reed e the wind consumption was 220 c.c.
per second, and the pressure 2 inches of water, so that (p po)/po = ^>
Hence
W = 00015 H.P.
A small hand fogsignal of Holmes' pattern, known as the " Little Squeaker,"
consumed a horsepower calculated on the basis of similar measurements to
be '03. For the very effective " Manual " of the Trinity House Service the
horsepower was about 3'0.
These examples may all be classed under the head of reeds, the harmonium
reed being " free " and probably in consequence less efficient, and the others
"striking." To them may be added the case of a whistle of high pitch*, for
which the wind consumption represented 1'8 x 10* ergs per second, or '00024
horsepower, practically the same as for the small horn above. The latter
was certainly the more powerful of the two, considered as a source of audible
sound.
It may now be instructive to consider the case of a large siren, such as
the 7inch disk siren experimented upon at St Catherine's in 1901. The
wind consumption here was 29 cubic feet, or 810 litres, per second. This
average current, for the purposes of a rough calculation, may be analysed
into a steady current of the same amount and an alternating current whose
extremes are represented by +810 litres per second, the latter being alone
effective for the production of sound. The first question that arises is to
what pressure does this correspond, and is it a reasonable fraction of the
actual pressure employed ?
The answer must depend upon the other circumstances of the case, such
as the character of the cone or other tubular resonator associated with the
siren. We shall begin by supposing that there is nothing of this kind, so
that the above alternating flow takes place at the surface of a sphere of radius
r situated in the open. The velocitypotential and the rate of total flow
being given by (1) and (2) with o> equal to 4?r, we have for the maximum
rate of that flow A \/{l + k 2 r*}, or with sufficient approximation for our purpose
A simply. If s be the " condensation,"
Bp = a?ps p d<f>/dt,
so that by (1)
kA A (8)
* Proc. Roy. Soc. XXVL p. 248 (1877) ; Scientific Papers, i. p. 329.
R. V. 9
130 ON THE PRODUCTION AND DISTRIBUTION OF SOUND [290
To obtain a numerical result we must make some supposition as to the
magnitude of r. Let us take &r= J. Then
in which A is the maximum flow, X the wavelength, and a the velocity of
sound, i.e. 3 x 10 4 cm. per second. In the experiments referred to, the pitch
was low and such that
X = 8 feet = 240 cm.,
whence with A = 8*1 x 10* c.c. per second we find
1
The maximum condensation corresponding to the assumed introduction of
air is thus only fa of an atmosphere. The pressure is in the same
proportion, and we see that it is but an insignificant fraction of the pressure
actually employed (25 Ibs. per square inch)*. We infer that no moderate
pressure can be utilized in this way, and that some cone or resonating tube
is a necessity. It may be remarked that the radius r of the sphere, on which
the introduction of air is supposed to take place, is 1/4A; or X/STT, that is in
the case taken 4 inches or 10 cms.
Cones and Resonators.
The next question is what improvement in the direction of utilizing
higher pressures can be attained by the association of cones and resonators ?
But to this it is at present difficult to give a satisfactory answer. Theory
shows that, apart from friction and other complications perhaps not very
important, the efficiency of a small source may by these means be increased
to any extent. Thus, in the case of the cone, if u be the maximum [particle]
velocity [in] a progressive wave at a point where the section is <r, conservation
of energy requires that <ru? be constant. The maximum total flow (<ru) is
therefore proportional to <r , i.e. to the linear dimension of the section. If
the vibrations are infinitesimal, we may begin with as small a diameter as
we please and end with a large one, and thus obtain any desired multiplication
of the source. For it is the total flow at the open end of the cone which
measures the power of the source for external purposes. If, however, the
quantity of air periodically introduced at the small end can no longer be
treated as infinitesimal, this argument fails; and it is probable that the
advantage derivable from the cone diminishes. In an extreme case we can
easily recognize that this must be so. For the most that the cone could do
would be to add its own contents to that of the air forcibly introduced. As
* [Atmosphere =15 Ibs. per square inch.]
1903] ON THE PRODUCTION AND DISTRIBUTION OF SOUND 131
the latter increases without limit, the addition must at last become relatively
unimportant, and then the cone might as well be dispensed with. Similar
considerations apply to the use of a resonator.
There is no reason to doubt that great advantage accrues from the use
of the conical trumpet in existing fogsignalling apparatus, although probably
it falls short of what would be expected according to the theory of infinitely
small vibrations. If it be a question of striving to augment still further the
force of the sound, we must remember that the application of power has
already been carried to great lengths. The utilization of more power might
demand an increase in the scale of the apparatus. This in itself would present
no particular difficulty, but we must not forget that everything has relation
to the wavelength of the sound, and that this is to a great extent fixed for
us by the nature of the ear. It may well be that we are trying to do more
than the conditions allow, and that further advance would require a different
kind of apparatus. As matters stand, it seems to be generally admitted that
the instruments using great power are not proportionally effective.
If, as I incline to believe, a large proportion of the power applied to
important instruments is not converted into sound, there should be an
opening for reducing the very large demands now made. We have to
consider what becomes of the power wasted. I have long thought that
it is spent in the eddies consequent upon the passage of the air through
the comparatively narrow ports of the siren, and in this opinion my friend
Sir O. Lodge, with whom I have recently had an opportunity of discussing
the matter, concurs. If indeed it were a question of steady flow, one might
pronounce with certainty that a great improvement would ensue from a better
shaping of the passages, which on the downstreamside should cone out
gradually from the narrowest place. And although the intermittent character
of the stream is an important element, this conclusion can hardly be altogether
disturbed. The advantage of an enlargement of the ports themselves should
also be kept in sight.
The conical trumpets at present employed must act to some extent as
resonators, so that the precise relation of the pitch or speed of the siren
to the trumpet cannot be a matter of indifference. Although the relation
in question is liable to be disturbed by changes of temperature, it would
appear that a better adjustment than is feasible with the present governors
should be arrived at. To effect this an instrument capable of indicating the
vigour of the vibration within the trumpet, as the speed of the siren varies,
would be useful.
92
132
ON THE PRODUCTION AND DISTRIBUTION OF SOUND
[290
Vibration Indicator.
Experiments that I have tried appear to prove that the problem above
proposed can be solved in a very simple manner. The principle is that
of the unsymmetrical formation of jets when an alternating aircurrent flows
through an aperture coned upon one side. An experiment given in Theory
of Sound, 322, may be quoted in illustration :" When experimenting with
one of Ko'nig's brass resonators of pitch c', I noticed that when the corre
sponding fork, strongly excited, was held to the mouth, a wind of considerable
force issued from the nipple at the opposite side. This effect may rise to
such intensity as to blow out a candle upon whose wick the stream is
directed.... Closer examination revealed the fact that
at the sides of the nipple the outward nWing Fig 1.
stream was replaced by one in the opposite direction, *
so that a tongue of flame from a suitably placed
candle appeared to enter the nipple at the same ^ \j
time that another candle situated immediately in
front was blown away. The two effects are of course
in reality alternating, and only appear to be simul I)
taneous in consequence of the inability of the eye to
follow such rapid changes."
The application of the principle for the present
purpose is very simple. The candle is replaced by
a U pressuregauge (fig. 1), the jet from a contracted
nozzle playing into one limb. The whole is inclosed
airtight in a testtube, so that no permanent
pressure or suction has any effect upon the gauge.
The nozzle communicates by means of a flexible
tube with the space where the vibration is to be
measured. Some throttling to check the vibration
of the liquid may be convenient.
The small Holmes apparatus gave an indication
of about 3 inches of water, and the Trinity House " Manual " one inch of
mercury. The sensitiveness may be lessened by contracting the nozzle or
probably by insertion of water to diminish the airspace within the testtube.
Reeds.
[In comparison with sirens] reeds have the advantage of working without
a governor, and the pitch once properly fixed is more likely to be maintained.
I do not know whether reeds have been tried for very large scale instruments.
1903] ON THE PRODUCTION AND DISTRIBUTION OF SOUND 133
In the Barker apparatus* three reeds are combined with one trumpet. At
first sight it may seem doubtful whether the tongues would vibrate in the
same phase, but upon examination I think it will appear that this is the
only way in which they could vibrate. On a large scale either the reeds
must be multiplied, or an entirely different shape must be adopted, out of
all proportion broader than at present. Some experiments that I have tried
seem to show that the latter alternative is not impracticable.
Trumpets of Elongated Section.
In the trumpets at present employed the section is of circular form and
the greater part of the axis is vertical. This disposition has its conveniences,
but it entails bending the axis at the wide end of the cone if the mouth is to
face horizontally. The effect of such a bending upon the propagation of the
wave within the trumpet is hard to estimate. When, as in the case of certain
rockstations, the sound is required to be heard in all directions, a symmetrical
form is adopted in the Trinity House Service, the mouth of the trumpet
which faces vertically being partially stopped by an obstacle known as the
" mushroom." The intention is to cut off the sound in a vertical direction
while allowing it to spread in horizontal directions through the annular
aperture between the bellmouth of the trumpet and the mushroom.
Considering the case of the axis horizontal throughout, we may inquire
into the probable distribution of the sound. The ratio between the diameter
of the mouth and the wavelength is here of essential importance. If the
diameter much exceed the half wavelength, the sound is concentrated in
the prolongation of the axis. If on the other hand the diameter do not
exceed the abovementioned quantity, we may expect a tolerably equable
distribution of sound, at any rate through angles with the axis less than 80.
It follows that the behaviour of the various components of a compound
sound may be quite different. The fundamental tone may spread fairly
well, while the octave and higher elements are unduly concentrated in the
neighbourhood of the axis.
It appears then that a limitation must be imposed upon the size of the
mouth, if it be desired that the sound should spread. But since the spreading
is required only in the horizontal plane, the limitation applies only to the
horizontal dimension of the mouth. There is no corresponding limitation
upon the vertical diameter. We are thus led to prefer an elongated form
of section, the horizontal dimension being limited to the half wavelength,
while the vertical dimension may amount if desired to many wavelengths.
* Report of Trinity House FogSignal Committee on Experiments conducted at St Catherine's
Point, Isle of Wight, 1901.
134 ON THE PRODUCTION AND DISTRIBUTION OF SOUND [290
This subject was explained and illustrated in a lecture before the Royal
Institution*, the source of (inaudible) sound being a "birdcall" giving
waves of 3 cms. length, which issued from a flattened trumpet whose mouth
measured 5 cms. by 1^ cm. The indicator was a highpressure sensitive
flame, and it appeared very clearly that when the long dimension of the
section stood vertical the sound was approximately limited to the horizontal
plane, but within that plane spread without much loss through all directions
less inclined than 80.
In order to carry the demonstration a little nearer to what would be
required in practice, I have lately experimented with the sound from a reed
organpipe, giving waves of length equal to 8 inches (20 cms.) and thus easily
audible. The trumpet is of wood, pyramidal in form, and the section at the
mouth is 36 x 4 inches (91 x 10 cms.). The
length (OB) is 6 feet (183 cms.). These
dimensions were chosen so that OA, 00 should
exceed OB by $\, A larger difference might
entail too great a discrepancy of phase in the
waves at A and B ; a less difference might lead
to an unnecessary prolongation of the cone
along the axis. The trumpet was so mounted
that its mouth just projected from an open window, and that it could be
readily turned round OB as horizontal axis so as to allow the length of the
section (AC) to be either horizontal or vertical. The observers took up
various positions on a lawn at a moderate distance from the window.
To observers in the line of the axis OB it should make no difference how
the trumpet is rotated. On the whole this expectation was confirmed, but
a little precaution is required. As usual the phenomenon was complicated
by reflected sounds (doubtless from the ground). It was well in every case
to make sure, by slightly raising or lowering the head, that the maximum
sound was being heard.
When the point of observation deviated (in the horizontal plane) from
the axis, the difference due to rotation was soon apparent. At 30 obliquity
the sound appears greatly increased as AC passes from the horizontal to the
vertical position. At higher obliquities with AC horizontal the sound falls
off greatly, but recovers when AC is made vertical. Altogether the effects
are very striking, and carry conviction to the mind more fully than experi
ments with sensitive flames where one is more or less in doubt as to the
magnitude of the differences indicated by the flame. It will be remarked
that to carry out this experiment upon a practical scale will mean a very
large structure, the linear magnification being that (6 times) required to pass
from an 8inch wavelength to one (say) of 48 inches.
' Proc. Roy. Irut. Jan. 1902 ; Nature, 66, p. 42 (1902) [Vol. v. p. 1].
1903] ON THE PRODUCTION AND DISTRIBUTION OF SOUND 135
Work done by Detached Sources.
In the case of a single source the pressures to be overcome are proportional
to the magnitude of the source, and thus the work done is proportional to the
square of the magnitude of the source, as is indeed otherwise evident. If, as
is usually the case in practice, the object be to emit sound in one (horizontal)
plane only, an economy may be effected by distributing the source. If sources
all in one phase be distributed along a vertical line, the effect is the same
at distant points in the horizontal plane as if they were all concentrated in
one point, but the work required to be done may be much less, the saving
corresponding to the fact that in directions other than horizontal the sound
is now diminished. We will begin by considering two unit sources in the
same phase.
If <f>, i/r be the potentials of these sources at a point whose distances are
r, r', we have, as in (1),
ooB*(atr) _c
4nr 47T/
Thus, when r = 0, 47rr 2 f = cos kat kr sin kat,
dr
dd) d^lr ka sin kat ka . . , , ~ x
^ + Ji = ~~ ~ + J n sm fc ( at ~ ^)>
dt dt 4nrr 4nrU
if D denote the distance between the sources. The work done by the source
at r = is accordingly proportional to
()
and an equal work is done by the source at r' = 0. If D be infinitely great,
the sources act independently, and thus the scale of measurement in (11)
is such that unity represents the work done by each source when isolated.
If .D = 0, the work done by each source is doubled, and the two sources
become equivalent to one of doubled magnitude.
If D be equal to ^X, or to any multiple thereof, amkD = Q, and we see
from (11) that the work done by each source is unaffected by the presence
of the other. This conclusion may be generalized. If any number (n) of
equal sources in the same phase be arranged in a vertical line so that the
distance between immediate neighbours is \, the work done by each is the
same as if the others did not exist. The whole work accordingly is n, whereas
the work to be done at a single source of magnitude n would be n a . Thus
if the sound be only wanted in the horizontal plane, the distribution into
n parts effects an economy in the proportion of n : 1. It is not necessary
that all the possible places between the outer limits be actually occupied.
136 ON THE PRODUCTION AND DISTRIBUTION OF SOUND [290
All that is necessary is that there be n equal sources altogether, and that the
distance between any pair of them be a multiple of \.
Returning for the moment to the case of two sources only, we may be
interested to estimate the work consumed by following the law of emission
of sound to a distance in the various directions. If p be the cosine of the
angle between any direction and the vertical, the relative retardation due to
the difference of situation is pD. If the potential at any great distance due
to one source is cos kat*, that due to the other may be represented by
cos k(at pD) For the aggregate potential we have
cos kat (1 + cos pkD) + sin kat sin fjJcD,
or for the intensity
2(l+cos/tM)) (12)
This is in direction fi. For the total intensity over angular space we must
integrate with respect to p. from  1 to + 1. The mean intensity is thus
...(13)
The scale of measurement is at once recovered by supposing D = 0, in which
case the intensity in various directions would be uniform. The ratio of the
mean intensities, which is also that of the work done, is thus
This is the ratio in which the work done is diminished when a source is
divided into two parts and these parts separated to a distance D.
While from the theoretical point of view there is no doubt as to the
saving that might arise from the use of a number of separated sources, it is
to be noticed that the saving is in the pressure. Since at the present time
most of the pressure employed with a single source appears to be wasted, we
are left in doubt whether with the existing arrangements economy would be
attained by breaking up the source.
We will now investigate the expression for the energy radiated from any
number of sources of the same pitch situated at finitely distant points in the
neighbourhood of the origin 0. The velocitypotential </> of the motion due
to one of the sources at (x, y, z) is at Q
(15)
where R is the distance between Q and (x, y, z). At a great distance from
the origin we may identify R in the denominator with OQ, or p ; while under
the cosine we write
) ........ ................ (16)
* It is not necessary to exhibit the dependence on r.
1903] ON THE PRODUCTION AND DISTRIBUTION OF SOUND 137
X, p, v being the directioncosines of OQ. On the whole
4vrp<f> = 24 cos [nt + e  kp + k(\x + py + vz)}, (17)
in which p is a constant for all the sources, but 4, e, x, y, z vary from one
source to another. The intensity in the direction X, fi, v is thus represented
by
[24 cos {e + k (\x + fiy + vz)}] 2 + [24 sin {e + k (\x + jiy + vz)}]*,
or by
24 2 + 224,4 2 cos [e,  e 2 + k {X (a?, acj + p (y, y,) + v (z,  *,)}], . . .(18)
the second summation being for every pair of sources of which 4,, 4 2 are
specimens. We have now to integrate (18) over angular space.
It will suffice if we effect the integration for the specimen term; and
this we shall do most easily if we take the line through the points (x lt y lt z^,
(#2, 2/2, #2) as ax i g f reference, the distance between them being denoted by
D. If X, //., v make an angle with D whose cosine is p,
Dfj, = \(x l x 2 ) + fj,(y 1 y 2 ) {v(z l z 2 ), (19)
and the mean value of the specimen term is
r+i
4,4 cosfe,
r+i
1 4J
J i
9 A A
that is ^^ 2 sinA; J Dcos(6 1 e 2 ) ......................... (20)
The mean value of (18) over angular space is thus
^ + 2S A ^ C s( ^ 62)8in ^ > .. ............. (21)
where D denotes the distance between the specimen pair of sources. If all
the sources are in the same phase cos (e! e 2 ) = 1. If the distance between
every pair of sources is a multiple of \, sinM) = and (21) reduces to
its first term.
We fall back upon a former particular case if we suppose that there are
only two sources, that these are units, and are in the same phase. (21) then
becomes
agreeing with (11), which represents the work done by each source.
If the question of the phases of the two unit sources be left open (21)
gives
2 + 2cos(e 1 6 a ) 8i ^ ...................... (22)
If D be small, this reduces to
2 + 2cos(e 1 e 2 ),
138 ON THE PRODUCTION AND DISTRIBUTION OF SOUND [290
which is zero if the sources be in opposite phases, and is equal to 4 if the
phases be the same.
If, however, sinkD be equal to 1, the case is altered. Thus when
DfX, we get
2 J cosfc*),
and this is a minimum (and not a maximum) when the phases are the same.
In (22) if the phases are 90 apart, the cosine vanishes. The work done
is then simply the double of what would be done by each source acting alone,
and this whatever the distance D may be.
Continuous Distributions.
If the distribution of a source be continuous, the sum in (21) is to be
replaced by a double integral. As an example, consider the case of a source
all in one phase and uniformly distributed over a complete circular arc of
radius c. If D be the distance of two elements dd, d6', we have to consider
the integral
where D = 2csm(0 &} ............................ (24)
Since every element dd' contributes equally, it suffices to take 6' = 0, so that
the integral to be evaluated is
/
J
o ~
or,
The integral (26) may be expressed by means of Bessel's function J , for
2 fi T
J (x) =  I cos (x sin <) d(f>,
2 [I* sin (x sin d>)
that J (x) dx = ~ ^  "
.'o TTJO 8in<
Thus, if constant factors be disregarded, we get
(27)
, ,
so that = ~  "
in which x = 2kc. Since (27) reduces to unity when c, or a, vanishes, it
represents the ratio in which the work done is diminished when a source,
1903] ON THE PRODUCTION AND DISTRIBUTION OF SOUND 139
originally concentrated at the centre, is distributed over a circular arc of
radius c.
The case of a source uniformly distributed over a circular disk of radius c
is investigated in my book on the Theory of Sound*. According to what is
there proved, the factor, analogous to (27), expressing the ratio in which the
work done is diminished when a source originally concentrated at the centre
is expanded over the disk, has the form
where, as usual,
Another case of interest is when the distribution takes place over the
surface of a sphere of radius c. In (25) we have merely to introduce the
factor sin 0, equal to 2 sin \Q cos \6, so that we get instead of (26)
. /"* sin (x sin <b) . ,, 4 /n
4 ?' cos<j>d<j> = (I cosx).
J o & 3?
The factor, corresponding to (27), is therefore simply
No work at all is done if c be such that kc is a multiple of TT, or 2c a multiple
of A.
By the method of the Theory of Sound (loc. cit.) we might in like manner
investigate the effect of distributing a source of sound uniformly throughout
the volume of a sphere, but the above examples will suffice for our purpose.
Experimental Illustrations.
There is no difficulty in illustrating upon a small scale the results above
deduced from theory. The simplest experiment is with an ordinary open
organpipe, gently blown, so as to exclude overtones as much as may be.
The open ends act as two equal sources of sound in the same phase.
Connected by a long flexible tube with a wellregulated bellows, the pipe
can be held in any position and be observed in the open air from a moderate
distance. When the length of the pipe is perpendicular to the line of
observation, the two sources are at the same distance and the effects conspire.
But if the pipe point toward the observer, the two sources, being at about
\ apart, are in antagonism and the sound is much enfeebled.
* Macmillan, 1st edition, 1878 ; 2nd edition, 1896, 302.
140
ON THE PRODUCTION AND DISTRIBUTION OF SOUND
[290
In order to exemplify the principle further, a multiple pipe was constructed
(fig. 2). This consisted of a straight lead tube 31 inches long and "35 inch
bore*, open at the ends A, B. At four points D, E, F, G, distant 6^ inches
from the ends and 6 inches from one another, the tube was perforated, and
the holes were blown by four streams of wind from the branched supplytube
C. The whole was cemented to a framework of wood, so that it could be
turned round without relative displacement. The intention was that all six
apertures A, D, E, F, G, B should act as sources of sound in the same phase,
but one could hardly be sure d priori that this behaviour would be observed.
Would the simultaneous motions of the aircolumn on the two sides of E
(for example) be both towards E or both from El Might it not rather
Fig. 2.
Seer/on THROUGH G
Wooo
be that the motions would take the same absolute direction, in which case
E, F, &c., would fail to act as sources ? A little observation, however, sufficed
to prove that the apparatus really acted as intended. By listening through
a rubber tube whose outer end was brought into proximity with the apertures,
it was easy to satisfy oneself that D, E, F, G were effective sources of sound,
and were in fact more powerful than the open ends A, B, as was to be
expected. The half wavelength of the actual sound was 8 inches, showing
that the " openness " of the pipe at D, E, F, G was rather imperfect, owing
Inch = 254 cms.
1903] ON THE PRODUCTION AND DISTRIBUTION OF SOUND 141
to the smallness of the holes. Still the apparatus afforded a combination
of six sources of sound, all in the same phase and at about half a wave
length apart.
The observations were made upon a lawn ; and, as the sound was rather
feeble, a very moderate distance sufficed. When AB was vertical, the sources
cooperated and the maximum sound was heard. But when AB was [turned
round], so as no longer to be perpendicular to the (horizontal) line of observation,
the sound was less, a deviation of 30 causing a very great falling off. The
effect was as if the sound had suddenly gone away to a great distance. The
success of the experiment no doubt depended a good deal upon the absence
of overtones, a condition of things favoured by the feebleness of the sound
and also by the high pitch.
291.
ON THE WORK DONE BY FORCES OPERATIVE AT ONE OR
MORE POINTS OF AN ELASTIC SOLID.
[Philosophical Magazine, Vol. vi. pp. 385392, 1903.]
AN investigation of the waves generated in an isotropic elastic solid
by a periodic impressed force, localized in the neighbourhood of a point, was
first given by Stokes*. A simpler treatment of the problem will be found
in my paper "On the Light from the Sky, &c.f," and more fully in Theory
of Sound, 378. It will be desirable to recapitulate the principal steps.
If a, $, 7 be the displacements at any point of the solid, X', Y', Z' the
impressed forces reckoned per unit of mass, we have equations such as
in which a and b are the velocities of dilatational and transverse waves
respectively, and 8 represents the dilatation expressed by
If, as throughout the present paper, a, fi, y, &c. be proportional to
(Pa/dt* =  p*a, and (1) &c. become
' = 0, .................. (3)
'0, .................. (4)
(a*  b*) do/dz + b*V*y + p* y + Z' = ................... (5)
These are the fundamental equations. For our purpose we may suppose
that X', Y' vanish throughout, and that Z' is finite only in the neighbourhood
of the origin. It will be convenient to write
k = p/b, h=p/a ............................... (6)
Cam*. Phil. Tram. Vol. ix. p. 1 (1849); Collected Workt, Vol. n. p. 243.
+ Phil. Mag. XLI. pp. 107, 274 (1871) ; Scientific Papers, i. p. 96.
1903] ON THE WORK DONE BY FORCES, ETC. 143
The dilatation S is readily found. Differentiating (3), (4), (5) with respect
to #, y, z and adding, we get
V 2 S + h?S + a*dZ'jdz = (7)
The solution of (7) is
1
r denoting the distance between the element at x, y, z near the origin (0)
and the point (P) under consideration. If we integrate partially with respect
to z, we find
the integrated term vanishing in virtue of the condition that Z' is finite only
within a certain space T. Moreover, since the dimensions of T are supposed
to be very small in comparison with the wavelength, d (r~ l e~ ihr )/dz may be
removed from under the integral sign.
It will be convenient also to change the meaning of x, y, z, so that they
shall represent as usual the coordinates of P relatively to 0. Thus, if Z l e**
denote the whole force applied at the origin, so that
Z^pfffZ'dxdydz,
in which p is the density,
giving the dilatation at the point P.
In like manner we may find the rotations ra' , sr", tar'", defined by
*^, .*.  = **>'"
dy dz dz dx dx dy
For from (3), (4), (5) we have
whence w'" = 0, and
These are the results given in my paper of 1871.
The values of 3, BT', vr", ty'" determine those of o, /3, y. If we take
a= *, 0. *x_, 7 *af + B , ...... (is)
dxdz ddz dz 1
144 ON THE WORK DONE BY FORCES OPERATIVE [291
ifcr /pi*r gtt\
where tM, X^ ................ (16)
it is easy to verify that these forms give the correct values to S, vr', *r", vr'".
As regards the dilatation,
in which ^*X =
This reproduces (9).
As regards the rotations, we see that x does not influence them. In fact
w ~_ , / = *^, "?;
2 dy dx'
and these agree with (14). The solution expressed by (15), (16), (17) is thus
verified, and it applies whether the solid be compressible or not.
In the case of incompressibility, h = 0. If we restore the timefactor e ipt
and throw away the imaginary part of the solution, we get
Afcxz [( 3 \ 3 . 3
cos (vt kr)T sm (vt kr)  ^
...... (18)
the value of $ differing from that of a merely by the substitution of y for x,
The value of ^1 is given by (17), and Z^ cospt is the whole force operative at
the origin at time t.
At a great distance from the origin (18), (19) reduce to
xzcos(ptkr) Z l

...... (20,21)
Upon this (Theory of Sound, 378) I commented :" W. Konig (Wied.
Ann. xxxvn. p. 651, 1889) has remarked upon the nonagreement of (18),
(19), first given in a different form by Stokes, with the results of a somewhat
similar investigation by Hertz (Wied. Ann. xxxvi. p. 1, 1889), in which the
terms involving cospt, sinpt do not occur, and he seems disposed to regard
Stokes's results as affected by error. But the fact is that the problems
treated are essentially different, that of Hertz having no relation to elastic
1903] AT ONE OR MORE POINTS OF AN ELASTIC SOLID 145
solids. The source of the discrepancy is in the first terms of (3) &c., which
are omitted by Hertz in his theory of the aether. But assuredly in a theory
of elastic solids these terms must be retained. Even when the material
is supposed to be incompressible, so that 8 vanishes, the retention is still
necessary, because, as was fully explained by Stokes in the memoir referred
to, the factor (a 2  6 2 ) is infinite at the same time."
Although the substance of the above comment appears to be justified,
I went too far in saying that Hertz's solution has no relation to elastic solids.
It is indeed not permissible to omit the first terms of (3) &c. merely because
the solid is incompressible ; but if, though the solid is compressible, it be in
fact not compressed, these terms disappear. Now Hertz's solution, corre
sponding to the omission of the second part of ^ in (16), makes
...... (22)
and accordingly 8 = T + r + r = 0:
dx dy dz
values which satisfy (V 2 + fc 2 ) (a, , 7) = ......................... (23)
Thus (3), (4), (5) are satisfied, and the solution applies to an elastic solid
upon which no forces act except at the origin. The only question remaining
open is as to the character of the forces which must be supposed to act
at that place. This is rather a delicate matter; but it is evident at any
rate that the forces are not of the simple character contemplated in the
preceding investigation. It would appear that they must be double or
multiple, and have components parallel to x and y as well as z. By a double
force is meant the limit of a couple of given moment when the components
increase and their mutual distance decreases, analogous to the double source
of acoustics.
I now propose to calculate the work done by the force Z l at the
origin as it generates the waves represented by (18), (19). For this
purpose we require the part of 7 in the neighbourhood of the origin which
is in quadrature with the force, i.e. is proportional to sin pt. From (19)
we get
the last term (in cos pt) not contributing. Expanding sin kr, cos kr and
retaining the terms of order kr, we get for the square bracket
Thus
B. v. 10
146
ON THE WORK DONE BY FORCES OPERATIVE
[291
when r is small, so that the part proportional to sin pt is in the limit finite
and independent of tfr. If IT be the work done in time dt,
JIMT
 = Z, cospt . IfcpA cospt;
and by (6), (17)
dW
mean r =
.(25)
The righthand member of (25) is thus the work done (on the average) in
unit of time.
This result may be confirmed by a calculation of the energy radiated
away in unit time, for which purpose we may employ the formulae (20), (21)
applicable when r is great. The energy in question is the double of the
kinetic energy to be found in a spherical shell whose thickness (r 2 r^ is
the distance travelled by transverse waves in the unit of time, viz. b. In the
expression for the kinetic energy the resultant (velocity) 2 at any point x, y, z
is by (20), (21) proportional to
a quantity symmetrical with respect to the axis. Also sin 2 (p kr) is to be
replaced by its mean value, viz. . Thus the kinetic energy is
the double of which is identical with (25).
We will now form the expression for the resolved displacement at P due
to Z l cos pt acting at (parallel to OZ\ the displacement being resolved in
Fig. i.
a direction PT in the plane ZOP making an angle ff with OP (fig. 1).
angle between PT and OZ is denoted by <, so that $=0+0'.
The
1903] AT ONE OR MORE POINTS OF AN ELASTIC SOLID 147
The resolved displacement is
7cos< + asin<, (26)
a and 7 being given by (18), (19), in which we write
zjr = cos 6, x/r = sin 0.
We find
/ /i/ cos <i> 3 / /w~I
sm sm # + cos ^ cos ^
. L r , ~ j , , [cos <f>  3 cos cos 0'\
A//* A/ 7*
a symmetrical function of and 0' as required by the general principle of
reciprocity (Theory of Sound, 108). The value of A is given by (17) in
which, however, we will now write $ for Z 1} so that
A = ^JF (27)
The above equation gives the resolved displacement at P in a direction
making an angle 0' with r due to a force % cos pt at acting in a direction
inclined to r at angle 0. If we suppose that a force ' cos p acts at P in
direction PT and inquire as to the work done by this force upon the motion
due to %, we have to retain that part of the resolved displacement due to $
which is in quadrature with 5' cospt, viz. the part proportional to sin pt. The
mean work is given by the symmetrical expression
5' (cos (0 + 6') 3 cos cos 0' ( , smkr\ sin kr .
 \ *  cos kr . ^ sm sin ff\ .
p\ ^r 2 V kr J kr }
(28)
If the forces are parallel, = 0, & = 0, and (28) becomes
^Yl (29)
If we further suppose that kr is very small, the square bracket reduces to
the value , and we get
m .................................. (30)
12TT&/3
Comparing with (25) we see that the work done by 5' on the motion due
to an equal g is the same as that done by $ itself, as should evidently be.
If in (29) = 90, so that the forces are perpendicular to the line joining
the two points of application, we get
102
148 WORK GENERATED AT POINTS OF AN ELASTIC SOLID [291
As we have seen, when kr is small (31) is finite and positive. It vanishes
when
ftr
and this occurs first in the second quadrant.
In general, when there are a number of forces acting at detached points,
the whole work done must be obtained by a double summation of (28). If
the forces are continuously distributed, the sum becomes a double integral.
A particular case, of interest in connexion with the problem of electrical
vibrations along a circular wire*, occurs when the forces act tangentially at
the various points of a circular arc. Here & = 0, and (28) becomes
&r . \ sin A^ . J
< 33 >
* Compare Pocklington, Proc. Camb. Phil. Soc. ix. p. 324 (1897); Nature, LXII. p. 486 (1903).
It would seem that (33) must lead to a more complicated expression for the energy radiated than
that in Dr Pocklington's investigation.
292.
ON THE ACOUSTIC SHADOW OF A SPHERE.
WITH AN APPENDIX*, GIVING THE VALUES OF LEGENDRE'S FUNCTIONS
FROM P n TO P,o AT INTERVALS OF 5 DEGREES. BY PROFESSOR A. LODGE.
[Phil. Trans. 203 A, pp. 87110, 1904.]
IN my book on the Theory of Sound, 328, I have discussed the effect
upon a source of sound of a rigid sphere whose surface is close to the source.
The question turns upon the relative magnitudes of the wavelength (\)
and the radius (c) of the sphere. If kc be small, where k 27T/X, the
presence of the sphere has but little effect upon the sound to be perceived
at a distance.
The following table was given, showing the effect in three principal
directions of somewhat larger spheres :
kc
f
*'S + G
1
294291
\
1
259729
231999
1
502961
1
1
285220
236828
1
6898
2
1
3182
3562
* [1911. Only the Table is reproduced here. In the original paper will be found a description
of the process of calculation.]
150 ON THE ACOUSTIC SHADOW OF A SPHERE [292
Here F* + G* represents the intensity of sound at a great distance from the
sphere in directions such that /* is the cosine of the angle between them and
that radius which passes through the source. Upon the scale of measurement
adopted, F* + G* = \ for all values of /*, when kc = 0, that is, when the
propagation is undisturbed by any obstacle. The increased values under
/i = 1 show that the sphere is beginning to act as a reflector, the intensity in
this direction being already more than doubled when kc = 2. " In looking
at these figures, the first point which attracts attention is the comparatively
slight deviation from uniformity in the intensities in different directions.
Even when the circumference of the sphere amounts to twice the wave
length, there is scarcely anything to be called a sound shadow. But what is,
perhaps, still more unexpected is that in the first two cases the intensity
behind the sphere [/* = !] exceeds that in a transverse direction [/z, = 0].
This result depends mainly on the preponderance of the term of the first
order, which vanishes with fi. The order of the more important terms
increases with kc ; when kc is 2, the principal term is of the second order.
" Up to a certain point the augmentation of the sphere will increase the
total energy emitted, because a simple source emits twice as much energy
when close to a rigid plane as when entirely in the open. Within the limits
of the table this effect masks the obstruction due to an increasing sphere, so
that when /*= !, the intensity is greater when the circumference is twice
the wavelength than when it is half the wavelength, the source itself
remaining constant."
The solution of the problem when kc is very great cannot be obtained by
this method, but it is to be expected that when /* = 1 the intensity will be
quadrupled, as when the sphere becomes a plane, and that when p is
negative the intensity will tend to vanish. It is of interest to trace
somewhat more closely the approach to this state of things to treat, for
example, the case of kc= 10*. In every case where it can be carried out
the solution has a double interest, since in virtue of the law of reciprocity it
applies when the source and point of observation are interchanged, thus
giving the intensity at a point on the sphere due to a source situated at
a great distance.
But before proceeding to consider a higher value of kc, it will be well to
supplement the information already given under the head of kc = 2. The
original calculation was limited to the principal values of /u, corresponding to
the poles and the equator, under the impression that results for other values
of p would show nothing distinctive. The first suggestion to the contrary
was from experiment. In observing the shadow of a sphere, by listening
* See Rayleigh, Proc. Roy. Soc. Vol. LXMI. p. 40 [Scientific Papers, VoL v. p. 114] ; also
Maodonald, Vol. LKI. p. 251 ; Vol. LXZII. p. 69 ; Poincart, Vol. LXIII. p. 42.
1904]
ON THE ACOUSTIC SHADOW OF A SPHERE
151
through a tube whose open end was presented to the sphere, it was found
that the somewhat distant source was more loudly heard at the antipole
(/i = 1) than at points 40 or 50 therefrom. This is analogous to Poisson's
experiment, where a bright point is seen in the centre of the shadow of a
circular disc an experiment easily imitated acoustically* and it may be
generally explained in the same manner. This led to further calculations for
values of//, between and 1, giving numbers in harmony with observation.
The complete results for this case (kc = 2) are recorded in the annexed table.
In obtaining them, terms of Legendre's series, up to and including P 8 , were
retained. The angles d are those whose cosine is p.
Arc = 2.
e
F + iG
F*+fr
4(F 2 + G 2 )
+ 7968 + 2342V
6898
2759
15
+ 8021 + 1775 i
6749
2700
30
+ 7922 + 0147 i
6278
2511
45
+ '7139 '2287 *
5619
2248
60
+ 51 14 4793 i
4912
1965
75
+ 1898 6247 t
4263
1705
90
 1538  5766
3562
1425
105
3790 3413 i
2601
1040
120
 3992  0243
1600
0640
135
2401 +'2489 i
1196
0478
150
0088 + 4157 i
1729
0692
165
+ 1781 + 4883 i
2701
1080
180
+ 2495 + 5059 t
3182
1273
A plot of 4 (F* + G' 2 ) against 6 is given in fig. 1, curve A.
The investigation for kc 10 could probably be undertaken with success
upon the lines explained in" Theory of Sound ; but as it is necessary to
include some 20 terms of the expansion in Legendre's series, I considered
that it would be advantageous to use certain formulae of reduction by which
the functions of various orders can be deduced from their predecessors, and
this involves a change of notation. Formulae convenient for the purpose
have been set out by Professor Lambf. The velocitypotential i/r is supposed
to be proportional throughout to e ikat , but this timefactor is usually omitted.
The general differential equation satisfied by ^ is
(1)
of which the solution in polar coordinates applicable to a divergent wave of
the nth order in Laplace's series may be written
+ n = S n r Xn (kr) ............................... (2)
* Phil. Mag. Vol. ix. p. 278, 1880 ; Scientific Papers, Vol. i. p. 472.
t Hydrodynamics, 267 ; Camb. Phil. Trans. Vol. xvm. p. 350, 1900.
152
ON THE ACOUSTIC SHADOW OF A SPHERE
[292
<3Q
1904J ON THE ACOUSTIC SHADOW OF A SPHERE 153
For the present purpose we may suppose without loss of generality that
k=l. The differential equation satisfied by Xn (r) i 8
and of this the solution which corresponds to a divergent wave is
d \ n e~ ir
Putting n = and n = 1, we have
It is easy to verify that (4) satisfies (3). For if Xn satisfies (3), r 1 X ' n
satisfies the corresponding equation for Xn+1 . And r~ l e~ ir satisfies (3)
when n = 0.
From (3) and (4) the following formulae of reduction may be verified :
Xn(r)=r Xn+1 (r), (6)
(7)
(8)
By means of the last, X2 , X ^, &c. may be built up in succession from
and .
From (2) d^ n /dr = ^ n (nr^ Xn + r X ' n ),
or, with use of (7),
(9)
Thus, if U n be the nth component of the normal velocity at the surface of
the sphere (r = c),
................ (10)
Whenn=0, U = S oX ' (c) =  S c Xl (c) ...................... (11)
The introduction of S n from (10), (11) into (2) gives ^ n in terms of U n
supposed known.
When r is very great in comparison with the wavelength, we get
from (4)
so that ^. = S n ............................... (13)
154 ON THE ACOUSTIC SHADOW OF A SPHERE [292
In order to find the effect at a great distance of a source of sound
localised on the surface of the sphere at the point /*=!, we have only to
expand the complete value of U in Legendre's functions. Thus
= * (2n + 1) P n (M)/ +  Udp, = 2 ^ X P n 00 JJ UdS, ...(14)
in which fflfdS denotes the magnitude of the source, i.e., the integrated
value of U over the small area where it is sensible. The complete value
of ^r may now be written
.^ffUdS.e* ** v (2n + 1) iP. QQ ....
47rr *t <*w { Xn _, (c)  (n + 1) Xn (c)} '
When n = 0, ^ n _ 1 (c)  (n + 1) % n (c) is to be replaced by  c 2 ^i (c).
If we compare (15) with the corresponding expression in Theory of Sound,
(3), 238, we get
c) .......... (16)
Another particular case of interest arises when the point of observation,
as well as the source, is on the sphere, so that, instead of r = oo , we have
r = c. Equation (15) is then replaced by
.
It may be remarked that, since ^ in (17) is infinite when p 4 1 and
accordingly P n = 1, the convergence at other points can only be attained
in virtue of the factors P n . The difficulties in the way of a practical
calculation from (17) may be expected to be greater than in the case
of (15).
We will now proceed to the actual calculation for the case of c = 10,
or fa =10. The first step is the formation of the values of the various
functions x>(10), starting from % (10), XiC 1 ^) ^ or tnese we ^ ave ^ rom (5)
lO^o (10) = cos 10  i sin 10,
10 s xi (10) = & cos 10 + sin 10 + 1 (cos 10  ^ sin 10).
The angle (10 radians) = 540 + 32 57H68 ; thus
sin 10 =  5440210, cos 10=  "8390716,
and 10 X =  8390716 + 5440210 i,
10 Xl =  6279282  7846695 1.
1904]
ON THE ACOUSTIC SHADOW OF A SPHERE
155
From these, % s , ^ 3 , ... are to be computed in succession from (8), which
may be put into the form
For example,
10 3 Xa = 3 (lO'xO  lO^o = + '6506931  7794218 i.
When the various functions 10 w+1 ^ n have been computed, the next step
is the computation of the denominators in (15). We write
D n = 10 n+1 {% n i (n + 1) X n ] = 10 x 10 n ^ n _! (n + 1) 10 w+1 ^ n , . . .(18)
and the values of D n are given along with 10 n+1 ^ n in the annexed table.
n
10+i x (10)
D n
083907 + 054402 i
+ 62793 + 78467;
1
062793 078467 t
71349 + 70095 i
2
+ 065069077942
82314 55084 i
3
+ 095327 + 039496 t
+ 2693893741 i
4
+ 001 660 + 1 05589 t
+ 944981 3299 i
5
093834+ 055534 t
+ 57960 + 72269;
6
104877044501 t
20420+ 86685 i
7
0425061 13386 t
70872 + 46208t
8
+ 0411171255781
79512 00366 t'
9
+ 1 124061 00096 t
7128825482;
10
+ l72454064605i
7729329031;
11
+ 249747035574
12724321916 i
12
+ 401964017216t
27280713194;
13
+ 755164007465t
65526506764;
14
+ 1636978002941i
170030 03054;
15
+ 3992071 OO1062 i
 475033 0124;
16
+ 1073844 0003531
 142633 0047;
17
+ 31445  00010 i
 45862 0017;
18
+ 99319 0000 ;
 157250  0010 t'
19
+ 33603
 57274
20
+ 12112
220750
21
+ 46299
897460
22
+ 186974
38374X10 2
It will be seen that the imaginary part of 10 n+1 Xn(10) tends to zero,
as n increases. It is true that if we continue the calculation, having used
throughout, say, 5 figures, we find that the terms begin to increase again.
This, however, is but an imperfection of calculation, due to the increasing
value of 1 ! ?T (2n + l) in the formula and consequent loss of accuracy, as each
term is deduced from the preceding pair. Any doubt that may linger will
be removed by reference to (4), according to which the imaginary term
in question has the expression
156
ON THE ACOUSTIC SHADOW OF A SPHERE
[292
Now, if we expand ' sin r and perform the differentiations, the various
terms disappear in order. For example, after the 25th operation we have
d \sinr 50. 48. ..4. 2 52. 50. ..6. 4 54... 6 .
Tdr) ~ ! 51! 53! 55!
the first term being in every case positive and the subsequent terms
alternately negative and positive. The series is convergent, since the
numerical values of the terms continually diminish, the ratio of consecutive
terms being (when r = 10)
100 100 100^
2753' 4.55' 6.57'
Accordingly the first term gives a limit to the sum of the series. On
introduction of the factor 10"+', this becomes
10*
1.3. 5. ..49". 51'
i.e., approximately 10 ~ 8 x 3'0. A fortiori, when n is greater than 25, the
imaginary part of 10 n+1 % n (10) is wholly negligible.
We can now form the coefficients of P n under the sign of summation
in (15), i.e., the values of
i(2nfl)D n  1 (19)
For a reason that will presently appear, it is convenient to separate the
odd and even values of n.
n
i(2n + l)D n i
n
i(2n + l)D n  1
+ 0062170077691
1
+ 021020 021396 t
2
+041955028076;
3
+068978 019822 i'
4
+093391 + 013143;
5
+ 092629+ 074289 i'
6
+033 469 + 1 42083 t
7
096831+148517
8
 21 3800 +000984 i'
9
084474 236328 i'
10
+ 238104089430i
11
+ 030236 + 175549 t
12
091426 + 004422 t
13
 000425 041200 t'
14
+017056000031 t
15
+000002 + 006526 i
16
 00231 4 + 000000 t
17
000000 000762 t'
18
+ 000235
19
+ 000068 t
20
000018
21
000005 t
22
+ 000001
In the case of = 0, or p = + 1, the P's are all equal to + 1, and we have
nothing more to do than to add together all the terms in the above table.
When 6= 180, or /* = !, the even P's assume (as before) the value +1,
but now the odd P's have a reversed sign and are equal to 1. If we add
together separately the even and odd terms, and so obtain the two partial
sums 2, and 2 a , then 2, + 2 S will be the value of 2 for = 0, and 2,  2 2 will
be the value of 2 for 6 = 180. And this simplification applies not merely to
the special values and 180, but to all intermediate pairs of angles. If
2, + 2, corresponds to 0, 2,2 2 will correspond to 1800.
1904]
ON THE ACOUSTIC SHADOW OF A SPHEKK
157
For and 180 we find
S a = + 122870 I 35326 i,
whence for 6 =
+ 031135 + 85436 t ;
and for 0=180
= + 154005 + 120762 i,
2 = 2 (F + iG) = + 091735  050110 t.
When = 90, the odd P's vanish, and the even ones have the values
P.l,
1.3
4 ~2.4'
For other values of we require tables of P n (0) up to about n = 20.
That given by Professor Perry* is limited to n less than 7, and the results
are expressed only to 4 places of decimals. I have been fortunate enough
to interest Professor A. Lodge in this subject, and the Appendix to this
paper gives a table calculated by him containing the P's up to n = 20
inclusive, and for angles from to 90 at intervals of 5. As has already
been suggested, the range from to 90 practically covers that from 90
to 180, inasmuch as
P (90 4 ff) = P m (90  6), P M+1 (90 + d) =  P 2n+1 (90  6).
In the table of coefficients it will be observed that the highest entry
occurs at n = 10, in accordance with an anticipation expressed in a former
paper.
As will readily be understood, the multiplication by P n and the sum
mations involve a good deal of arithmetical labour. These operations, as
well as most of the preliminary ones, have been carried out in duplicate with
the assistance of Mr C. Boutflower, of Trinity College, Cambridge.
kc = 10.
e
2 (F+ iG)
i(J* k +G?
+ 154005 + 120762 t
38300
5
+ 158407 + 114959
38309
10
+ 170186 + 096603
38295
15
+ l'84773 + 063523i
38176
30
+ 152622117708 t
37148
45
 113754 148453 t
34978
60
074695 + 1 59746 t
31098
75
+ 145160 062553 i
24984
90
131954009924*'
175104
105
+ 094204 + 041681 t
106117
120
057769048417 t
056815
135
+ 029444 + 043841
027890
150
 008 146 035600 t
013338
165
012081+028341 i
009492
170
+035454 + 001457
012591
175
+076023034059i
069395
180
+ 091 735 0501 10 i
109263
Phil. Mag. Vol. xxxn. p. 516, 1891 ; see also Farr, Vol. XLIX. p. 572, 1900.
158
ON THE ACOUSTIC SHADOW OF A SPHERE
[292
The results are recorded in the annexed table and in curve B, fig. 1.
The intention had been to limit the calculations to intervals of 15, but
the rapid increase in (F* + G*) between 165 and 180 seemed to call for the
interpolation of two additional angles. This increase, corresponding to the
bright point in Poisson's experiment of the shadow of a circular disc, is
probably the most interesting feature of the results. A plot is given in
fig. 1, snowing the relation between the angle 0, measured from the pole, and
the intensity, proportional to F 1 + G*. It should, perhaps, be emphasised
that the effect here dealt with is the intensity of th'e pressure variation,
to which some percipients of sound, e.g., sensitive flames, are obtuse. Thus
at the antipole a sensitive flame close to the surface would not respond to
a distant source, since there is at that place no periodic motion, as is evident
from the symmetry.
I now proceed to consider the case where the source, as well as the place
of observation, are situated upon the sphere ; but as this is more difficult
than the preceding, I shall not attempt so complete a treatment. It will be
supposed still that kc= 10.
The analytical solution is expressed in (17), which we may compare with
(15). Restricting ourselves for the present to the factors under the sign of
summation, we see that the coefficient of P n in (17) is
while the corresponding coefficient in (15) is (2n + l)t n /Z) n .
If these coefficients be called C n , C' n respectively, we have
C' n = tc +1 x n (c).G" n ......................... (20)
in which the complex factors c n+1 y n (c), G' n , for c=10, have already been
tabulated. We find
(2n + l)10" XB
(2n + l)10**x
M
D n
n
A,
00099 + 00990 t
i
00306+029991
2
00542 + 05097 i
a
00835 + 07358 i
4
01233 +09883 '
5
01827+12817 i
6
02813 + 1 6390 t
7
04666 + 20956 i
8
08667 + 26889t
9
l8111+33152t
10
 35284 + 30805 i
11
42766 + 137961
12
36673 + 03351 i
13
31110+OO629 t
14
 27920 +OO100
15
26051+0OOHt
16
24844 + 00001 t
17
23998
18
23369
. 19
22881
20
22496
21
22183
22
21925
23
21711
24
21528
25
21374
26
21240
1904]
ON THE ACOUSTIC SHADOW OF A SPHERE
159
The product above tabulated shows marked signs of approaching the
limit 2, as n increases ; so that the series (17) is divergent when P n = 1,
i.e., when 8 = 0, as was of course to be expected. The interpretation may be
followed further. By the definition of P n , we have
{1  2a . cos e + a 2 }
so that, if we put a= 1,
1 + P 1 . a + P 2 . a 2 + ... + P n . a" + ... ; ...(21)
Thus, when 6 is small, and the series tends to be divergent, we get
from (17)
ffUdS.e^ .
'
and this is the correct value, seeing that 2c sin (^ 6} represents the distance
between the source and the point of observation, and that on account of the
sphere the value of ty is twice as great in the neighbourhood of the source as
it would be were the source situated in the open.
When 0=180, i.e., at the point on the sphere immediately opposite
to the source, the series converges, since P n takes alternately the values
+ 1 and 1. It will be convenient to retabulate continuously these
values from n = 18 onwards without regard to sign and to exhibit the
differences.
R
Function
First difference
Second difference
Third difference
18
23369
19
22881
 0488
._
20
22496
 0385
+ '0103
21
22183
 0313
+ 0072
0031
22
21925
O258
+ 0055
0017
23
21711
0214
+ 1)044
0011
24
21528
 0183
+ 0031
0013
25
21374
0154
+ 0029
0002
26
21240
 0134
+ 0020
0009
In summing the infinite series, we have to add together the terms as
they actually occur up to a certain point and then estimate the value of the
remainder. The simple addition is carried as far as n = 21 inclusive, and the
result is for the even values of n
 183939 + 93506 i,
and for the odd values
194734 + 91721 i,
or, with signs reversed to correspond with
+ 194734  91721 1.
160
ON THE ACOUSTIC SHADOW OF A SPHERE
[292
The complete sum up to n  21 inclusive is thus
+ 10795 + 1 785 (24)
The remainder is to be found by the methods of Finite Differences. The
formula applicable to series of this kind may be written
in which we may put
<(0) = 21925, </>(!) = 21711, &c.
Thus
(0)  <J> (1) + ... = + 10962 + 0054 + 0004 = 11020,
and for the actual remainder this is to be taken negatively. The sum of the
infinite series for = 180 is accordingly
 0225 + 1785 i ............................ (25)
from which the intensity, represented by (0225) 2 f (1785) 2 , is proportional
to 03237. Referring to (17), we see that the amplitude of ^ is in this case
 x x/(03237) ......................... (26)
We may compare this with the amplitude of the vibration which would
occur at the same place if the sphere were removed. Here
ffUdS
47T/'
.(27)
since r = 2c. The effect of the sphere is therefore to reduce the intensity in
the ratio of 25 to '03237.
In like manner we may treat the case of 9 = 90, i.e., when the point of
observation is on the equator. The odd P's now vanish and the even P's
take signs alternately opposite. The following table gives the values required
for the direct summation, i.e., up to n = 21 inclusive :
n
(fc + DWWjU.J'.iW)
n
(2n + l)10'H 1 x B .P l ,(90)
J>.
A*
4
8
12
16
20
 0099+ 0990
 0462+ 3706i
 2370+ 7353 i
 8273+ 0756 f
 4879+ OOOOi
 3964
2
6
10
14
18
+ 0271 2548 i
+ 0879 5122*
+ 8683  7581 i
+ 5848 0021 i
+ 4334
20047 + 128051
+ 20015l5272i
1904]
ON THE ACOUSTIC SHADOW OF A SPHERE
161
The next three terms, written without regard to sign, and their differences
are as follows :
22
3688
24
3470
 '0218
26
3292
 '0178
+ 0040
The remainder is found, as before, to be
+ i (3688) + i (0218) + (0040) = + 1903.
The sum of the infinite series from the beginning is accordingly
4 '1871  2467 i, (28)
in which (187 1) 2 + ('2467) 2 = '09588.
The distance between the source and the point of observation is now
2c sin 45 = c V2.
The intensity in the actual case is thus '09588 as compared with '5 if the
sphere were away.
For other angular positions than those already discussed, not only would
the arithmetical work be heavier on account of the factors P n , but the
remainder would demand a more elaborated treatment.
R. v.
11
162
ON THE ACOUSTIC SHADOW OF A SPHERE
[292
APPENDIX. BY PROFESSOR A. LODGE.
Table of Zonal Harmonics; i.e., of the Coefficients of the Powers of x
as far as P x in the Expansion of (1 2#cos + .c 2 )~z in the form
1 + P 1 * + P t a i I ... 4 PnX* + ... for 5 Intervals in the Values of 6 from
to 90. The Table is calculated to 7 decimal places, and the last
figure is approximate.
6
~y
5
10
1ft
P,( = C080)
*>
PS
P*
P 6
10000000
9961947
9848078
9659258
10000000
9886059
9547695
8995191
10000000
9772766
9105688
8041639
10000000
9622718
8532094
6846954
10000000
9436768
7839902
:.t71259
20
25
30
9396926
9063078
8660254
8245333
7320907
6250000
6648847
5016273
3247595
4749778
2465322
+ 0234375
2714918
+ 0008795
 2232722
35
40
45
8191520
7660444
7071068
5065151
3802362
2500000
+ 1454201
 0252333
 1767767
 1714242
 3190044
 4062500
 3690967
 4196822
 3756505
50
55
60
6427876
5735764
5000000
1197638
 0065151
 1250000
 3002205
 3886125
 4375000
 4275344
 3851868
 2890625
 2544885
 0867913
+ 0898437
65
70
75
4226183
3420201
2588190
 2320907
 3245333
 3995191
 4452218
 4130083
 3448846
 1552100
 0038000
+ 1434296
2381072
3280672
3427278
80
85
90
5
10
19
1736482
0871557
Nil
 4547695
 4886059
 5000000
 2473819
 1290785
Nil
+ 2659016
3467670
3750000
2810175
1576637
Nil
P 6
P
PS
P 9
Pio
iooooooo
9215975
7044712
3983060
10000000
8961595
6164362
+ 2455411
TOOOOOOO
8675072
52 18462
+ 0961844
10000000
8358030
+ 4227908
 O427679
10000000
8012263
+ 3214371
 1650562
20
25
30
+ 0719030
 2039822
 3740235
 10722<>2
 3440850
 4101780
 2518395
 "4062285
 3387755
 35 UK Mil!
 3895753
 1895752
 4012(592
 3052371
 0070382
35
40
45
50
55
60
 4114480
 3235708
 1484376
+ 0563782
2297230
3232421
 3095600
 1006016
+ 1270581
2854345
3190966
2231445
 1154393
+ 1386270
2983398
294<!>^ 1
+ 1421667
 0736389
+ 0965467
2900130
2855358
+ 1040702
 1296151
 2678985
+ 2541595
2! 173452
+ 1151123
 1381136
 2692039
 18S2286
65
70
75
3138270
2088770
+ 0431002
+ 0422192
 '1485259
 2730500
 2411439
 2780153
 1702200
 2300283
 0475854
+ 1594939
+ 0323225
+ 2192910
2316302
80
85
90
 1321214
 2637801
 3125000
 2834799
 '1778359
Nil
! + 0233080
2017462
2734376
2596272
1912893
Nil
+ 0646821
 1498947
 2460938
1904]
ON THE ACOUSTIC SHADOW OF A SPHERE
163
Table of Zonal Harmonics ; i.e., of the Coefficients of the Powers of x
as far as P^ in the Expansion of (1 2#cos 6 + #") in the form
1 + Pj# + Patf 2 + ... + P n x n + ... for 5 Intervals in the Values of from
to 90. The Table is calculated to 7 decimal places, and the last
figure is approximate continued.
6
PH
Pl2
Pl3
PH
Pis
6
5
10
15
10000000
7639723
+ 2199746
 2654901
10000000
7242508
+ 1205620
 3402156
10000000
6822849
+ 0252742
 3868998
10000000
+ 6383094
 0639478
 4048245
10000000
+ 5925694
 1453436
 3948856
20
25
30
 4001361
 1739692
+ 1607048
 3528461
 0223995
+ 2732027
 2682722
+ 1215469
+ 3066580
 1585374
+ 2332489
+ 2584895
 0376336
+ 2952537
+ 1465789
35
40
45
3096940
+ 1712040
 1041843
+ 2532528
 0211959
 2467193
+ 1130760
 1892595
 2393239
 0565267
 2599246
 0972709
 1950586
 2083112
+ O903925
50
55
60
 2640939
 1769491
+ 0638713
 1987621
+ 0522404
2337529
 0019170
+ 2209602
+ 1658041
+ 1821884
+ 1959135
 0571737
+ "2281988
+ 0110216
 2100185
65
70
75
+ 2351950
+ 1864450
 0305439
+ 1608831
 0787947
 2274796
 0863490
 2239288
 0850288
 2197701
 O745390
+ 1687887
 0989734
+ 1597121
+ 1638193
80
85
90
5
10
15
 2145820
 1988401
Nil
 1307104
+ 1041876
+ 2255858
+ 1544264
+ 2010073
Nil
+ 1730902
 0629592
 2094726
 0860215
 1982155
Nil
Pie
PIT
Pl8
f*
Pa,
10000000
+ 5453192
 2173739
 3594981
10000000
+ 4968206
 2787566
 3024136
10000000
+ 4473403
 3284945
 2284640
10000000
+ 3971492
 3658960
 1432466
iooooooo
+ 3465207
 3905880
 0527721
20
25
30
+ 0801110
2997862
+ 0036143
+ 1815511
+ 2495290
 1318805
+ 2560661
+ 1566049
 2254922
+ 2965867
+ O399984
 2553464
+ 3002029
 0780855
 2169986
35
40
45
 2565851
 0654984
+ 2150310
 2244163
+ 0986597
+ 2100803
 1151189
+ 2088162
+ 0857609
+ 0289684
+ 2180388
 O809310
+ 1556356
+ 1273281
 1930653
50
55
60
+ 1133974
 1714205
 1498551
 0732822
 2012353
+ 0522168
 1986904
 0625381
+ 1922962
 1792842
+ 1207910
+ 1377671
 0359655
+ 1945128
 0483584
65
70
75
+ 1249926
+ 1757158
 0760903
+ 1956924
 0336558
 1924117
+ 0427633
 1883363
 O249700
 1501989
 O935549
+ 1696995
 1644051
+ 1165241
+ 1093683
80
85
90
 1912133
+ 0255526
+ 1963808
+ 0165069
+ 1908789
Nil
+ 1861639
+ 0082151
 1854706
+ 0473145
 1794383
Nil
 1608344
 0383005
+ 1761970
112
164 ON THE ACOUSTIC SHADOW OF A SPHERE [292
NOTE BY LORD RAYLEIGH.
Professor Lodge's comparison of P x with Laplace's approximate value*
suggests the question whether it is possible to effect an improvement in the
approximate expression without entailing too great a complication. The
following, on the lines of the investigation in Todhunter's Functions of
Laplace, <fec.f, 89, shows, I think, that this can be done.
We have
When n is great, approximate values may be used for the coefficients of
the sines in (a). To obtain Laplace's expression it suffices to take
1 1.3 1.3.5
' 2' 2T4' 2TTT6' &C ' ;
but now we require a closer approximation. Thus
1 . (2n + 3) 2\ 2n + 2
1 . 3 . ( + 1) : (n + 2) !_._ / 1 1 \
1.2. (2n + 3) . (2n + 5) ~ 2 . 4 \ 2n + 2 2T+4/ '
and so on. If we write
the coefficients are approximately
1 1.3 1.3.5
*' 2*' 274*"' 2T476**' ^
;md the series takes actually the form assumed by Todhunter for analytical
convenience. In his notation
= *co8 + ^cos 30 + ^ # cos 50 + .
2.4
and P n = ^ . .. [C'a
where ultimately t is to be made equal to unity.
t Macmillan and Co., London, 1875.
1904]
ON THE ACOUSTIC SHADOW OF A SPHERE
165
Bv summation of the series (t < 1 ),
= J cos (0

where
1  2 2 cos 26 + V, tan (f>
F sin 20
1  t 3 cos 20 '
For our purpose it is only necessary to write C/t and S/t for C and S
respectively, and to identify P with a; in (7). Thus
4
to
p and <f> being given by (8). We find, with t= 1 l/4n,
P 2 = 4sin 2 0(ll/2n),
so that V P = V(2 sin 0) . (1 l/8n) ;
sin 20
2 sin 2 + l/2n '
7T COt
and
whence
Using (), (77), (y8) in (e) we get
P,
cot0
V(7rn sin 0)
which is the expression required.
By this extension, not only is a closer approximation obtained, but the
logic of the process is improved.
A comparison of values according to (0) with the true values may be
given in the case of n equal to 20.
Values of P~.
e
True value
j
According to (0)
15
 05277
 05320
30
21700
21712
45
 19307
 19306
60
 04836
 04834
75
+ 10937
+ 10937
90
+ 17620
+ 17618
[1911. I find that (0) had been given some years earlier and in a more
general form by Hobson, Phil. Trans. A, Vol. 187, p. 490, 1896.]
293.
SHADOWS.
[Royal Institution Proceedings, Jan. 15, 1904.]
MY subject is shadows, in the literal sense of the word shadows thrown
by light, and shadows thrown by sound. The ordinary shadow thrown by
light is familiar to all. When a fairly large obstacle is placed between
a small source of light and a white screen, a welldefined shadow of the
obstacle is thrown on the screen. This is a simple consequence of the
approximately rectilinear path of light. Optical shadows may be thrown
over great distances, if the light is of sufficient intensity : in a lunar eclipse
the shadow of the earth is thrown on the moon : in a solar eclipse the shadow
of the moon is thrown on the earth. Acoustic shadows, or shadows thrown
by sound, are not so familiar to most people; they are less perfect than optical
shadows, although their imperfections are usually overestimated in ordinary
observations. The ear is able to adjust its sensitiveness over a wide range,
so that, unless an acoustic shadow is very complete, it often escapes detection
by the unaided ear, the sound being sufficiently well heard in all positions.
In certain circumstances, however, acoustic shadows may be very pronounced,
and capable of easy observation.
The difference between acoustic and optical shadows was considered of so
much importance by Newton, that it prevented him from accepting the wave
theory of light. How, he argued, can light and sound be essentially similar
in their physical characteristics, when light casts definite shadows, while
sound shadows are imperfect or nonexistent ? This difficulty disappears
when due weight is given to the consideration that the lengths of light
waves and sound waves are of different orders of magnitude. Visible light
consists of waves of which the average length is about one fortythousandth
of an inch. Audible sound consists of waves ranging in length from about
an inch to nearly forty feet : the wave length corresponding to the middle C
of the musical scale is roughly equal to four feet. It is, therefore, no matter
for wonder that the effects produced by sound waves and by light waves
differ in important particulars.
1904] ON SHADOWS 167
Moreover, the wavelength is not the only magnitude on which the
perfection of the shadow depends ; the size of the obstacle, and the distance
across which the shadow is thrown, must also be taken into consideration.
The optical shadow of a small object, thrown across a considerable distance,
partakes of the imperfections generally observed in connection with sound
shadows.
It was calculated by the French mathematician, Poisson, that, according
to the wave theory of light, there should be a bright spot in the middle of the
shadow of a small circular disc a result that was thought to disprove the
wave theory by a reductio ad absurdum. Although unknown to Poisson, this
very phenomenon had actually been observed some years earlier, and was
easily verified when a suitably arranged experiment was made.
Under suitable conditions a bright spot can be observed at the centre of
the shadow of a threepenny bit. The coin may be supported by three
or four very fine wires, and its shadow thrown by sunlight admitted at
Fig. 1.
Beproduction of a Photograph of the Shadow of a Silver Penny Piece.
a pinhole aperture placed in the shutter of a darkened room. The coin
may be at a distance of about fifteen feet from the aperture, and the screen
at about fifteen feet beyond the coin. To obtain a more convenient
illumination, a larger aperture in the shutter may be filled by a short
focus lens, which forms a diminutive image of the sun ; this image serves
as a point source of light. A smaller disc has some advantages. Fig. 1
is reproduced from a photograph of the shadow of a silver penny piece,
struck at the time of the Coronation. The shadow, formed in the manner
just described, was allowed to fall directly on a photographic plate; after
development a negative was obtained, in which the dark parts of the shadow
were represented by transparent gelatine, while the bright parts were
represented by opaque deposits of silver. To obtain a correct representation,
a contact print was formed from the negative in the usual way, upon a lantern
plate ; and from this fig. 1 has been reproduced.
168 ON SHADOWS [293
It is at once evident that at the centre of the shadow, where one would
expect the darkness to be most complete, there is a distinct bright spot.
This result has always been considered a valuable confirmation of the wave
theory of light.
I now propose to speak of acoustic shadows shadows thrown by sound.
The most suitable source of sound for the following experiments is the bird
call*, which emits a note of high pitch so high, indeed, that it is inaudible
to most elderly people. The sound emitted has two characteristics, valuable
for our purpose the wavelength is very short ; and the sound is thrown
forward, without too much tendency to spread, thus differing from sounds
produced by most other means.
Since the sound emitted is nearly inaudible, some objective method of
observing it is required. For this purpose we may utilize the discovery
of Barrett and Tyndall, that a gas flame issuing under somewhat high
pressure from a pinhole burner flares when sound waves impinge on it, but
recovers and burns steadily when the sound ceases. The sensitiveness of
the flame depends on the pressure of the gas, which should be adjusted
so that flaring just does not occur in the absence of sound. If the birdcall
is directed towards the sensitive flame, the latter flares so long as the
call is sounded and no obstacle intervenes. On interposing the hand about
midway between the two, the flame recovers and burns steadily. Thus
the sound emitted by the birdcall casts a shadow, and to this extent
resembles light.
It will now be shown that the sensitive flame flares when it is placed
at the centre of the acoustic shadow thrown from a circular disc, but recovers
in any other position within the shadow ; thus proving that there is sound at
the centre of the shadow, although at a small distance from this point there
is silence. The part of the flame which is sensitive to sound is that just
above the pinhole orifice, so that it is necessary to arrange the birdcall, the
centre of the disc, and the pinhole orifice in a straight line. .For the disc, it
is convenient to use a circular plate of glass about 18 inches in diameter
with a piece of black paper pasted over its middle portion, a small hole being
cut in the paper exactly at the centre of the disc. The glass disc is hung by
two wires, and the positions of the birdcall and sensitive flame can be
adjusted by sighting through the hole in the paper. If the disc is caused to
oscillate in its own plane, the flame flares every time that the disc passes
through its position of equilibrium, and recovers whenever the disc is not in
that position. The analogy between this experiment, and that in which
a bright spot is formed at the centre of the optical shadow of a small disc, is
sufficiently obvious.
See Proc. Roy. lnt. Jan. 17. 1902. [This Collection, Vol. v. p. 1.]
1004] ON SHADOWS 169
The approximate theory of the shadow of the circular disc is easily given,
and it explains the leading features of the phenomenon. But, even in the
simpler case of sound, an exact calculation which shall take full account
of the conditions to be satisfied at the edge, has so far baffled the efforts
of mathematicians. When the obstacle is a sphere, the problem is more
tractable, and, in a recent memoir in the Philosophical Transactions a
solution is given, embracing the cases where the circumference of the sphere
is as great as two or even ten wavelengths. When the sphere is small
relatively to the wavelength, the calculation is easy, but the difficulty
rapidly increases as the diameter rises. The diagram* gives the intensity
in various positions on the surface of the sphere when plane waves of sound,
i.e. waves proceeding from a distant source, impinge upon it. The intensity
is a maximum at the point nearest to the source, which may be called the
pole. From the pole to the equator, distant 90 from it, the intensity falls
off, and the fall continues as we enter the hinder hemisphere. But at an
angular distance from the pole of about 135 in one case and 165 in the
other, the intensity reaches a minimum and thence increases towards the
antipole at 180.
In private experiments the distribution of sound over the surface of
the sphere may be explored with the aid of a small Helmholtz resonator
and a flexible tube, and in this way evidence may be obtained of the rise of
sound in the neighbourhood of the antipole. A more satisfactory demon
stration is obtained by the method already employed in the case of the disc,
the disc being replaced by a globe (about 12 inches in diameter), or by
a croquetball of about 3J inches diameter. In the former case the burner
may be situated behind the sphere at such a distance as 5 inches. In the
latter a distance of 1^ inches (from the surface) suffices. By a suitable
adjustment of the flame, flaring ensues when everything is exactly in line,
but the flame recovers when the ball is displaced slightly in a transverse
direction. Since the wavelength of the sound is 3 cm. and the circumference
of the croquetball is about 30 cm., this case corresponds to the curve B of
our diagram.
In connection with the mathematical investigation which led to the
results represented graphically in fig. 2*, there is a point of interest which
I should like to mention. The investigation was carried out upon the
supposition that the source of sound is at a considerable distance, so that
the waves reaching the sphere are plane ; and that the receiver, by which
the sound is detected, is situated on the surface of the sphere. At any
given position on the surface of the sphere, the receiver will indicate the
reception of sound of a certain intensity, which may be read off from fig. 2*.
Now the final results assume a form which shows that, if the positions of the
* [This Collection, Vol. v. p. 152.]
170 ON SHADOWS [293
source and the receiver are interchanged, the latter will indicate the reception
of sound of the same intensity as in the original arrangement. Thus each of
the curves in fig. 2* represents the solution of two distinct problems: the
intensity of the sound derived from a distant source and detected at any
point on the surface of the sphere ; and the intensity of the sound derived
from a source on the surface of the sphere, and observed at a distant point.
This result forms an interesting example of a principle of very wide application,
which I have termed the Principle of Reciprocity. Some special cases were
given many years ago by Helmholtz.
It is a matter of common observation that if one person can see another,
either directly or by means of any number of reflections in mirrors, then the
second person can equally well see the first. The same law applies to
hearing, apparent exceptions being easily explained. For instance, such
is the case of a lady sitting in a closed carriage, listening to a gentleman
talking to her through the open window. If the street is noisy, the lady can
hear what the gentleman says very much more distinctly than he can hear
what she replies. This is due to the fact that the gentleman's ears are
assailed by noises of the street from which the lady's ears are shielded by the
walls of the carriage.
Another instance may be mentioned, which will appeal to electricians.
In the arrangement known as Wheatstone's bridge, resistances are joined
in the form of a lozenge, a galvanometer being connected between two
opposite angles of the lozenge, while a battery is connected between the
other two angles. When the resistances are suitably adjusted, no current
flows through the galvanometer ; but a slight want of adjustment produces
a deflection of the galvanometer, thus indicating the passage of a small
current. Now, if the positions of the battery and the galvanometer are
interchanged, without alteration of resistance, the same current as before
will flow through the galvanometer, and therefore the deflection will be
the same as before. Thus with a given cell, galvanometer and set of
resistances, the sensitiveness of the Wheatstone's bridge arrangement is
the same whichever pair of opposite angles of the lozenge are joined
by the galvanometer. If a source of alternating E.M.F. is used instead
of the battery, and a telephone is substituted for the galvanometer, then the
principle of reciprocity still applies, whether the resistances are inductive
or noninductive.
A simple illustration, of a mechanical nature, is now shown. Fig. 3
represents a straightened piece of watchspring clamped at one end to
a firm support. A weight can be hung at either of the points A or B of the
spring, when it may be observed that the deflection at B due to the suspension
of the weight at A, is exactly equal to the deflection at A due to the
* [This Collection, Vol. v. p. 152.]
1904]
ON SHADOWS
171
suspension of the weight at B. This result is equally true wherever the
points A and B may be situated ; it applies not only to a loaded spring, which
has been chosen as suitable for a simple lantern demonstration, but also to
any sort of beam or girder.
It will have become clear, from what has been said, that waves encounter
considerable difficulty in passing round the outside of a curved surface.
I wish now to refer to a complementary phenomenon the ease with which
To Illustrate a Simple Mechanical Application of the Principle of Reciprocity.
waves travel round the inside of a curved surface. This is the case of the
whispering gallery, of which there is a good example in St Paul's Cathedral.
The late Sir George Airy considered that the effect could be explained as an
instance of concentrated echo, the sound being concentrated by the curved
walls, just as light may be brought to a focus by a concave mirror. From
my own observations, made in St Paul's Cathedral, I think that Airy's
Fig. 4.
Model Illustrating the Peculiarities of a Whispering Gallery.
explanation is not the true one ; for it is not necessary, in order to observe
the effect, that the whisperer and the listener should occupy particular
positions in the gallery. Any positions will do equally well. Again,
whispering is heard more distinctly than ordinary conversation, especially
if the whisperer's face is directed along the gallery towards the listener.
It is known that a whisper has less tendency to spread than the fullspoken
voice ; thus a whisper, heard easily in front of the whisperer, is inaudible
172 ON SHADOWS [293
behind that person's head. These considerations led me to form a fairly
satisfactory theory of the whispering gallery, nearly twentyfive years ago*.
The phenomenon may be illustrated experimentally by the small scale
arrangement represented diagrammatically in fig. 4. A strip of zinc, about
2 feet wide and 12 feet long, is bent into the form of a semicircle ; this forms
the model of the whispering gallery. The birdcall B [\ = 2 cm.] is adjusted
so that it throws the sound tangentially against the inner surface of the zinc :
it thus takes the place of the whisperer. The sensitive flame F takes
the place of the listener. A flame is always more sensitive to sound reaching
it in one direction than in others; the flame F is therefore adjusted so that
it is sensitive to sounds leaving the gallery tangentially. The flaring of the
flame shows that sound is reaching it : if an obstacle is interposed in the
straight line FB the flame flares as before ; but if a lath of wood W, which
need not be more than 2 inches wide, is placed against the inner surface
of the zinc, the flame recovers, showing that the sound has been inter
cepted. Thus the sound creeps round the inside surface of the zinc, and
there is no disturbance except at points within a limited distance from that
surface f.
' Theory of Sound, 287.
f [1911. For a theoretical treatment of the question see Phil. Mag. Vol. K. p. 1001. 1910.]
294.
SIR GEORGE GABRIEL STOKES, BART., 18191903.
[Royal Society YearBook, 1904.]
IN common with so many distinguished men Sir George Stokes was the
son of a clergyman. His father, Gabriel Stokes, who was Rector of Skreeu,
County Sligo, married Elizabeth Haughton, and by her had eight children
of whom George was the youngest. The family can be traced back to Gabriel
Stokes, born 1680, a well known engineer in Dublin and Deputy Surveyor
General of Ireland, who wrote a treatise on Hydrostatics and designed the
Pigeon House Wall in Dublin Harbour. This Gabriel Stokes married
Elizabeth King in 1711 and among his descendants in collateral branches
there are several mathematicians, a Regius Professor of Greek, two Regius
Professors of Medicine, and a large sprinkling of scholars of Trinity College,
Dublin. In more recent times Margaret Stokes, the Irish Antiquary, and
the Celtic scholar, Whitley Stokes, children of the eminent physician,
Dr William Stokes, have, among others, shed lustre on the name.
The home at Skreen was a very happy one. In the excellent sea air the
children grew up with strong bodies and active minds. Of course great
economy had to be practised to meet the educational needs of the family ;
but in the Arcadian simplicity of a place where chickens cost sixpence and
eggs were five or six a penny, it was easy to feed them. They were all deeply
attached to their mother, a beautiful and severe woman who made herself
feared as well as loved.
Stokes was taught at home ; he learnt reading and arithmetic from the
parish Clerk, and Latin from his father who had been a scholar of Trinity
College, Dublin. The former used to tell with great delight that Master
George had made out for himself new ways of doing sums, better than the
book. In 1832, at 13 years of age, he was sent to Dr Watts' school in
Dublin ; and in 1835 for two years to Bristol College, of which Dr Jerrard
was Principal. There is a tradition that he did many of the propositions of
Euclid, as problems, without looking at the book. He considered that he
174 (JEOKGK GABRIEL STOKES [294
owed much to the teaching of Francis Newman, brother of the Cardinal, then
mathematical master at Bristol College and a man of great charm of
character as well as of unusual attainments.
On the first crossing to Bristol the ship nearly foundered ; and his brother,
who was escorting him, was much impressed by his coolness in face of danger.
His habit, often remarked in after life, of answering with a plain " yes " or
" no," when something more elaborate was expected, is supposed to date from
this time, when his brothers chaffed him and warned him that if he gave
"long Irish answers" he would be laughed at by his schoolfellows.
It is surprising to learn that as a little boy he was passionate, and liable
to violent, if transitory, fits of rage. So completely was this tendency
overcome that in after life his temper was remarkably calm and even. He
was fond of botany, and when about sixteen or seventeen, collected butterflies
and caterpillars. It is narrated that one day while on a walk with a friend
he failed to return the salutation of some ladies of his acquaintance, afterwards
explaining his conduct by remarking that his hat was full of beetles !
In 1837, the year of Queen Victoria's accession, he commenced residence
at Cambridge, where he was to find his home, almost without intermission,
for sixtysix years. In those days sports were not the fashion for reading
men, but he was a good walker, and astonished his contemporaries by the
strength of his swimming. Even at a much later date he enjoyed encounters
with wind and waves in his summer holidays on the north coast of Ireland.
At Pembroke College his mathematical abilities soon attracted attention, and
in 1841 he graduated as Senior Wrangler and first Smith's Prizeman. In
the same year he was elected Fellow of his College.
After his degree, Stokes lost little time in applying his mathematical
powers to original investigation. During the next three or four years there
appeared papers dealing with hydrodynamics, wherein are contained many
standard theorems. As an example of these novelties, the use of a stream
function in three dimensions may be cited. It had already been shown by
Lagrange and Earnshaw that in the motion of an incompressible fluid in two
dimensions the component velocities at any point may be expressed by means
of a function known as the streamfunction, from the property that it remains
constant along any line of motion. It was further shown by Stokes that
there is a similar function in three dimensions when the motion is symmetrical
with respect to an axis. For many years the papers now under consideration
were very little known abroad, and some of the results are still attributed by
Continental writers to other authors.
A memoir of great importance on the " Friction of Fluids in Motion, etc.,"
followed a little later (1845). The most general motion of a medium in the
neighbourhood of any point is analysed into three constituents a motion of
1904] GEORGE GABRIEL STOKES 175
pure translation, one of pure rotation, and one of pure strain. These results
are now very familiar; it may assist us to appreciate their novelty at the
time, if we recall that when similar conclusions were v put forward by
Helmholtz twentythree years later, their validity was disputed by so acute
a critic as Bertrand. The splendid edifice, concerning the theory of inviscid
fluids, which Helmholtz raised upon these foundations, is the admiration of
all students of Hydrodynamics.
In applying the above purely kinematical analysis to viscous fluids, Stokes
lays down the following principle : " That the difference between the
pressure on a plane passing through any point P of a fluid in motion and the
pressure which would exist in all directions about P if the fluid in its neigh
bourhood were in a state of relative equilibrium depends only on the relative
motion of the fluid immediately about P ; and that the relative motion due
to any motion of rotation may be eliminated without affecting the differences
of the pressures above mentioned." This leads him to general dynamical
equations, such as had already been obtained by Navier and Poisson, starting
from more special hypotheses as to the constitution of matter.
Among the varied examples of the application of the general equations
two may be noted. In one of these, relating to the motion of fluid between
two coaxial revolving cylinders, an error of Newton's is corrected. In the
other, the propagation of sound, as influenced by viscosity, is examined. It
is shown that the action of viscosity (/*) is to make the intensity of the sound
diminish as the time increases, and to render the velocity of propagation less
than it would otherwise be. Both effects are greater for high than for low
notes ; but the former depends on the first power of p, while the latter
depends only on ft 2 , and may usually be neglected.
In the same paragraph there occur two lines in which a question, which
has recently been discussed on both sides, and treated as a novelty, is disposed
of. The words are " we may represent an arbitrary disturbance of the
medium as the aggregate of series of plane waves propagated in all directions."
In the third section of the memoir under consideration, Stokes applies
the same principles to find the equations for an elastic solid. In his view
the two elastic constants are independent and not reducible to one, as in
Poisson's theory of the constitution of matter. He refers to indiarubber
as hopelessly violating Poisson's condition. Stokes' position, powerfully
supported by Lord Kelvin, seems now to be generally accepted. Otherwise,
many familiar materials must be excluded from the category of elastic solids.
In 1846 he communicated to the British Association a Report on Recent
Researches in Hydrodynamics. This is a model of what such a survey should
be, and the suggestions contained in it have inspired many subsequent
investigations. He greatly admired the work of Green, and his comparison
176 GEORGE GABRIEL STOKES ['294
of opposite styles may often recur to the reader of mathematical lucubrations.
Speaking of the Reflection and Refraction of Sound, he remarks that " this
problem had been previously considered by Poisson in an elaborate memoir.
Poisson treats the subject with extreme generality, and his analysis is
consequently very complicated. Mr Green, on the contrary, restricts himself
to the case of plane waves, a case evidently comprising nearly all the
phenomena connected with this subject which are of interest in a physical
point of view, and thus is enabled to obtain his results by a very simple
analysis. Indeed Mr Green's memoirs are very remarkable, both for the
elegance and rigour of the analysis, and for the ease with which he arrives
at most important results. This arises in a great measure from his divesting
the problems he considers of all unnecessary generality ; where generality is
really of importance he does not shrink from it. In the present instance
there is one important respect in which Mr Green's investigation is more
general than Poisson's, which is, that Mr Green has taken the case of any two
fluids, whereas Poisson considered the case of two elastic fluids, in which
equal condensations produce equal increments of pressure. It is curious,
that Poisson, forgetting this restriction, applied his formulae to the case of
air and water. Of course his numerical result is quite erroneous. Mr Green
easily arrives at the ordinary laws of reflection and refraction. He obtains
also a very simple expression for the intensity of its reflected sound...." As
regards Poisson's work in general there was no lack of appreciation. Indeed,
both Green and Stokes may be regarded as followers of the French school
of mathematicians.
The most cursory notice of Stokes' hydrodynamical researches cannot close
without allusion to two important memoirs of somewhat later date. In 1847
he investigated anew the theory of oscillatory waves, as on the surface of the
sea, pursuing the approximation so as to cover the case where the height is
not very small in comparison with the wavelength. To the reprint in Math,
and Phys. Papers are added valuable appendices pushing the approximation
further by a new method, and showing that the slopes which meet at the
crest of the highest possible wave (capable of propagation without change of
type) enclose an angle of 120.
The other is the great treatise on the Effect of Internal Friction of Fluids
on the Motion of Pendulums. Here are given the solutions of difficult
mathematical problems relating to the motion of fluid about vibrating solid
masses of spherical or cylindrical form ; also, as a limiting case, the motion of
a viscous fluid in the neighbourhood of a uniformly advancing solid sphere,
and a calculation of the resistance experienced by the latter. In the
application of the results to actual pendulum observations, Stokes very
naturally assumed that the viscosity of air was proportional to density.
After Maxwell's great discovery that viscosity is independent of density
1904] GEORGE GABRIEL STOKES 177
within wide limits, the question assumed a different aspect; and in the
reprint of the memoir Stokes explains how it happened that the comparison
with theory was not more prejudiced by the use of an erroneous law.
In 1849 appeared another great memoir on the Dynamical Theory of
Diffraction, in which the luminiferous aether is treated as an elastic solid so
constituted as to behave as if it were nearly or quite incompressible. Many
fundamental propositions respecting the vibration of an elastic solid medium
are given here for the first time. For example, there is an investigation of
the disturbance due to the operation at one point of the medium of a periodic
force. The waves emitted are of course symmetrical with respect to the
direction of the force as axis. At a distance, the displacement is transverse
to the ray and in the plane which includes the axis, while along the axis
itself there is no disturbance. Incidentally a general theorem is formulated
connecting the disturbances due to initial displacements and velocities. " If
any material system in which the forces acting depend only on the positions
of the particles be slightly disturbed from a position of equilibrium, and then
left to itself, the part of the subsequent motion which depends on the initial
displacements may be obtained from the part which depends upon the initial
velocities by replacing the arbitrary functions, or arbitrary constants, which
express the initial velocities by those which express the corresponding initial
displacements, and differentiating with respect to the time."
One of the principal objects of the memoir was to determine the law of
vibration of the secondary waves into which in accordance with Huygens'
principle a primary wave may be resolved, and thence by a comparison with
phenomena observed with gratings to answer a question then much agitated
but now (unless restated) almost destitute of meaning, viz., whether the
vibrations of light are parallel or perpendicular to the plane of polarisation.
As to the law of the secondary wave Stokes' conclusion is expressed in the
following theorem: "Let = 0, 17 = 0, %=f(bt x) be the displacements
corresponding to the incident light; let 0j be any point in the plane P,
dS an element of that plane adjacent to O l ; and consider the disturbance
due to that portion only of the incident disturbance which passes continually
across dS. Let be any point in the medium situated at a distance from
the point O l which is large in comparison with the length of a wave ; let
00i = r, and let this line make angles 6 with the direction of propagation of
the incident light, or the axis of x, and <f> with the direction of vibration, or
the axis of z. Then the displacement at will take place in a direction
perpendicular to 00!, and lying in the plane Z0 1 ', and if ' be the
displacement at O l reckoned positive in the direction nearest to that in which
the incident vibrations are reckoned positive,
' = T (1 + cos 6) sin < . f (bt r).
R v. 12
178 GEORGE GABRIEL STOKES [294
In particular, if
f(btx) = c sin ^ (bt  x\
\
we shall have
f ' = ^? (i + cos 0) sin <f> . cos ^ (fc  7)."
1_ \ /' A*
Stokes' own experiments on the polarisation of light diffracted by a
grating led him to the conclusion that the vibrations of light are perpendicular
to the plane of polarisation.
The law of the secondary wave here deduced is doubtless a possible one,
but it seems questionable whether the problem is really so definite as Stokes
regarded it. A merely mathematical resolution may be effected in an
infinite number of ways ; and if the problem is regarded as a physical one,
it then becomes a question of the character of the obstruction offered by an
actual screen.
As regards the application of the phenomena of diffraction to the question
of the direction of vibration, Stokes' criterion finds a better subject in the
case of diffraction by very small particles disturbing an otherwise uniform
medium, as when a fine precipitate of sulphur falls from an aqueous solution.
The work already referred to, as well as his general reputation, naturally
marked out Stokes for the Lucasian Professorship, which fell vacant at this
time (1849). It is characterised throughout by accuracy of thought and
lucidity of statement. Analytical results are fully interpreted, and are
applied to questions of physical interest. Arithmetic is never shirked.
Among the papers which at this time flowed plentifully from his pen, one
" On Attractions, and on Clairaut's Theorem " deserves special mention. In
the writings of earlier authors the law of gravity at the various points of the
earth's surface had been deduced from more or less doubtful hypotheses as
the distribution of matter in the interior. It was reserved for Stokes to
point out that, in virtue of a simple theorem relating to the potential, the
law of gravity follows immediately from the form of the surface, assumed to
be one of equilibrium, and that no conclusion can be drawn concerning the
internal distribution of attracting matter.
From an early date he had interested himself in Optics, and especially
in the Wave Theory. Although, not long before, Herschel had written
ambiguously, and Brewster, the greatest living authority, was distinctly
hostile, the magnificent achievements of Fresnel had converted the younger
generation ; and, in his own University, Airy had made important applications
of the theory, e.g., to the explanation of the rainbow, and to the diffraction of
objectglasses. There is no sign of any reserve in the attitude of Stokes.
He threw himself without misgiving into the discussion of outstanding
1904] GEORGE GABRIEL STOKES 179
difficulties, such as those connected with the aberration of light, and by
further investigations succeeded in bringing new groups of phenomena within
the scope of the theony.
An early example of the latter is the paper " On the Theory of certain
Bands seen in the Spectrum." These bands, now known after the name of
Talbot, are seen when a spectrum is viewed through an aperture half covered
by a thin plate of mica or glass. In Talbot's view the bands are produced
by the interference of the two beams which traverse the two halves of the
aperture, darkness resulting whenever the relative retardation amounts to an
odd number of halfwave lengths. This explanation cannot be accepted as
it stands, being open to the same objection as Arago's theory of stellar
scintillation. A body emitting homogeneous light would not become invisible
on merely covering half the aperture of vision with a half wave plate. That
Talbot's view is insufficient is proved by the remarkable observation of
Brewster that the bands are seen only when the retarding plate is held
towards the blue side of the spectrum. By Stokes' theory this polarity is
fully explained, and the formation of the bands is shown to be connected
with the limitation of the aperture, viz., to be akin to the phenomena of
diffraction.
A little later we have an application of the general principle of reversion
to explain the perfect blackness of the central spot in Newton's rings, which
requires that when light passes from a second medium to a first the coefficient
of reflection shall be numerically the same as when the propagation is in the
opposite sense, but be affected with the reverse sign the celebrated " loss of
half an undulation." The result is obtained by expressing the conditions
that the refracted and reflected rays, due to a given incident ray, shall on
reversal reproduce that ray and no other.
It may be remarked that on any mechanical theory the reflection from an
infinitely thin plate must tend to vanish, and therefore that a contrary
conclusion can only mean that the theory has been applied incorrectly.
A not uncommon defect of the eye, known as astigmatism, was first
noticed by Airy. It is due to the eye refracting the light with different
power in different planes, so that the eye, regarded as an optical instrument,
is not symmetrical about its axis. As a consequence, lines drawn upon a
plane perpendicular to the line of vision are differently focussed according to
their direction in that plane. It may happen, for example, that vertical lines
are well seen under conditions where horizontal lines are wholly confused,
and vice versd. Airy had shown that the defect could be cured by cylindrical
lenses, such as are now common ; but no convenient method of testing had
been proposed. For this purpose Stokes introduced a pair of pianocylindrical
lenses of equal cylindrical curvatures, one convex and the other concave, and
122
180 GEORGE GABRIEL STOKES [294
so mounted as to admit of relative rotation. However the components may
be situated, the combination is upon the whole neither convex nor concave.
If the cylindrical axes are parallel, the one lens is entirely compensated by
the other, but as the axes diverge the combination forms an astigmatic lens
of gradually increasing power, reaching a maximum when the axes are
perpendicular. With the aid of this instrument, an eye, already focussed as
well as possible by means (if necessary) of a suitable spherical lens, convex or
concave, may be corrected for any degree or direction of astigmatism ; and
from the positions of the axes of the cylindrical lenses may be calculated, by
a simple rule, the curvatures of a single lens which will produce the same
result. It is now known that there are comparatively few eyes whose vision
may not be more or less improved by an astigmatic lens.
Passing over other investigations of considerable importance in themselves,
especially that on the composition and resolution of streams of polarised light
from different sources, we come to the great memoir on what is now called
Fluorescence, the most farreaching of Stokes' experimental discoveries. He
"was led into the researches detailed in this paper by considering a very
singular phenomenon which Sir J. Herschel had discovered in the case of
a weak solution of sulphate of quinine and various other salts of the same
alkaloid. This fluid appears colourless and transparent, like water, when
viewed by transmitted light, but exhibits in certain aspects a peculiar blue
colour. Sir J. Herschel found that when the fluid was illuminated by a beam
of ordinary daylight, the blue light was produced only throughout a very thin
stratum of fluid adjacent to the surface by which the light entered. It was
unpolarised. It passed freely through many inches of the fluid. The
incident beam after having passed through the stratum from which the blue
light came, was not sensibly enfeebled or coloured, but yet it had lost the
power of producing the usual blue colour when admitted into a solution of
sulphate of quinine. A beam of light modified in this mysterious manner
was called by Sir J. Herschel epipolised.
" Several years before, Sir D. Brewster had discovered in the case of an
alcoholic solution of the green colouring matter of leaves a very remarkable
phenomenon, which he has designated as internal dispersion. On admitting
into this fluid a beam of sunlight condensed by a lens, he was surprised by
finding the path of the rays within the fluid marked by a bright light of
a bloodred colour, strangely contrasting with the beautiful green of the fluid
itself when seen in moderate thickness. Sir David afterwards observed the
same phenomenon in various vegetable solutions and essential oils, and in
some solids. He conceived it to be due to coloured particles held in suspension.
But there was one circumstance attending the phenomenon which seemed
very difficult of explanation on such a supposition, namely, that the whole or
a great part of the dispersed beam was unpolarised, whereas a beam reflected
1904] m GEORGE GABRIEL STOKES 181
from suspended particles might be expected to be polarised by reflection.
And such was, in fact, the case with those beams which were plainly due to
nothing but particles held in suspension. From the general identity of the
circumstances attending the two phenomena, Sir D. Brewster was led to
conclude that epipolic was merely a particular case of internal dispersion,
peculiar only in this respect, that the rays capable of dispersion were dispersed
with unusual rapidity. But what rays they were which were capable of
affecting a solution of sulphate of quinine, why the active rays were so
quickly used up, while the dispersed rays which they produced passed freely
through the fluid, why the transmitted light when subjected to prismatic
analysis showed no deficiences in those regions to which, with respect to
refrangibility, the dispersed rays chiefly belonged, were questions to which
the answers appeared to be involved in as much mystery as ever."
Such a situation was well calculated to arouse the curiosity and enthusiasm
of a young investigator. A little consideration showed that it was hardly
possible to explain the facts without admitting that in undergoing dispersion
the light changed its refrangibility, but that if this rather startling supposition
were allowed, there was no further difficulty; and experiment soon placed
the fact of a change of refrangibility beyond doubt. " A pure spectrum from
sunlight having been formed in air in the usual manner, a glass vessel
containing a weak solution of sulphate of quinine was placed in it. The rays
belonging to the greater part of the visible spectrum passed freely through
the fluid, just as if it had been water, being merely reflected here and there
from motes. But from a point about halfway between the fixed lines G and
H to far beyond the extreme violet, the incident rays gave rise to a light of
a skyblue colour, which emanated in all directions from the portion of the
fluid which was under the influence of the incident rays. The anterior
surface of the blue space coincided, of course, with the inner surface of the
vessel in which the fluid was contained. The posterior surface marked the
distance to which the incident rays were able to penetrate before they were
absorbed. This distance was at first considerable, greater than the diameter
of the vessel, but it decreased with great rapidity as the refrangibility of the
incident rays increased, so that from a little beyond the extreme violet to the
end, the blue space was reduced to an excessively thin stratum adjacent to
the surface by which the incident rays entered. It appears, therefore, that
this fluid, which is so transparent with respect to nearly the whole of the
visible rays, is of an inky blackness with respect to the invisible rays, more
refrangible than the extreme violet. The fixed lines belonging to the violet
and the invisible region beyond were beautifully represented by dark planes
interrupting the blue space. When the eye was properly placed these planes
were, of course, projected into lines."
At a time when photography was of much less convenient application
182 GEORGE GABRIEL STOKES . [294
than at present even wet collodion was then a novelty the method of
investigating the ultraviolet region of the spectrum by means of fluorescence
was of great value. The obstacle presented by the imperfect transparency of
glass soon made itself apparent, and this material was replaced by quartz in
the lenses and prisms, and in the mirror of the heliostat. When the electric
arc was substituted for sunlight a great extension of the spectrum in the
direction of shorter waves became manifest.
Among the substances found " active " were the salts of uranium an
observation destined after nearly half a century to become in the hands of
Becquerel the starting point of a most interesting scientific advance, of which
we can hardly yet foresee the development.
In a great variety of cases the refrangibility of the dispersed light was
found to be less than that of the incident. That light is always degraded by
fluorescence is sometimes referred to as Stokes' law. Its universality has
been called in question, and the doubt is perhaps still unresolved. The
point is of considerable interest in connection with theories of radiation
and the second law of Thermodynamics.
Associated with fluorescence there is frequently seen a " false dispersion,"
due to suspended particles, sometimes of extreme minuteness. When a
horizontal beam of falsely dispersed light was viewed from above in a vertical
direction, and analysed, it was found to consist chiefly of light polarised in
the plane of reflection. On this fact Stokes founded an important argument
as to the direction of vibration of polarised light. For " if the diameters of
the (suspended) particles be small compared with the length of a wave of
light, it seems plain that the vibrations in a reflected ray cannot be
perpendicular to the vibrations in the incident ray." From this it follows
that the direction of vibration must be perpendicular to the plane of
polarisation, as Fresnel supposed, and the test seems to be simpler and more
direct than the analogous test with light diffracted from a grating. It should
not be overlooked that the argument involves the supposition that the effect
of a particle is to load the aether.
It was about this time that Lord Kelvin learned from Stokes " Solar and
Stellar Chemistry." " I used always to show [in lectures at Glasgow] a spirit
lamp flame with salt on it, behind a slit prolonging the dark line D by bright
continuation. I always gave your dynamical explanation, always asserted
that certainly there was sodium vapour in the sun's atmosphere and in the
atmospheres of stars which show presence of the Z)'s, and always pointed out
that the way to find other substances besides sodium in the sun and stars
was to compare bright lines produced by them in artificial flames with dark
lines of the spectra of the lights of the distant bodies*."
Letter to Stokes, published in Edinburgh addresg, 1871.
1904] GEORGE GABRIEL STOKES 18$
Stokes always deprecated the ascription to him of much credit in this
matter ; but what is certain is that had the scientific world been acquainted
with the correspondence of 1854, it could not have greeted the early memoir
of Kirchhoff (1859) as a new revelation. This correspondence will appear
in Vol. IV of Stokes' collected papers, now being prepared under the editor
ship of Prof. Larmor. The following is from a letter of Kelvin, dated
March 9, 1854: "It was Miller's experiment (which you told me about a long
time ago) which first convinced me that there must be a physical connection
between agency going on in and near the sun, and in the flame of a spirit
lamp with salt on it. I never doubted, after I learned Miller's experiment,
that there must be such a connection, nor can I conceive of any one knowing
Miller's experiment and doubting.... If it could only be made out that the
bright line D never occurs without soda, I should consider it perfectly certain
that there is soda or sodium in some state in or about the sun. If bright
lines in any other flames can be traced, as perfectly as Miller did in his case,
to agreement with dark lines in the solar spectrum, the connection would be
equally certain, to my mind. I quite expect a qualitative analysis of the
sun's atmosphere by experiments like Miller's on other flames."
By temperament, Stokes was overcautious. " We must not go too fast,"
he wrote. He felt doubts whether the effects might not be due to some
constituent of sodium, supposed to be broken up in the electric arc or flame,
rather than to sodium itself. But his facts and theories, if insufficient to
satisfy himself, were abundantly enough for Kelvin, and would doubtless have
convinced others. If Stokes hung back, his correspondent was ready enough
to push the application and to formulate the conclusions.
It is difficult to restrain a feeling of regret that these important advances
were no further published than in Lord Kelvin's Glasgow lectures. Possibly
want of time prevented Stokes from giving his attention to the question.
Prof. Larmor significantly remarks that he became Secretary of the Royal
Society in 1854. And the reader of the Collected Papers can hardly fail to
notice a marked falling off in the speed of production after this time. The
reflection suggests itself that scientific men should be kept to scientific work,
and should not be tempted to assume heavy administrative duties, at any
rate until such time as they have delivered their more important messages
to the world.
But if there was less original work, science benefited by the assistance
which, in his position as Secretary of the Royal Society, he was ever willing
to give to his fellow workers. The pages of the Proceedings and Transactions
abound with grateful recognitions of help thus rendered, and in many cases
his suggestions or comments form not the least valuable part of memoirs
which appear under the names of others. It is not in human nature for an
author to be equally grateful when his mistakes are indicated, but from the
184 GEORGE GABRIEL STOKES [294
point of view of the Society and of science in general, the service may be
very great. It is known that in not a few cases the criticism of Stokes was
instrumental in suppressing the publication of serious errors.
No one could be more free than he was from anything like an unworthy
jealousy of his comrades. Perhaps he would have been the better for a little
more wholesome desire for reputation. As happened in the case of Cavendish,
too great an indifference in this respect, especially if combined with a morbid
dread of mistakes, may easily lead to the withholding of valuable ideas and
even to the suppression of elaborate experimental work, which it is often
a labour to prepare for publication.
In 1857 he married Miss Robinson,daughter of Dr Romney Robinson, F.R.S., .
astronomer of Armagh. Their first residence was in rooms over a nursery
gardener's in the Trumpington Road, where they received visits from Whewell
and Sedgwick. Afterwards they took Lensfield Cottage, where they resided
until her death in 1899. Though of an unusually quiet and silent disposition,
he did not like being alone. He was often to be seen at parties and public
functions, and, indeed, rarely declined invitations. In later life, after he had
become President of the Royal Society, the hardihood and impunity with
which he attended public dinners were matters of general admiration. The
nonsense of fools, or rash statements by men of higher calibre, rarely provoked
him to speech ; but if directly appealed to, he would often explain his view
at length with characteristic moderation and lucidity.
His experimental work was executed with the most modest appliances.
.Many of his discoveries were made in a narrow passage behind the pantry of
his house, into the window of which he had a shutter fixed with a slit in it
and a bracket on which to place crystals and prisms. It was much the
same in lecture. For many years he gave an annual course on Physical
Optics, which was pretty generally attended by candidates for mathematical
honours. To some of these, at any rate, it was a delight to be taught by a
master of his subject, who was able to introduce into his lectures matter fresh
from the anvil. The present writer well remembers the experiments on the
spectra of blood, communicated in the same year (1864) to the Royal Society.
There was no elaborate apparatus of tanks and " spectroscopes." A testtube
contained the liquid and was held at arm's length behind a slit. The prism
was a small one of 60, and was held to the eye without the intervention of
lenses. The blood in a fresh condition showed the characteristic double
band in the green. On reduction by ferrous salt, the double band gave
place to a single one, to reassert itself after agitation with air. By such
simple means was a fundamental reaction established. The impression left
upon the hearer was that Stokes felt himself as much at home in chemical
and botanical questions as in Mathematics and Physics.
1904] GEORGE GABRIEL STOKES 185
At this time the scientific world expected from him a systematic treatise
on Light, and indeed a book was actually advertised as in preparation.
Pressure of work, and perhaps a growing habit of procrastination, interfered.
Many years later (1884 1887) the Burnett Lectures were published. Simple
and accurate, these lectures are a model of what such lectures should be, but
they hardly take the place of the treatise hoped for in the sixties. There
was, however, a valuable report on Double Refraction, communicated to the
British Association in 1862, in which are correlated the work of Cauchy,
MacCullagh and Green. To the theory of MacCullagh, Stokes, imbued with
the ideas of the elastic solid theory, did less than justice. Following Green,
he took too much for granted that the elasticity of aether must have its origin
in deformation, and was led to pronounce the incompatibility of MacCullagh's
theory with the laws of Mechanics. It has recently been shown at length by
Prof. Larmor that MacCullagh's equations may be interpreted on the
supposition that what is resisted is not deformation, but rotation. It is
interesting to note that Stokes here expressed his belief that the true
dynamical theory of double refraction was yet to be found.
In 1885 he communicated to the Society his observations upon one of the
most curious phenomena in the whole range of Optics a peculiar internal
coloured reflection from certain crystals of chlorate of potash. The seat of
the colour was found to be a narrow layer, perhaps onethousandth of an inch
in thickness, apparently constituting a twin stratum. Some of the leading
features were described as follows :
(1) If one of the crystalline plates be turned round in its own plane,
without alteration of the angle of incidence, the peculiar reflection vanishes
twice in a revolution, viz., when the plane of incidence coincides with the
plane of symmetry of the crystal.
(2) As the angle of incidence is increased, the reflected light becomes
brighter, and rises in refrangibility.
(3) The colours are not due to absorption, the transmitted light being
strictly complementary to the reflected.
(4) The coloured light is not polarised.
(5) The spectrum of the reflected light is frequently found to consist
almost entirely of a comparatively narrow band. In many cases the reflection
appears to be almost total.
Some of these peculiarities, such for example as the evanescence of the
reflection at perpendicular incidence, could easily be connected with the
properties of a twin plane, but the copiousness of the reflection at moderate
angles, as well as the high degree of selection, were highly mysterious. There
is reason to think that they depend upon a regular, or nearly regular,
alternation of twinning many times repeated.
186 GEORGE GABRIEL STOKES [294
It is impossible here to give anything more than a rough sketch of Stokes'
optical work, and many minor papers must be passed over without even
mention. But there are two or three contributions to other subjects as to
which a word must be said.
Dating as far back as 1857 there is a short but important discussion on
the effect of wind upon the intensity of sound. That sound is usually ill
heard up wind is a common observation, but the explanation is less simple
than is often supposed. The velocity of moderate winds in comparison with
that of sound is too small to be of direct importance. The effect is attributed
by Stokes to the fact that winds usually increase overhead, so that the front
of a wave proceeding up wind is more retarded above than below. The
front is thus tilted ; and since a wave is propagated normally to its front,
sound proceeding up wind tends to rise, and so to pass over the heads of
observers situated at the level of the source, who find themselves, in fact, in
a sound shadow.
In a more elaborate memoir (1868) he discusses the important subject of
the communication of vibration from a vibrating body to a surrounding gas.
In most cases a solid body vibrates without much change of volume, so that
the effect is represented by a distribution of sources over the surface, of
which the components are as much negative as positive. The resultant is
thus largely a question of interference, and it would vanish altogether were it
not for the different situations and distances of the positive and negative
elements. In any case it depends greatly upon the wavelength (in the gas)
of the vibration in progress. Stokes calculates in detail the theory for
vibrating spheres and cylinders, showing that when the wavelength is large
relatively to the dimensions of the vibrating segments, the resultant effect is
enormously diminished by interference. Thus the vibrations of a pianostring
are communicated to the air scarcely at all directly, but only through the
intervention of the sounding board*.
On the foundation of these principles he easily explains a curious
observation by Leslie, which had much mystified earlier writers. When a
bell is sounded in hydrogen, the intensity is greatly reduced. Not only so,
but reduction accompanies the actual addition of hydrogen to rarified air.
The fact is that the hydrogen increases the wavelength, and so renders more
complete the interference between the sounds originating in the positively
and negatively vibrating segments.
The determination of the laws of viscosity in gases was much advanced
by him. Largely through his assistance and advice, the first decisive
determinations at ordinary temperatures and pressures were effected by
* It may be worth notice that similar conclusions are more simply reached by considering
the particular case of A plane vibrating surface.
1904] GEORGE GABRIEL STOKES 187
Tomlinson. At a later period he brilliantly took advantage of Crookes'
observations on the decrement of oscillation of a vibrator in a partially
exhausted space to prove that Maxwell's law holds up to very high exhaustion
and to trace the mode of subsequent departure from it. Throughout the
course of Crookes' investigations on the electric discharge in vacuum tubes,
in which he was keenly interested and closely concerned, he upheld the
British view that the cathode stream consists of projected particles which
excite phosphorescence in obstacles by impact : and accordingly, after the
discovery of the Rontgen rays, he came forward with the view that they
consisted of very concentrated spherical pulses travelling through the aether,
but distributed quite fortuitously because excited by the random collisions of
the cathode particles.
A complete estimate of Stokes' position in scientific history would need
a consideration of his more purely mathematical writings, especially of those
on Fourier series and the discontinuity of arbitrary constants in semi
convergent expansions over a plane, but this would demand much space and
another pen. The present inadequate survey may close with an allusion to
another of those " notes," suggested by the work of others, where Stokes in
a few pages illuminated a subject hitherto obscure. By an adaptation of
Maxwell's colour diagram he showed (1891) how to represent the results of
experiments upon ternary mixtures, with reference to the work of Alder
Wright. If three points in the plane represent the pure substances, all
associations of them are quantitatively represented by points lying within
the triangle so defined. For example, if two points represent water and
ether, all points on the intermediate line represent associations of these
substances, but only small parts of the line near the two ends correspond to
mixture. If the proportions be more nearly equal, the association separates
into two parts. If a third point (off the line) represents alcohol, which is
a solvent for both, the triangle may be divided into two regions, one of
which corresponds to single mixtures of the three components, and the other
to proportions for which a single mixture is not possible.
A consideration of Stokes' work, even though limited to what has here
been touched upon, can lead to no other conclusion than that in many
subjects, and especially in Hydrodynamics and Optics, the advances which we
owe to him are fundamental. Instinct, amounting to genius, and accuracy
of workmanship are everywhere apparent ; and in scarcely a single instance
can it be said that he has failed to lead in the right direction. But, much as
he did, one can hardly repress a feeling that he might have done still more.
If the activity in original research of the first fifteen years had been maintained
for twenty years longer, much additional harvest might have been gathered
in. No doubt distractions of all kinds multiplied, and he was very punctilious
in the performance of duties more or less formal. During the sitting of the
188 GEORGE GABRIEL STOKES [294
last Cambridge Commission he interrupted his holiday in Ireland to attend
a single meeting, at which however, as was remarked, he scarcely opened his
mouth. His many friends and admirers usually took a different view from
his of the relative urgency of competing claims. Anything for which a date
was not fixed by the nature of the case, stood a poor chance. For example,
owing to projected improvements and additions, the third volume of his
Collected Works was delayed until eighteen years after the second, and fifty
years after the first appearance of any paper it included. Even this measure
of promptitude was only achieved under much pressure, private and official.
But his interest in matters scientific never failed. The intelligence of
new advances made by others gave him the greatest joy. Notably was this
the case in late years with regard to the Rontgen rays. He was delighted at
seeing a picture of the arm which he had broken sixty years before, and
finding that it showed clearly the united fracture.
Although this is not the place to dilate upon it, no sketch of Stokes can
omit to allude to the earnestness of his religious life. In early years he
seems to have been oppressed by certain theological difficulties, and was not
exactly what was then considered orthodox. Afterwards he saw his way
more clearly. In later life he took part in the work of the Victoria Institute :
the spirit which actuated him may be judged from the concluding words of
an Address on Science and Revelation. " But whether we agree or cannot
agree with the conclusions at which a scientific investigator may have arrived,
let us, above all things, beware of imputing evil motives to him, of charging
him with adopting his conclusions for the purpose of opposing what is revealed.
Scientific investigation is eminently truthful. The investigator may be
wrong, but it does not follow he is other than truthloving. If on some
subjects which we deem of the highest importance he does not agree with
us and yet he may agree with us more nearly than we suppose let us,
remembering our own imperfections, both of understanding and of practice,
bear in mind that caution of the Apostle : ' Who art thou that judgest
another man's servant? To his own master he standeth or falleth.'"
Scientific honours were showered upon him. He was Foreign Associate
of the French Institute, and Knight of the Prussian Order Pour le Mtrite.
He was awarded the Gauss Medal in 1877 ; the Arago on the occasion of the
Jubilee Celebration in 1899, and the Helmholtz in 1901. In 1889 he was
made a Baronet on the recommendation of Lord Salisbury. From 1887 to
1891 he represented the University of Cambridge in Parliament, in this, as
in the Presidency of the Society, following the example of his illustrious
predecessor in the Lucasian Chair. He was Secretary of the Society from
1854 to 1885, President from 1885 to 1890, received the Rumford medal in
1852, and the Copley in 1893.
1904] GEORGE GABRIEL STOKES 189
But the most remarkable testimony by far to the estimation in which he
was held by his scientific contemporaries was the gathering at Cambridge in
1899, in celebration of the Jubilee of his Professorship. Men of renown
flocked from all parts of the world to do him homage, and were as much
struck by the modesty and simplicity of his demeanour as they had previously
been by the brilliancy of his scientific achievements. The beautiful lines by
his colleague, Sir R. Jebb, cited below, were written upon this occasion.
There is little more to tell. In 1902 he was chosen Master of Pembroke.
But he did not long survive. At the annual dinner of the Cambridge
Philosophical Society, held in the College about a month before his death, he
managed to attend though very ill, and made an admirable speech, recalling
with charming simplicity and courtesy his lifelong intimate connection with
the College, to the Mastership of which he had recently been called, and with
the Society through which he had published much of his scientific work.
Near the end, while conscious that he had not long to live, he retained his
faculties unimpaired ; only during the last few hours he wandered slightly,
and imagined that he was addressing the undergraduates of his College,
exhorting them to purity of life. He died on the first of February, 1903.
Clear mind, strong heart, true servant of the light,
True to that light within the soul, whose ray
Pure and serene, hath brightened on thy way,
Honour and praise now crown thee on the height
Of tranquil years. Forgetfulness and night
Shall spare thy fame, when in some larger day
Of knowledge yet undream'd, Time makes a prey
Of many a deed and name that once were bright.
Thou, without haste or pause, from youth to age,
Hast moved with sure steps to thy goal. And thine
That sure renown which sage confirms to sage,
Borne from afar. Yet wisdom shows a sign
Greater, through all thy life, than glory's wage;
Thy strength hath rested on the Love Divine.
295.
ON THE MEASUREMENT OF CERTAIN VERY SHORT
INTERVALS OF TIME.
[Nature, Vol. LXIX, pp. 560, 561, 1904.]
ACCORDING to the discovery of Kerr, a layer of bisulphide of carbon,
bounded by two parallel plates of metal and thus constituting the dielectric
of a condenser or leyden, becomes doubly refracting when the leyden is
charged. The plates, situated in vertical planes, may be of such dimensions
as 18 cm. long, 3 cm. high, and the interval between them may be 0'3 cm.,
the line of vision being along the length and horizontal. If the polarising
and analysing nicols be set to extinction, with their principal planes at 45 to
the horizontal, there is revival of light when the leyden is charged. If the
leyden remain charged for some time and be then suddenly discharged, and
if the light under observation be sensibly instantaneous, it will be visible if
the moment of its occurrence be previous to the discharge ; if, however, this
moment be subsequent to the discharge, the light will be invisible. The
question now suggests itself, what will happen if the instantaneous light be
that of the spark by which the leyden is discharged ? It is evident that the
conditions are of extraordinary delicacy, and involve the duration of the
spark, however short this may be. The effect requires the simultaneity
of light and double refraction, whereas here, until the double refraction
begins to fail, there is no light to take advantage of.
The problem thus presented has been very skilfully treated by MM. Abra
ham and Lemoine (Ann. de Chimie, t. xx, p. 264, 1900). The sparks are
those obtained by connecting the leyden with a deflagrator and with the
terminals of a large Ruhmkorff coil fed with an alternating current. It is
known that if the capacity be not too small, several charges and discharges
occur during the course of one alternation in the primary, and that while the
charges are gradual, the discharges are sudden in the highest degree. If, as
in the present case, the capacity is small, it is necessary to submit the poles
of the deflagrator to a blast of air, otherwise the leyden goes out of action and
the discharge becomes continuous. Under the blast, the number of sparks
may amount to several thousands per second of time. In this way the
intensity of the light is much increased and the impression upon the eye
1904] MEASUREMENT OF CERTAIN VERY SHORT INTERVALS OF TIME 191
becomes continuous, but in other respects the phenomenon is the same as if
there were but one spark.
In order to obtain a measure of the double refraction, which is rapidly
variable in time, somewhat special arrangements are necessary. At the
receiving end the light, after emergence from the trough containing the
bisulphide of carbon, falls first upon a double image prism, of somewhat feeble
separating power, so held that one of the images is extinguished when the
leyden is out of action. The other image would be of full brightness, but
this, in its turn, is quenched by an analysing nicol. When there is double
refraction to be observed, the nicol is slightly rotated until the two images
are of equal brightness. This equality occurs in two positions, and the angle
between them may be taken as a measure of the effect. A full discussion is
given in the paper referred to.
The finiteness of the angle, which in my experiments amounted to 12,
is a proof that the light on arrival at the CS 2 still finds it in some degree
doubly refracting. To obtain the greatest effect the leads from the leyden
to the deflagrator should be as short as the case admits, and the course of
the light from the sparks to the CS 2 should not be unnecessarily prolonged.
The measure of the double refraction, and in an even greater degree the
brightness of the light as received, are favoured by connecting a very small
leyden directly with the spark terminals, but the advantage is hardly sufficient
to justify the complication.
The observations of Abraham and Lemoine bring out the striking fact
that if the course of the light be prolonged with the aid of reflectors so as to
delay by an infinitesimal time the arrival at the CS 2 , the opportunity to pass
afforded by the double refraction is in great degree lost, and the angular
measure of the effect is largely reduced. There is here no change in the
electrical conditions under which the spark occurs, but merely a delay in the
arrival of the light.
The optical arrangements which I found most convenient in repeating
the above experiment differ somewhat from those of the original authors.
The sparks are taken at a short distance from the polarising nicol and
somewhat on one side, and in both cases they are focussed upon the analysing
nicol. When the course is to be a minimum, the light is reflected obliquely
by a narrow strip of mirror situated in the axial line, and focussed by a lens
of short focus placed near the first nicol. This lens and mirror are so
mounted on stands that they can be quickly withdrawn, and by means of
suitable guidance and stops as quickly restored to their positions. In this
case the distance travelled by the light from its origin to the middle of the
length of CS 2 is about 30 cm.
The arrangements for a more prolonged course are similar, and they
remain undisturbed during one set of comparisons. The mirror is larger, and
192 MEASUREMENT OF CERTAIN VERY SHORT INTERVALS OF TIME [295
reflects nearly perpendicularly ; it is placed upon the axial line at a sufficient
distance behind the sparks. The light is rendered nearly parallel by a
photographic portrait lens of about 18 cm. focus, the aperture of which suffices
to fill up the field of view unless the distance is very long. In all cases the
eye of the observer is focussed upon the double image of the interval between
the plates of the CS 9 leyden.
The earlier experiments were made at home somewhat under difficulties.
For the blast nothing better was available than a glassblowing foot bellows ;
but nevertheless the results were fairly satisfactory. Afterwards at the
Royal Institution the use of a larger coil in connection with the public
supply of electricity, and of an automatic blowing machine, gave steadier
sparks and facilitated the readings. An increase of about one metre in the
total distance travelled by the light reduced the measured angle from 12 to
6, so that the time occupied by light in traversing one metre was very
conspicuous.
It is principally with the view of directing attention to the remarkable
results of Abraham and Lemoine that I describe the above repetition of their
experiment, but I have made one variation upon it which is not without
interest. In this case the spark is placed directly in the axial line and at
some distance behind, which involves the use of longer leads, and therefore
probably of a lower degree of instantaneity. The additional retardation is
now obtained by the insertion of a 60 cm. long tube containing CS 2 between
the sparks and the first nicol, and the comparison relates to the readings
obtained with and without this column, all else remaining untouched. The
difference is very distinct, and it represents the time taken in traversing the
CS a over and above that taken in traversing the same length of air. It
should be remarked that what we are here concerned with is not the wave
velocity in the CS 2 , but the <jrrowpvelocity, which differs from the former on
account of the dispersion.
In the above experiments the leyden, where the Kerr effect is produced,
is charged comparatively slowly and only suddenly discharged. For some
purposes the scope of the method would be extended if the whole duration of
the double refraction were made comparable with the above time of discharge.
This could be effected somewhat as in Lodge's experiments, where a spark,
called the Bspark, occurs between the outer coatings of two jars at the same
moment as the Aspark between their inner coatings. The outer coatings
remain all the while connected by a feeble conductor, which does not prevent
the formation of the Bspark under the violent conditions which attend the
passage of the Aspark. The plates of the Kerr leyden would be connected
with the outer coatings of the jars, or themselves constitute the " outer "
plates of two leydens replacing the jars.
296.
NOTE ON THE APPLICATION OF POISSON'S FORMULA TO
DISCONTINUOUS DISTURBANCES.
[Proceedings of the London Mathematical Society, Ser. 2, Vol. n.
pp. 266269, 1904.]
IN a recent paper* Prof. Love draws attention to " the discovery of an
oversight in Stokes's justly famous memoir on the 'Dynamical Theory of
Diffraction.'" The dilatation A satisfies the partial differential equation
d 2 A/dt* = a 2 V 2 A, and is calculated from it by means of Poisson's integral
formula. "According to this formula any function / which satisfies an
equation of the form d*f/dt* = a 2 V 2 f can be expressed in terms of initial
values by the equation
in which the integration refers to angular space about the point at which
f is estimated, and / (at) and / (at) denote the initial values of f and df/dt
on a sphere of radius at with its centre at the point."
"...it will be seen that all Stokes's results depend upon the employment
of Poisson's integral formula to express the dilatation and the components
of the rotation. In a recent paper I have pointed out that this formula
does not in general yield correct expressions for these quantities. In the
same paper I identified the formula (A) with one which has been used by
Poincar6 and others, viz.,
where r denotes distance from the point at which / is estimated."
"The reason for the failure of such formulae as (B) to represent the
dilatation and the components of rotation is clear from an inspection of (B).
When the point at which the disturbance is estimated is near the front of
an advancing wave the sphere described about the point penetrates but a
little way into the region within which the initial disturbance is confined,
and the part of the sphere which is included in the integration is very small.
Thus the formula cannot express any quantity which has a value different
from zero at the front of an advancing wave. Now there is no kinematical
* Proc. London Math. Soc., Ser. 2, Vol. i. p. 291, 1903.
R. V. 13
194
NOTE ON THE APPLICATION OF POISSON'S
[296
or dynamical reason why the dilatation and rotation in an elastic solid should
be supposed to vanish at the front of an advancing wave, and it appears
therefore that Stokes's analysis is adequate to express the effects of particular
types of initial disturbance, but not those of an arbitrary initial disturbance
confined to a finite portion of the medium."
Having myself on a former occasion* applied Poisson's formula to the
forbidden case of a uniform initial condensation limited to the slice bounded
by two parallel planes without meeting any difficulty, I was naturally rather
taken aback by the above criticism, although it is true that I then contem
plated f as representing the velocitypotential rather than the dilatation.
But the argument for the dilatation assumes much the same form, and it
may be desirable to set it out in full.
Let us suppose then that a gaseous medium is initially undisturbed
except between the parallel planes A, B, and that within AB there is
initially no velocity, but only a uniform dilatation (or condensation). We
know, of course, what will happen from the theory of plane waves. The
initial state of things is equivalent to the superposition of two progressive
waves between which the dilatation is shared. These advance in opposite
directions, and in each the particle velocity is uniform and in the direction of
propagation. We have now to inquire what account of the matter Poisson's
formula will give.
In this / represents the initial dilatation, confined to AB. The initial
velocity of dilatation / is zero both
within and without the slice, but this
is not of itself sufficient to establish the
evanescence of the first integral in (A)
after the sphere has reached the slice.
We have also to consider what may
happen at the boundary planes A and B
themselves. Taking the plane A, we see
that immediately in front of it the
dilatation rises suddenly from to %f , but
the effect of this is compensated in the
integral by the drop from / to /
which occurs symmetrically behind. In
like manner the boundary B can con
tribute nothing, and we may equate the first integral to zero.
In the second integral, f has a constant value over the portion QPQ'
of the sphere and vanishes over the remainder. Also, if denote the
angle AOQ,
Theory of Sound, 274.
1904] FORMULA TO DISCONTINUOUS DISTURBANCES
and OP=at. Thus
195
since AP increases with velocity a; and accordingly the dilatation is
correctly given by Poisson's formula.
When the wave has passed, the sphere
cuts completely through the slice, and
AB AB
so that tffd<7 = constant,
and consequently /= 0. In all respects
the passage of the wave is correctly repre
sented.
It is clear that the objection is really
directed not against (A), but against (B),
which, so far as I know, was not used at
all by Poisson or by Stokes. And indeed this is recognized by Prof. Love
himself in a later passage (p. 321), where he seems to shift the ground and
advances against (A) an entirely distinct objection. With this I am not at
present concerned, though I have my doubts whether it is more substantial
than the first.
Even as regards (B), the charge of failure preferred against it seems to
ignore the fact that the integrand becomes infinite in the case supposed
of a discontinuous initial condition. Let us apply (B) to the circumstances
already defined of a dilatation limited to the slice between two planes A
and B. The first term in / vanishes as before. The second term in f , also
as before, assumes the value / (1  cos 0). The only difficulty is in the
third term, where df /dr, vanishing within and without the slice, becomes
infinite on the circle of transition whose diameter is QQ'. If # be a coordinate
measured parallel to AB, f is a function of x only, and
*&*
dr dx
Thus the third integral
= cos 6 ./ ;
J WH
and altogether (B) gives / , the correct result. According to Prof. Love's
indictment the formula yields zero when the sphere only cuts a little into
the slice.
132
297.
FLUID FRICTION ON EVEN SURFACES.
[Note to a paper by Prof. Zahm, Phil. Mag., Vol. vm. pp. 66, 67, 1904.]
IN connexion with such experiments as those of Froude and Zahm
respectively on flat surfaces moving tangentially through water or air, it is
of interest to inquire how much can be inferred from the principle of
dynamical similarity. Dynamical similarity includes, of course, geometrical
similarity, so that in any comparisons the surfaces must be similar, not only
in respect of their boundaries, but also in respect of their roughnesses, at any
rate until it is proved that the restriction may be dispensed with. Full
geometrical similarity being presupposed, the tangential force per unit
area F may be regarded as a function of a the linear dimension of the solid,
p the density of the fluid, v the velocity, and v the kinematic coefficient of
viscosity. It is assumed that the compressibility of the fluid does not come
into account. As in a similar problem relating to the flow of fluid along
pipes*, the method of dimensions shows that, if it be a function of the above
quantities only,
F=pv*.f(av/v), (1)
where /denotes an arbitrary function, whose form must be obtained, if at all,
from other considerations. If F be independent of v, f is constant, and F is
proportional to the density and the simple square of the velocity. Conversely,
if F be not proportional to v j , / is not constant, and the viscosity and the
linear dimension enter.
If the general formula be admitted, several conclusions of importance can
be drawn, even though the form of / be entirely unknown. For example, in
the case of a given fluid (/> and v constant), F is strictly proportional to v 8 ,
provided a be taken inversely as v. Again, if the fluid be varied, we may
make comparisons relating to the same surface (a constant). For if v be
taken proportional to v, f remains unaffected ; so that F is proportional to pv*
PkU. Mag. ZJUUT. p. 59 (1892) ; Scientific Papers, in. p. 575.
1904] FLUID FRICTION ON EVEN SURFACES 197
simply. For air the kinematic viscosity is about 10 times greater than for
water, so that to obtain comparable cases the velocity through air would
need to be 10 times greater than through water, a remaining unchanged.
This condition being satisfied, the frictions per unit area would be as pv*, or
since water is about 1000 times as dense as air, the actual friction would be
about 10 times greater for the water than for the air.
According to Dr Zahm's experiments, where the velocity alone is varied,
the form of f is within certain limits determined to be
Within the same limits (1) gives as the complete expression for F,
( v /ay i &. ........................... (2)
[1911. In a series of experiments designed to determine the form of F,
there would be advantage in keeping v (and a) constant, while v is varied.]
298.
ON THE ELECTRICAL VIBRATIONS ASSOCIATED WITH
THIN TERMINATED CONDUCTING RODS.
[Philosophical Magazine, Vol. vin. pp. 105107, 1904.]
IN his discussion of this subject Prof. Pollock* rejects the simple
theoretical result of Abraham and others, according to which the wave
length (X) of the gravest vibration is equal to twice the length (I) of the rod,
in favour of the calculation of Macdonald which makes X = 2'53 I. On this
I would make a few remarks, entirely from the theoretical point of view.
The investigation of Abraham f is a straightforward one; and though
I do not profess to have followed it in detail, I see no reason for distrusting
it. It relates to the vibration about a perfect conductor in the form of an
elongated ellipsoid of revolution ; and the abovementioned conclusion follows
when the minor axis (26) of the ellipse vanishes in comparison with the
major axis /. As a second approximation Abraham finds
\ = 21 (1 + 56 e 2 ), (1)
where l/e= 41og(Z/6) (2)
But a question arises as to whether a result obtained for an infinitely
thin ellipsoid can be applied to an infinitely thin rod of uniform section. So
far as I see, it is not discussed by Abraham, though he refers to his conductor
as rodshaped. The character of the distinction may be illustrated by
considering the somewhat analogous case of aerial vibrations within a cavity
having the shape of the conductor. If the section be uniform, the wave
length of the longitudinal aerial vibration is exactly twice the length ; but if
an ellipsoidal cavity of the same length be substituted, then, however narrow
it may be, the wavelength will be diminished in a finite ratio on account of
the expansion of the section towards the central partsj. This example may
Phil. Mag. vn. p. 635 (1904).
t Wied. Ann. LXVI. p. 435 (1898).
J See Theory of Sound, 265.
1904] ELECTRICAL VIBRATIONS ASSOCIATED WITH CONDUCTING RODS 199
suffice to show that no general extension can be made from the ellipsoidal
to the cylindrical shape, however attenuated the section may be.
When we ask whether the extension is justifiable in the present case, we
shall find, I think, that the answer is in the affirmative so far as the first
approximation is concerned, but in the negative for the second approximation.
Let us commence with the consideration of the known solution for an
infinite conducting cylinder of radius r l enclosed in a coaxal sheath of radius
r z *. If in Maxwell's notation P, Q, R be the components of electromotive
intensity; a, b, c those of magnetic induction; V the velocity of light;
we have
P, Q, R = cospt.cosmz( z , , 0\ ,
in which z is measured along the axis and m=p/V. These expressions are
independent of r, and r 2 , and thus the nodal distance corresponding to a
given frequency of vibration is the same whatever may be the diameters of
the cylinders. But although the relation of m to p is unchanged, as ^ is
supposed to be reduced, the corresponding energies increase and that without
limit. Apart from the first two factors, the value of (e.g.) the resultant
electromotive intensity varies as 1/r; so that the integrated square is
proportional to log (r^r^. We see that when ^ is reduced without limit, the
phenomenon is ultimately dominated by the infinite energies associated with
the immediate neighbourhood of the attenuated conductor. Moreover,
when 7*1 is already infinitesimal, its further reduction in a finite ratio causes
only a vanishing relative change in the value of the energy of vibration.
In the problem for which the solution is above analytically expressed the
section of the rod is circular and uniform, but the considerations already
advanced point to the conclusion that when the section is infinitesimal
neither the circularity nor the uniformity is essential. So long, at any rate,
as all diameters, whether of the same section or of different sections, are in
finite ratios to one another, the relation of nodal interval to frequency
remains undisturbed.
The same line of argument further indicates that the conclusion may be
extended to a terminated rod of infinitesimal section. For the infinite energy
associated with the neighbourhood of the conductor is unaffected except at
points infinitely near the ends. It appears therefore that the wavelength of
the electrical vibration associated with a straight terminated rod of in
finitesimal section, is equal to twice the length of the rod, whether the shape
* See, for example, Phil. Mag. XLIV. p. 199 (1897) ; Scientific Papert, iv. p. 327.
200 ELECTRICAL VIBRATIONS ASSOCIATED WITH CONDUCTING RODS [298
be cylindrical so that the radius is constant, or ellipsoidal so that the radius
varies in a finite ratio at different points of the length. And this conclusion
still remains undisturbed, even though the shape be not one of revolution.
Whether the conditions of the limit can be sufficiently attained in
experiment is a question upon which I am not prepared to express a decided
opinion. From the logarithmic character of the infinity upon which the
argument is founded, one would suppose that there might be practical
difficulty in reducing the section sufficiently. Even if an adequate reduction
were possible mechanically, the conductivity of actual materials might fail.
We must remember that in the theory the conductivity is supposed to be
perfect.
299.
ON THE DENSITY OF NITROUS OXIDE.
[Proceedings of the Royal Society, Vol. LXXIV. pp. 181183, 1904.]
IN the Proceedings, Vol. LXXII. p. 204, 1897 *, I have given particulars of
weighings of nitrous oxide purified by two distinct methods. In the first
procedure, solution in water was employed as a means of separating less
soluble impurities, and the result was 3'6356 grammes. In the second
method a process of fractional distillation was employed. Gas drawn from
the liquid so prepared gave 3*6362. These numbers may be taken to repre
sent the corrected weight of the gas which fills the globe at C. and at the
pressure of the gauge (at 15), and they correspond to 2'6276 for oxygen.
Inasmuch as nitrous oxide is heavier than the impurities likely to be
contained in it, the second number was the more probable. But as I thought
that the first method should also have given a good result, I contented
myself with the mean of the two methods, viz. 3'6359, from which I calcu
lated that referred to air (free from H 2 O and CO a ) as unity, the density of
nitrous oxide was 152951.
The corresponding density found by M. Leduc is 1*5301, appreciably
higher than mine ; and M. Leduc argues that the gas weighed by me must
still have contained one or two thousandths of nitrogen f*. According to him
the weight of the gas contained in my globe should be 3'6374, or 1*5 milli
grammes above the mean of the two methods.
Wishing, if possible, to resolve the question thus raised, I have lately
resumed these researches, purifying the nitrous oxide with the aid of liquid
air kindly placed at my disposal by Sir J. Dewar, but I have not succeeded
in raising the weight of my gas by more than a fraction of the discrepancy
(15 milligramme). I have experimented with gas carefully prepared in the
laboratory from nitrate of ammonia, but as most of the work related to
material specially supplied in an iron bottle I will limit myself to it.
Or Scientific Papers, Vol. iv. p. 350.
t Recherche* xnr leg Gaz, Paris, 1898.
202 ON THE DENSITY OF NITROUS OXIDE [299
There are two ways in which the gas may be drawn from the supply.
When the valve is upwards, the supply comes from the vapourous portion
within the bottle, but when the valve is downwards, from the liquid portion.
The latter is the more free from relatively volatile impurities, and accordingly
gives the higher weight, and the difference between the two affords an indi
cation of the amount of impurity present. After treatment with caustic
alkali and sulphuric acid, the gas is conducted through a tap, which is closed
when it is desired to make a vacuum over the frozen mass, and thence over
phosphoric anhydride to the globe. For the details of apparatus, &c., refer
ence must be made to former papers. The condensing chamber, which can
be immersed in liquid air, is in the form of a vertical tube, 2 cm. in diameter
and 22 cm. long, closed below and above connected laterally to the main
channel.
The first experiment on July 13 was upon gas from the top of the bottle
as supplied, and without treatment by liquid air, with the view of finding
out the worst. The weight was 3*6015, about 35 milligrammes too light.
The stock of material was then purified, much as in 1896. For this purpose
the bottle was cooled in ice and salt* and allowed during about one hour to
blow off half its contents, being subjected to violent shaking at frequent
intervals. Subsequently three weighings were carried out with gas drawn
from the bottom, but without treatment by liquid air. The results stand :
July 18 36368
July 20 36360
July 25 36362
Mean 363633
Next followed experiments in which gas, still drawn from the bottom
of the bottle, was further purified by condensation with liquid air. The gas,
arriving in a regular stream, was solidified in the condensing chamber.
When it was judged that a sufficiency had been collected, the tap behind
was turned and a high vacuum established over the solid mass with the aid
of the Topler pump. The pump would then be cut off and the gas allowed
to evaporate and accumulate in the globe. A reapplication of liquid air
caused the gas to desert the globe for the condensing chamber, until in
a surprisingly short time a vacuum was reestablished. Little or nothing
could now be drawn off by the pump, and it was thought that a distinct
difference could be perceived between the first and second operations,
indicating that in the first condensation a little impurity remained gaseous.
If desired, the condensation could be repeated a third time. On one occasion
(August 7) the condensed gas was allowed to liquefy, for which purpose the
The lower the temperature below the critical point, the more effective is this procedure
likely to be.
1904] ON THE DENSITY OF NITROUS OXIDE 203
pressure must rise to not far short of atmospheric, and to blow off part of its
contents :
August 1 36363
Augusts 36367
August 7 36366
Mean 3'63653
The treatment with liquid air raised the weight by only 0'2 milligramme,
but the improvement is probably real. That the stock in the bottle still
contained appreciable impurity is indicated by a weighing on August 13,
in which without liquid air the gas was drawn from the top of the bottle.
There appeared
August 13 36354,
about 1 milligramme short of the proper weight.
It will be seen that the result without liquid air is almost identical with
that found by the same method in 1896, and that the further purification by
means of liquid air raises the weight only to 3'6365. I find it difficult to
believe that so purified the gas still contains appreciable quantities of nitrogen.
The corresponding weight of air being 2'3772*, we find that, referred to
air as unity, the density of nitrous oxide is
3 ' 6365  15297
23772"
Again, if oxygen be taken as 16, the density of nitrous oxide will be
36365 x 16
26276
22143.
The excess above 22 is doubtless principally due to the departure of
nitrous oxide from Boyle's law between atmospheric pressure and a condition
of great rarefaction. I hope shortly to be in a position to apply the cor
rection which will allow us to infer what is the ratio of molecular weights
according to Avogadro's rulef.
* Roy. Soc. Proc. Vol. LIII. p. 134, 1893 ; Scientific Papers, Vol. iv. p. 47.
t [1911. See Art. 303 below.]
300.
NOTE TO A PAPER BY PROF. WOOD ON THE ACHROMATI
ZATION OF APPROXIMATELY MONOCHROMATIC INTER
FERENCE FRINGES BY A HIGHLY DISPERSIVE MEDIUM.
[Phil. Mag., Vol. vm. pp. 330, 331, 1904.]
HAVING had an opportunity of seeing the above paper in proof, I append
with Prof. Wood's permission a lew remarks.
The remarkable shift of the bands of helium light when a layer of sodium
vapour is interposed in the path of one of the interfering pencils, is of the
same nature as the displacement of the white centre found by Airy and
Stokes to follow the insertion of a thin plate of glass. If D denote the
thickness of the plate and p its refractive index, (/u, 1)D is the retardation
due to the insertion of the plate, and if R be the relative retardation due to
other causes, the whole relative retardation is
JR + 0* !)/>, .............................. (1)
in which R and D are supposed to be independent of the wavelength X,
while p does depend upon it. The order of the band (n) is given by
(2)
For the achromatic band in the case of white light, or for the place of
greatest distinctness when the bands are formed with light approximately
homogeneous, n must be stationary as X varies, i.e.
dn/d\ = .................................. (3)
For a small range of wavelength we may write
x = x fl + 8x,
so that
1904] NOTE TO A PAPER BY PROF. WOOD 205
The achromatic band occurs, not when the whole relative retardation (1)
vanishes, but when
(5)
If D be great enough, there is no limit to the shift that may be caused by
the introduction of the dispersive plate.
As Schuster has especially emphasised, the question here is really one
of the groupvelocity. Approximately homogeneous light consists of a train
of waves in which the amplitude and wavelength slowly vary. A local
peculiarity of amplitude or wavelength travels in a dispersive medium with
the group and not with the wave velocity ; and the relative retardation with
which we are concerned is the relative retardation of the groups. From this
point of view it is obvious that, what is to be made to vanish is not (1) in
which p, is the ratio of wavevelocities V (V, but that derived from it by
replacing p by U Q fU, or V /U, where U is the groupvelocity in the dispersive
medium. In vacuum the distinction between U and V disappears, but in
the dispersive medium
U=d(KV)(dic, .............................. (6)*
K being the reciprocal of the wavelength in the medium. If we denote as
usual the wavelength in vacuo by \,
Accordingly
V V d K d(u/\) dp
= = = t "' x
Substituting this for /u, in (1), we see that the position of the most
distinct band is given by
0, ........................ (9)
in agreement with (5).
* Theory of Sound, 191, 1877.
301.
ON THE OPEN ORGANPIPE PROBLEM IN TWO DIMENSIONS.
[Philosophical Magazine, Vol. vm. pp. 481487, 1904.]
IN the usual symmetrical organpipe of radius R, supposed to be provided
at the mouth with an infinite flange, we know that the correction (a) that
must be added to the length in order that the open end may be treated as
a loop, lies between irR and SR/^TT. The wavelength of vibration is here
supposed to be very great, so that in the neighbourhood of the mouth the
flow follows the electrical law. If we use this analogy and regard the walls
of the pipe and the flange as nonconductors, the question is one of the
resistance of the airspace, measured from a section well inside the pipe to
an infinite distance beyond the mouth. And in spite of the extension to
infinity this resistance is finite. For if r be the radius of a large sphere
whose centre is at the mouth, the resistance between r and r + dr is rfr/27rr 8 ;
and the part corresponding to the passage from a sufficiently great value of r
outwards to infinity may be neglected.
A parallel treatment of the problem in two dimensions, where inside the
mouth the boundary consists of two parallel planes, appears to fail. The
resistance to infinity, involving now I r~ l dr, instead of I r^dr, has no finite
limit; and we must conclude that when the wavelength (X) is very great,
the correction to the length becomes an infinite multiple of the width of the
pipe*. But it remains an open question how the correction to the length
compares with \, whether, for example, when X is given it would vanish
when the width of the pipe is indefinitely diminished.
The following consideration suggests an affirmative answer to the last
question. If we start with a pipe of circular form and suppose the section,
[1911. For a further discussion of the problem which arises when X is treated as infinitely
great see J. J. Thomson's Recent Retearche* in Electricity, 241, 1893.]
1904] ON THE OPEN ORGANPIPE PROBLEM IN TWO DIMENSIONS 207
while retaining its area, to become more and more elliptical, it would appear
that the correction to the length must continually diminish. But the
question has sufficient interest to justify a more detailed treatment.
In Theory of Sound, 302, the problem is considered of the reaction of
the air upon the vibratory motion of a circular plate forming part of an
infinite and otherwise fixed plane. For our present purpose the circular
plate is to be replaced by an infinite rectangular strip extending from
y = oo to y = + oo , and in the other direction of width x. If d<f>/dn be
the given normal velocity of the element dS of the plane and k = 27T/X,
gives the velocitypotential at any point P distant r from dS. In the present
case d<j)/dn is constant, where it differs from zero, and accordingly may be
removed from under the integral sign. If a denote the velocity of sound,
<f> varies as e ikat , and if cr be the density, we get for the whole variation of
pressure acting upon the plate
ffSpdS =  afJ<f>dS =  ika<rff<f> dS.
Thus by (1)
In the double sum
22 dSdS', (3)
which we have now to evaluate, each pair of elements is to be taken once
only, and the product is to be summed after multiplication by the factor
r i e t*r j depending on their mutual distance. The best method is that sug
gested by Maxwell for the common potential. The quantity (3) is regarded
as the work that would be consumed in the complete dissociation of the
matter composing the plate, that is to say, in the removal of every element
from the influence of every other, on the supposition that the potential of
two elements is proportional to r~ 1 e~ ikr . The amount of work required, which
depends only on the initial and final states, may be calculated by supposing
the operation performed in any way that may be most convenient. For this
purpose we suppose that the plate is divided into elementary strips, and that
on one side external strips are removed in succession.
To carry out this method we require first an expression for the potential
(F) at the edge of a strip. Here,
dxdy, (4)
oo r
where r = VC^ 2 + y 3 ) ; and therein
/too Or / ,, Her roc pVexv
dy = 2 dr 2 5 j<fr, (5)
r y J* y Ji \V 1
208 ON THE OPEN ORGANPIPE PROBLEM IN TWO DIMENSIONS [301
representing the potential of a linear source at a point distant x from it.
Convergent and semiconvergent series for (5), applicable respectively when
x is small and when x is great, are well known.
We have
f e ***dv _ / jr_\* ,( 1' l a .8 1
J, V(^l) " \2ikx) e 1 1 80w 1 . 2 . (Site) 1
where S m =l+* + i + ... + l/m ......................... (7)
and 7 is Euler's constant ('5772 . . .). A simple method of derivation, adequate
so far as the leading terms are concerned, will be found in Phil. Mag. XLIII.
p. 259, 1897 (Scientific Papers, iv. p. 290).
Confining ourselves for the present to the case where the total width
of the strip is small compared with the wavelength, we have to integrate
the second series in (6) with respect to x, for which purpose we have the
formula
(8)
In this way we obtain V, the potential at the edge of the strip of width x.
Afterwards we have to integrate again with respect to y and x. The inte
gration with regard to y introduces simply the factor y, representing the
(infinite) length of the strip. The integration with respect to a; is again to
be taken between the limits and x. Thus
.................. (9)
terms in x* being omitted. This is the equivalent of (3) in the case of an
infinite strip of length y and width x. Accordingly by (2)
(10)
The reaction of the air upon the plate may be divided into two parts,
of which one is proportional to the velocity of the plate and the other to the
acceleration. If f denote the displacement of the plate at time t, so that
rf fdt = d^jdn, we have
.. d .. dd>
= ika g = ika g ;
and therefore in the equation of motion of the plate, the reaction of the air
is represented by a dissipative force
<*...* .............................. (")
1904] ON THE OPEN ORGANPIPE PROBLEM IN TWO DIMENSIONS 209
retarding the motion, and by an accession to the inertia equal to
(12)
the first factor in each case denoting the area of the plate. The mass repre
sented by (12) is that of a volume of air having a base equal to the area of
the plate and a thickness
(13)
When x is given (13) increases without limit when \ (= 27T/&) is made
infinite, as we found .before. But if we regard X as given, and suppose x (the
width of the plate) to diminish without limit, we see that (13) also diminishes
without limit.
The application of these results to the problem of the open pipe in two
dimensions depends upon the imaginary introduction of a movable piston,
itself without mass, at the mouth of the pipe a variation without influence
upon the behaviour of the air at a great distance outside the mouth, with
which we are mainly concerned. The conclusion is that if the wavelength
or pipelength be given, the mouth or open end may be treated more and
more accurately as a loop as the width is diminished without limit. Both
parts of the pressure variation, corresponding the one to inertia and the other
to dissipative escape of energy, ultimately vanish.
So far we have considered the case where in (3), (4) the width of the
strip is small in comparison with the wavelength. It remains to say some
thing as to the other extreme case ; and it may be well to introduce the
discussion by a brief statement of the derivation of the semiconvergent
series in (6) by the method of Lipschitz*.
Consider the integral \j7+  ^., where w is a complex variable of the
form u + iv. If we represent, as usual, simultaneous pairs of values of u and
v by the coordinates of a point, the integral will vanish when taken round
any closed circuit not including the points w = + i. The circuit at present
to be considered is that enclosed by the lines u = 0, v = 1, and a quadrant at
infinity. It is easy to see that along this quadrant the integral ultimately
vanishes, so that the result of the integration is the same whether we
integrate from w = i to w = too , or from w = i to w = oo + i. Accordingly
I" e ~ irr>dv = f x e ~* e ~ du _ e ~ ir
Ji ^~l) ~ Jo V(2w + 7/ 2 ) ~ V(2ir) Jo
//, , _
V ( 2ir
I
__
\2irJ { 1 Sir 1 . 2 . (Sir) 2 1.2.3. (Sir) 3
* Crelle, Bd. LVI. (1859). See also Proc. Lond. Math. Soc. XLX. p. 504 (1888) ; Scientific
Papers, in. p. 44.
R. V. H
210 ON THE OPEN ORGANPIPE PROBLEM IN TWO DIMENSIONS [301
on expansion and integration by a known formula. This agrees with (6),
if kx be written for r.
In arriving at the value of // Vdxdy, we have to integrate (5) twice with
respect to x between the limits and x. Taking first an integration with
respect to x, we have
t*, f e^'dv
M, W=*)
in which
Again, for the second integration with respect to x,
M S
in which
Thus
When x is great, the outstanding integral in (15) may be treated in the
manner already explained. The integral
will yield the same value, whether taken from w = i to w = i + too , or from
w = i to w = oo + i. The first gives
e~ irv dv
,
_
J,
the second gives f
Jo (u + i
Accordingly
e~ ru e~ ir du
e~ ir
00 e^v e~ ir [
1 * V(f~ 1) ~ V(2tr) Jo
and the latter may be expanded in inverse powers of r and integrated as
before. For the leading term we have
/." '
1904] ON THE OPEN ORGANPIPE PROBLEM IN TWO DIMENSIONS 211
Thus approximately when x is great
/Y
JJ
2
if we confine ourselves to the leading real and imaginary terms.
From (2) we now get
ff* jo ikaod6 (ITT 2} d 2a d*
I SpdS=  J.xy. 1 j + T^xy.aa.r + xy.. j*. (17)
JJ * TT dn y \ k k*x] y dt y TT&X dP
When x is large, the inertia term ultimately vanishes in comparison with
the area of the plate. The reaction is then reduced to the dissipative term,
which is the same as would be obtained from the theory of plane waves of
infinite extent.
142
.
302.
EXTRACTS FROM NOBEL LECTURE.
[Given before the Royal Academy of Science at Stockholm, 1904.]
THE subject of the densities of gases has engaged a large part of my
attention for over 20 years. In 1882 in an address to the British Association
I suggested that the time had come for a redetermination of these densities,
being interested in the question of Prout's law. At that time the best
results were those of Regnault, according to whom the density of oxygen was
15'96 times that of hydrogen. The deviation of this number from the
integer 16 seemed not to be outside the limits of experimental error.
In my work, as in the simultaneous work of Cooke, the method of
Regnault was followed in that the working globe was counterpoised by a
dummy globe (always closed) of the same external volume as itself. Under
these conditions we became independent of fluctuations of atmospheric
density. The importance of this consideration will be manifest when it is
pointed out that in the usual process of weighing against brass or platinum
weights, it might make more apparent difference whether the barometer
were high or low than whether the working globe were vacuous or charged
with hydrogen to atmospheric pressure. Cooke's result, as at first announced,
was practically identical with that of Regnault, but in the calculations of
both these experimenters a correction of considerable importance had been
overlooked. It was assumed that the external volume of the working globe
was the same whether vacuous or charged to atmospheric pressure, whereas
of course the volume must be greater in the latter case. The introduction of
the correction reduced Cooke's result to the same as that which I had in the
meantime announced, viz. 15'88. In this case therefore the discrepancy from
Prout's law was increased, and not diminished, by the new determination.
Turning my attention to nitrogen, I made a series of determinations,
using a method of preparation devised originally by Harcourt, and recom
mended to me by Ramsay. Air bubbled through liquid ammonia is passed
through a tube containing copper at a red heat where the oxygen of the air
1904] EXTRACTS FROM NOBEL LECTURE 213
is consumed by the hydrogen of the ammonia, the excess of the ammonia
being subsequently removed with sulphuric acid. In this case the copper
serves merely to increase the surface and to act as an indicator. As long
as it remains bright, we have security that the ammonia has done its work.
Having obtained a series of concordant observations on gas thus prepared
I was at first disposed to consider the work on nitrogen as finished. After
wards, however, I reflected that the method which I had used was not that
of Regnault and that in any case it was desirable to multiply methods, so
that I fell back upon the more orthodox procedure according to which,
ammonia being dispensed with, air passes directly over red hot copper. Again
a series in good agreement with itself resulted, but to my surprise and disgust
the densities obtained by the two methods differed by a thousandth part
a difference small in itself but entirely beyond the experimental errors. The
ammonia method gave the smaller density, and the question arose whether
the difference could be attributed to recognized impurities. Somewhat
prolonged inquiry having answered this question in the negative, I was rather
at a loss how to proceed. It is a good rule in experimental work to seek to
magnify a discrepancy when it first presents itself, rather than to follow the
natural instinct of trying to get quit of it. What was the difference between
the two kinds of nitrogen ? The one was wholly derived from air ; the other
partially, to the extent of about onefifth part, from ammonia. The most
promising course for magnifying the discrepancy appeared to be the substi
tution of oxygen for air in the ammonia method, so that all the nitrogen
should in that case be derived from ammonia. Success was at once attained,
the nitrogen from the ammonia being now 1/200 part lighter than that from
air, a difference upon which it was possible to work with satisfaction. Among
the explanations which suggested themselves were the presence of a gas
heavier than nitrogen in the air, or (what was at first rather favoured by
chemical friends) the existence in the ammoniaprepared gas of nitrogen in
a dissociated state. Since such dissociated nitrogen would probably be
unstable, the experiment was tried of keeping a sample for eight months,
but the density was found to be unaltered.
On the supposition that the airderived gas was heavier than the
" chemical " nitrogen on account of the existence in the atmosphere of an
unknown ingredient, the next step was the isolation of this ingredient by
absorption of nitrogen. This was a task of considerable difficulty ; and it
was undertaken by Ramsay and myself working at first independently but
afterwards in concert. Two methods were available, the first that by which
Cavendish had originally established the identity of the principal component
of the atmosphere with the nitrogen of nitre and consisting in the oxidation
of the nitrogen under the influence of electric sparks with absorption of the
acid compounds by alkali. The other method was to absorb the nitrogen by
214 EXTRACTS FROM NOBEL LECTURE [302
means of magnesium at a full red heat. In both these ways a gas was
isolated of amount equal to about one per cent, of the atmosphere by volume
and having a density about half as great again as that of nitrogen. From
the manner of its preparation it was proved to be nonoxidisable and to
refuse absorption by magnesium at a red heat, and further varied attempts to
induce chemical combination were without result. On this account the name
argon was given to it. The most remarkable feature of the gas was the ratio
of its specific heats, which proved to be the highest possible, viz. T67,
indicating that sensibly the whole of the energy of molecular motion is
translational.
Argon must not be deemed rare. A large hall may easily contain
a greater weight of it than a man can carry.
In subsequent investigations Ramsay and Travers discovered small
quantities of new gases contained in the aggregate at first named argon.
Helium, originally obtained by Ramsay from clevite, is also present in
minute quantity.
Experiments upon the refractivity and viscosity of argon revealed nothing
specially remarkable, but the refractivity of helium proved to be unexpectedly
low, not attaining onethird of that of hydrogen the lowest then known.
As regards the preparation of argon, it is advantageous to begin with
liquid air, for preparing which a plant is now to be found in many laboratories.
[See Art. 288, p. 115.]
Although the preparation of a considerable quantity of argon is rather an
undertaking, there is no difficulty in demonstrating its existence with the
most ordinary appliances. By the use of a specially shaped tube and an
ordinary inductioncoil actuated by a small Grove battery, I was able to show
the characteristic spectrum of argon at atmospheric pressure, starting with
5 c.c. only of air.
Another question relating to the gases of the atmosphere has occupied
my attention namely the amount of free hydrogen. [See Art. 277, p. 49.]
Another branch of my work upon gases has relation to the law of pressure,
especially at low pressures. Under these circumstances the usual methods
are deficient in accuracy. Thus Amagat considers that under the best
conditions it is not possible to answer for anything less than O'Ol mm. of
mercury. By the use of a special manometer I was able to carry the accuracy
at least 50 times further than Amagat's standard, and thus to investigate
with fair accuracy the effect of pressures not exceeding O'Ol mm. in total
amount Boyle's law was fully verified, even in the case of oxygen, for which
C. Baur had found anomalies, especially in the neighbourhood of 0'7 mm.
pressure.
1904] EXTRACTS FROM NOBEL LECTURE 215
More recently I have made determinations of the compressibility of gases
between one atmosphere and half an atmosphere of pressure. For this
purpose two manometric gauges, each capable of measuring half an atmo
sphere, were employed. The equality of the gauges could be tested by using
them in parallel, to borrow an electrical term. One of the gauges alone would
thus serve for half an atmosphere, while the two combined in series gave the
whole atmosphere. In combination with these gauges volumes in the ratio
of 2 : 1 were needed. Here again the desired result was arrived at by the
use of two equal volumes, either alone or in combination. Any question as
to the precise equality of the two volumes is eliminated in each set of
observations by using the two single volumes alternately. The mean result
then necessarily corresponds to the half of the total volume, except in so far
as the capacities of the vessels may be altered by change of pressure.
The annexed table gives a summary of results for the various gases.
Gas B Temperature
100038 112
099974 107
100015 149
100026 138
100023 114
100279 150
100327 110
pv at atmos.
Here *, at! atmos."
the temperature being the same at both pressures and having the value
recorded. That B should be less than unity in the case of hydrogen and
exceed that value for the other gases, is what was to be expected from the
known behaviour at higher pressures.
The principal interest of these results is perhaps to calculate corrections
to ratios of densities, as found at atmospheric pressure, so as to infer what
the ratios would be in a state of great rarefaction. It is only under this
condition that Avogadro's law can be expected to apply accurately, as
I pointed out in 1892 in connection with oxygen and hydrogen.
In the case of nitrogen and oxygen, the correction is not important, and
the original comparison of densities* is sensibly unaffected. According to
this method the atomic weight of nitrogen is 1401, in opposition to the
1405 found by Stas.
* Rayleigh and Ramsay, Phil. Trans. 1895.
303.
ON THE COMPRESSIBILITY OF GASES BETWEEN ONE
ATMOSPHERE AND HALF AN ATMOSPHERE OF PRESSURE.
[Phil. Trans. A, Vol. cciv. pp. 351372, 1905.]
THE present* is the third of a series of memoirs in which are detailed
observations upon the compressibility of the principal gases at pressures from
one atmosphere downwards. In the first f of these the pressures dealt with
were exceedingly low, ranging from 1'5 millims. to O'Ol millim. of mercury,
and the use of a special and extraordinarily delicate manometer allowed the
verification of Boyle's law to be pushed to about ^rjW f a millimetre of
mercury.
In the second J memoir the products of pressure and volume at constant
temperature (that of the room) were compared when the pressure was changed
from 75 millims. to 150 millims. of mercury in the ratio of 2 : 1. The ratio
of the products (denoted by B) would be unity according to Boyle's law ; for
the more condensable gases, e.g. nitrous oxide, it exceeds unity. The following
were the final mean values :
Nature of gas
B
Air
99997
Hydrogen
99997
Oxygen . .
100024
Nitrous oxide
100066
Argon
100021
Carbonic oxide
100005
* A Preliminary Notice containing many of the results now recorded in greater detail was
published in Roy. Soc. Proc., February, 1904.
t Phil. Traru. A, cxcvi. pp. 205228, 1901 ; Scientific Papers, Vol. rv. p. 511.
* Phil. Tram. A, cxcviii. pp. 417430, 1902. On p. 428, line 8 from bottom, read 1 + mr
instead of 1  mr. [Scientific Papers, v. p. 27.]
The number for carbonic oxide was obtained subsequently to the publication of the memoir.
It is the mean of two sets of observations, giving severally 100003 and 100008. The gas was
prepared from ferrocyanide of potassium (see Roy. Soc. Proc. Vol. LXII. p. 204, 1897 ; Scientific
Papen, VoL nr. p. 347).
1905]
ON THE COMPRESSIBILITY OF GASES
217
The deviations from unity in the cases of oxygen and argon were thought
to exceed the errors of observation. The results presently to be given for
oxygen render it probable that the larger half of the deviation was, in fact,
error. At any rate, Boyle's law was sensibly observed by air, hydrogen and
carbonic oxide.
The method employed in this research appeared to be satisfactory, and
I was desirous of extending it to higher pressures, still, however, below the
atmospheric, as to which there seemed to be a great dearth of information.
I could find only some incidental observations by Amagat* on air and
carbonic acid, and these it may be well to quote :
Air
Acide carbonique
Pression initiale en
pv
Pression initiale en
pv
centimetres p' v >
centimetres
fr
37314
10003
37760
10026
37318
10002
37724
10022
31523
10000
37561
10025
30613
10027
24970
10015
25393
10010
24974
10020
25376
10005
24948
10015
24830
9997
24947
10021
24841
10002
The pressures were as 2 :1, and the "initial pressure" p was the smaller.
The temperature was from 17 C. to 19 C. The ratio pv/p'v' is what I have
denoted by B. It will be seen that the numbers for air exhibit considerable
discrepancies.
The earlier entries in Amagat's table correspond pretty closely with the
observations that I proposed to undertake. Besides the general elucidation
of the behaviour of gases at reduced pressures, the object in view was to
obtain material for comparing the densities of various gases at great rare
factions. In the actual weighings of gases the pressure in the containing
vessel is usually atmospheric, but the ratios of densities so obtained are not
immediately available for inferring molecular weights according to Avogadro's
rule. This rule can only be supposed to apply with rigour when the gases
are so far rarefied as to come within the range of Boyle's lawf. For this
* Ann. de Chimit, tome xxvm. 1883.
t The application of this idea to oxygen and hydrogen was made in my paper " On the
Eelative Densities of Oxygen and Hydrogen," Roy. Soc. Proc. Vol. L. p. 448, 1892 ; Scientific
Papers, Vol. in. p. 525. My hesitation then and later to push the investigation further, so aa
to obtain corrections to the relative densities observed at atmospheric pressure, arose from the
uncertainties in which the anomalous observations of Mendeleef and Siljerstrom had enveloped
the behaviour of gases at low pressures.
218 ON THE COMPRESSIBILITY OF OASES BETWEEN [303
purpose it is advisable that the range of pressure employed should be sufficient
to give accuracy, but not so high that the application to Avogadro's rule
involves too much extrapolation. The comparison of volumes at pressures
atmospheric and half atmospheric seems to meet these requirements, though
we must not forget that (apart from theory) the result is still of the nature
of an extrapolation. On this subject reference may be made to a paper by
Sir W. Ramsay and Dr B. Steele*.
The guiding idea in the present apparatus, as in that of 1902, is the use
of two manometric gauges combined in a special manner. " The object is to
test whether when the volume of a gas is halved its pressure is doubled, and
its attainment requires two gauges indicating pressures which are in the
ratio of 2 : 1. To this end we may employ a pair of independent gauges
as nearly as possible similar to one another, the similarity being tested by
combination in parallel, to borrow an electrical term. When connected
below with one reservoir of air and above with another reservoir, or with
a vacuum, the two gauges should reach their settings simultaneously, or at
least so nearly that a suitable connection can readily be applied. For brevity
we may for the present assume precise similarity. If now the two gauges be
combined in series, so that the lowpressure chamber of the first communicates
with the highpressure chamber of the second, the combination constitutes
a gauge suitable for measuring a doubled pressure."
The Manometers.
The construction of the gauges is modelled upon that used extensively in
my researches upon the density of gases f. An iron measuring rod, AB, is
actually applied to the two mercury surfaces, arranged so as to be vertically
superposed. This rod is of about 7 millims. diameter and is pointed below, A.
At the upper end, B, it divides at the level of the mercury into a sort of fork,
and terminates in a point similar to that at A, and, like it, directed down
wards. The coincidence of these points with their images reflected at the
mercury surfaces is observed with the aid of lenses of 20 millims. focus suitably
held in position. It is, of course, independent of any irregular refractions
which the walls of the tube may exercise. In each manometer the distance
between the points is 15 inches or 381 millims.
The internal diameter of the tubes, constituting the upper and lower
chambers of the manometers, is 22 millims. This is the diameter at the
level of the " points " to which the mercury surfaces are set. At the places
where the iron rod emerges above and below into the open, the glass is
contracted until it becomes an approximate fit, and airtightness is secured
with the aid of cement.
" On the VapourDensities of some Carbon Compounds ; an attempt to determine their
Molecular Weights," Phil. Mag. Oct. 1893.
t Roy. Soc. Proc. Vol. mi. pp. 134149, 1893 ; Scientific Papers, Vol. iv. p. 41.
1905] ONE ATMOSPHERE AND HALF AN ATMOSPHERE OF PRESSURE 219
Fig. 1.
(a * To mercury reservoirs)
220 ON THE COMPRESSIBILITY OF OASES BETWEEN [303
General Arrangement of Apparatus.
With one important difference to be explained presently the general
scheme is the same as in 1902 and is sketched in fig. 1. The lefthand
manometer can be connected above, through F, with the pump or with the
gas supply. The lower chamber A communicates with the upper chamber D
of the righthand manometer and with an intermediate reservoir E, to which,
as to the manometers, mercury can be supplied from below. The lower
chamber C of the righthand manometer is connected with the principal
gas reservoirs H, H. It is here that the novelty enters. In the 1902
apparatus the two equal bulbs were superposed, being connected by a narrow
neck. For the doubled volume, both bulbs were occupied by gas ; but for
the single volume, only the upper one was available. For the comparison
of the single and double volume, a principal factor of the final result, pre
liminary gauging had to be relied upon. In the new apparatus it was
desired largely to increase the volumes, and it was both more advantageous
and more convenient to place the two bulbs H side by side. The temperature
conditions are thereby improved ; but what I wish to emphasize at present
is the elimination, thereby rendered possible, of dependence upon preliminary
gauging, for either bulb is now available for the single volume ; and if both
are symmetrically employed in each set of observations, the mean necessarily
corresponds to half the total volume, whether or not the two single volumes are
precisely equal. The volumes are defined, as usual, by marks GG, II, upon
the associated tubes above and below. The use of the side tube JK will be
explained presently.
When, as shown, the mercury stands at the lower marks /, the double
volume is in use, and the pressure is such as will balance the mercury in one
(the righthand) manometer. A vacuum is established in the upper chamber
D, from which a way is open through ABF to the pump. When the mercury
is raised through one of the bulbs to the upper mark G, the volume is halved
and the pressure to be dealt with is doubled. Gas sufficient to exert the
single pressure (381 millims.) must be supplied to the intermediate chamber
E, which is now isolated from the pump by the mercury standing up in the
curved tube AB. Both manometers can now be set and the doubling of the
pressure verified.
The communication through to the pump is unobstructed, but on a side
tube a threeway tap is provided communicating on the one hand with the
gas supply and on the other with a vertical tube delivering under mercury,
by means of which a washout of the generating vessels can be effected when
it is not convenient to evacuate them. The six tubes of glass leading down
wards from the gas reservoirs, manometers, &c., are all well over a barometer
height in length, and are terminated by suitable indiarubber hoses and
1905] ONE ATMOSPHERE AND HALF AN ATMOSPHERE OF PRESSURE 221
reservoirs for the supply of mercury*. By this precaution the internal
pressure on the hoses is guaranteed to exceed the external atmospheric
pressure, and under this condition the use of indiarubber seems to be free
from serious objection. If, however, the external pressure be allowed to be
in excess, there are soon signs of the percolation of air and probably of
moisture.
When settings are actually in progress, the mercury in the hoses is
isolated from that in the reservoirs by pinchcocks, and the adjustment of
the supply is effected by squeezing the hoses. As explained in my first
paper, the final adjustment must be made by squeezers which operate upon
parts of the hoses which lie flat upon the large wooden tray underlying
the whole.
The Side Apparatus.
The use of this was fully explained in my former paper. By the employ
ment of manometric gauges we are enabled to dispense with scales and
cathetometers ; but since (save as to a small temperature correction) the
pressures are defined beforehand, the adjustment is thrown upon the volume.
The variable volume is introduced at the side tube JK, which, with its
associated bulb, allows of the elimination from the results of the volume
which cannot be directly gauged, including that over the mercury in the
lower chamber of the righthand manometer when set. The tubes above
and below the bulb were calibrated in the usual mannerf. It should be
remarked that the diagram shows the mercury in the side apparatus in
a position suitable for a measurement at the doubled pressure, while in the
rest of the apparatus the position of the mercury corresponds to the single
pressure.
General Sketch of Theory.
It will be convenient to repeat this, nearly as given in the former paper.
To save complication, it will be supposed that the temperature is constant,
not only throughout the whole apparatus at one time, but also at the four
different times concerned.
V l = volume of two large bulbs H, H together between GO, II (about
633 cub. centims.).
V 3 = volume between CJGG (the ungauged space).
F 4 = measured volume on upper part of JK from highest mark J down
wards.
* These reservoirs were protected from external moisture by tubes of chloride of calcium.
t The whole of the apparatus was made under my instructions by the late Mr Gordon, who
also took a large part in the observations. I take this opportunity of recording my indebtedness
to his faithful assistance over a long series of years.
2'2'2 ON THE COMPRESSIBILITY OF GASES BETWEEN
F 8 = measured volume, including bulb of side apparatus, from highest
mark J downwards.
P l = smaller pressure (height of mercury in righthand manometer).
P a = larger pressure (sum of heights of mercury in two manometers).
In the first pair of operations, when the large bulbs are in use, the
pressure P l corresponds to the volume ( F, + V s + V s ), and the pressure P a
corresponds to ( F, + V s + F 4 ), the quantity of gas being the same. Hence
the equation
B being a numerical quantity which would be unity according to Boyle's law.
In the second pair of operations with the same nature but with a different
quantity of gas, and with the same pressures, the mercury stands at GG
throughout, and we have
whence by subtraction
Pi (F> + F 5  F' D ) = BP 2 (i F, + F 4  F' 4 ).
From this equation F 3 has been eliminated, and B is expressed by means
of P,/P, and the actually gauged volumes F 5  F' 5) F 4  F' 4 . It is important
to remark that only the differences F 5  F' B , F 4 F' 4 are involved. The first
is measured on the lower part of the side apparatus, and the second on the
upper part ; while the capacity of the intervening bulb does not appear.
If Boyle's law be closely followed, there is nothing to prevent both
F B F' 5 and F 4  F' 4 from being very small. Except the preliminary com
parison of the manometers, the whole of the data required for the verification
are then contained in the observations of each set.
When the temperaturechanges are taken into account, V s , F 4 , F 6 are
not fully eliminated, but they appear with coefficients which are very small
if the temperature conditions are good.
Thermometers.
Of these four were employed. The first gave the temperature (T) of the
manometric columns ; the second gave the temperature (T) of F 3 ; the third
that of the bulb of the side apparatus (t). The temperature of the waterbath,
in which are contained the principal bulbs, is of course the most important.
The water was stirred continuously by a stream of air, and the tempera
ture was taken by a thermometer that could be read to y^ of a degree C.
No observations were begun until it had been ascertained that the tempera
ture of the water was slowly rising. It is important to understand what
really are the demands made upon this thermometer. It was arranged that
the mean temperature l when the double volume was in use should be
1905] ONE ATMOSPHERE AND HALF AN ATMOSPHERE OF PRESSURE 223
almost the same as for the single volumes 2 . The difference was usually
less than y^j of a degree and rarely exceeded ^ or yfoj. Under these
circumstances the use of the thermometer was practically only to identify
the same temperature on different days, and the actual error of its readings
and even of its scale of temperature were of but secondary importance.
Comparisons with other thermometers showed that there were no errors
which could possibly become sensible. The precautions necessary, in order
that the other thermometers should do their work satisfactorily, were indi
cated in the former paper. In the present work the number of intervening
screens was increased.
It is desirable to emphasise that most of the errors that could arise from
imperfect action of the thermometers is eliminated in the actual results,
which depend only upon a comparison between operations with and without
the large bulbs. For example, suppose that there is an error in the rather
illdefined temperature of the space V 3 . The conditions are the same whether
the large bulbs are in use or not ; and thus whatever error occurs in the one
case may be expected to repeat itself in the other. So far as this repetition
is complete, the error disappears in the comparison. Again, it might happen
that one of the large bulbs tended to be warmer than the other or than the
thermometer. But this, so far as it is constant, could lead to no error, the
effect when the bulb is used alone being compensated by the effect when
it forms one of the pair. Purely accidental errors are, in any case, eliminated
when the mean is taken of a number of observations.
The Large Reservoirs.
The tubes forming the principal parts are of glass, 25 centims. in length,
41 millims. in internal diameter, and about 2 millims. thickness in the walls.
There are prolongations above and below of narrow bore, upon which are
placed the marks defining the volumes.
As has been explained, the accurate comparison of these volumes is
unnecessary. As it happens, the actual volumes between the marks are so
nearly equal that it is difficult to say which is the larger. The total volume
Fj, required only to be roughly known for the sake of the subsidiary terms,
is 6326 cub. centims.
But there is another question to be considered. The single bulbs are
used under an internal pressure of an atmosphere. Under the same pressure
the combined volume of both bulbs would of course be exactly double the
mean of those of the bulbs used separately. But when the bulbs are in
combination, the internal pressure is reduced to half an atmosphere, and the
bulbs contract. A correction is thus necessary which runs similarly through
all the results calculated on the supposition that the ratio of volumes is
exactly 2:1.
224 ON THE COMPRESSIBILITY OF GASES BETWEEN [303
The amount of the correction has been determined in two ways. Direct
observation of the change of level of water filling the bulb and standing
in the small upper prolongation, when the internal pressure was changed
from one atmosphere to half atmosphere, gave a total relative alteration
of 44 x 10~ 8 per half atmosphere, of which 2'3 x 10~ s would be due to the
contraction of the water. The difference, viz. 2*1 x 10~ 5 , represents the
relative contraction or expansion of the volume per half atmosphere of
pressure.
A calculation founded upon the measured dimensions of the tubes,
including the thickness of the walls, combined with estimates of the elasticity
of the glass, gave 2*0 x 10~ 5 per half atmosphere, in better agreement than
could have been expected.
The real ratio of volumes with which we are concerned in these experi
ments is thus not 2 exactly, but
2 (1  000021).
The value of B calculated without allowance for this correction would be too
large, so that a gas which really obeyed Boyle's law exactly would appear to
be too condensable, like CO 2 . From a value of B so calculated we are to
subtract '000021.
Comparison of Manometers.
This comparison is effected by combining the manometers in parallel so
that the mercury at the lower levels is subject to the pressure of one con
tinuous quantity of gas, while the mercury at the upper levels is in vacuo, or
at any rate under the pressure of the same very rare gas. Any difference that
may manifest itself may be estimated by finding what change of gas volume
is required in order to pass from the pressure appropriate to one manometer
to that appropriate to the other.
The first matter requiring attention is the verticality of the measuring
rods, or rather of the lines joining the points actually applied to the mercury.
The points were visually projected upon a plumbline, hung a few centi
metres away, and were observed through a hole of 2 millims. or 3 millims.
diameter perforated in a black card. If the adjustment is perfect the same
position of the card allows accurate projection upon the thread of the upper
and lower points ; if not, the necessary motion of the card, perpendicular
to the line of sight, gives data for estimating the amount of the error. If
x and y be the linear horizontal deviations from true adjustment thus deter
mined in any two perpendicular planes, I the length of the rod, the angular
error is \/(o? t y 3 ) r 1, and the proportional error of height for the present
purpose is (of + tf)/2&. When the manometers were compared, no value of
905] ONE ATMOSPHERE AND HALF AN ATMOSPHERE OF PRESSURE 225
x or y exceeded millim., so that with I = 380 millims. the error of vertically
could be neglected.
In effecting the comparison of the manometers, both mercury levels must
be set below (in order to make the gas volume definite), while the settings
above are made alternately. It was at once apparent, when the righthand
manometer was set, that the rod on the left was a little too long, a perceptible
interval being manifest between the upper point and the mercury. In these
experiments the total gas volume was about 2845 cub. centims., the principal
part being the volume of a large bottle protected from rapid changes of
temperature by a packing of sawdust. The necessary changes were produced
by causing mercury to rise and fall in a vertical tube of small bore, the
position of the meniscus being noted at the moment when a setting was
judged to be good. The settings of the two manometers must be made
alternately in order to eliminate temperature changes; and the result of
each set of observations was derived from the means of four settings of one
manometer, and of five of the other. The lowering of the mercury in the
auxiliary tube required to pass from a setting of the lefthand manometer to
a setting of the righthand manometer, was found on three separate days
to be 50'3 millims., 51'3 millims., 49'9 millims., or, as a mean, 50'5 millims.
As regards the section of the auxiliary tube, it was found that a mercury
thread occupying 85 millims. of it weighed 5'335 grammes. The proportional
difference of volume is thus
505 x 5335
85x1354x2845"
and the same fraction represents the proportion by which the lefthand
measuring rod exceeds in length its fellow on the right. It would seem that
by this procedure the lengths of the rods are compared to about a millionth
part.
In the notation employed in the calculations
1000082 #o, whence = 1 '000041.
It may be observed that an error in the comparison of HI and T 2 enters
to only half its amount into the final result.
The Observations.
The manipulation necessary to imprison the right quantity of gas was
described in the former paper. When this has once been secured the
observations are straightforward. On each occasion six readings were taken,
extending over about an hour, during which time the temperature always
rose, and the means were combined into what was treated as one observation.
R. v. 15
ON THE COMPRESSIBILITY OF GASES BETWEEN
[303
A complete set usually included eight observations at the high pressure,
in four of which one large bulb was in use, and in the second four the other
bulb. Interpolated in the middle of these were the observations (usually
six in number) of the low pressure where both large bulbs were occupied
by gas. Further, each set included eight observations relating to the side
apparatus, in which the large bulbs stood charged with mercury. In this
way each set contained within itself complete material for the elimination
of F,, which might possibly vary from time to time with the character of the
contact between mercury and glass in the lower chamber of the righthand
manometer. Finally the means were taken of all the corresponding observa
tions, no further distinction being maintained between the two large bulbs.
The following table shows in the notation employed the correspondence
of volumes and temperatures :
I.
II.
III.
IV.
A
*
3
3
T 3
T t
\
v*
'r,
i
In the first observation V l is the volume of the two large bulbs together
and 6 l the temperature of the waterbath, reckoned from some convenient
neighbouring temperature as standard. V 3 is the ungauged volume already
discussed whose temperature T l is given by the upper thermometer. F 8 is
the (larger) volume in the side apparatus whose temperature $, is that of the
associated thermometer. In the second observation Fj is the (mean) volume
of a single bulb and 0? its temperature. F 4 is the volume in the side apparatus
whose temperature, as well as that of F s , is taken to be T 2 . III. and IV.
represent the corresponding observations when the large bulbs are not used.
The temperatures of the mercury in the manometric columns are represented
by TJ, TJ, T,, T 4 .
As an example of the actual quantities, the observations on hydrogen,
April 9 24, 1903, may be taken. The values of V 1 and F s are approximate.
F, = 6326, F,= 1102, F, = 13978, F 4 = 1'504,
F 8  F' 8 =  650, F 4  F' 4 =  245.
T! = + '43, T a = + '54, T, = + '05, T 4 = + 11.
The volumes are in cubic centimetres and the temperatures are in Centi
grade degrees, reckoned from 11.
1905] ONE ATMOSPHERE AND HALF AN ATMOSPHERE OF PRESSURE 227
The Reductions.
The simple theory has already been given, but the actual reductions are
rather troublesome, on account of the numerous temperature corrections.
These, however, are but small.
We have first to deal with the expansion of the mercury and of the iron
in the manometers. If the actual heights of the mercury (at the same
temperature) be H l} H z , we have for the relative pressures H/(l+mr),
where m = "00017. Thus in the notation already employed
\JL ^
1 + wr 3
and P 2 = * , or .
I + WT 2 1 + rar 4
The quantity of gas at a given pressure occupying a known volume is to
be found by dividing the volume by the absolute temperature. Hence each
volume is to be divided by 1 + fi0, 1 + fiT, 1 + fit, as the case may be, where
ft is the reciprocal of the absolute temperature chosen as a standard for the
set. Thus in the above example for hydrogen, ft = l/(273 + 11) = 1/284.
Our equations expressing that the quantities of gas are the same at the
single and at the doubled pressures accordingly take the form
H 2 ( V V s V 6 )(#! + #,)( JF, F3_+F 4 )
\ft0 1 1+ftT, 1+/8$J~. 1 + WT 2 (1+00, + i+~/3Tt)'
1+mr. l+j3T t '
where B is the numerical quantity to be determined according to Boyle's
law identical with unity.
By subtraction and neglect in the small terms of the squares of the small
temperature differences, we obtain
(1 + mr,) (1 + 0,) 2# 2 (1 + mr 2 ) (1 + /30 2 )
( Tl  T,  2r 2 + 2r 4 ) + (T,  T 3  2T 2 + 2T 4 )}
152
228
ON THE COMPRESSIBILITY OF GASES BETWEEN
The first three terms on the right, viz., those in F 8 , F 4 , F 8 , vanish if
TI = Tjj Tj = Tf> 7 t = T 9 , T, = T 4 , , = U If in general R denote the sum of
the five terms, we may write with sufficient approximation for the actual
experiments
It may be well to exhibit further the steps of the reduction in the case
of hydrogen above detailed. The five terms composing R are
Term in F 8 =  000038
F 4 =  000009
F 5 = + 000026
Term in F 4 F' 4 =  '000775
F 8 F' 8 = + 001027
R = + 000231
Thus 2# 2 /(# 1 + Hi) = 1  000041
m ( T2 _ Tl )= + 00001 9
0(0, 00= + 000028
E=  000231
B= 999775
In the above calculation the volumes of the principal capacities at the
two pressures have been assumed to be as 2 : 1 exactly. As has already been
explained, the value of B so obtained is subject to correction of '000021 to be
subtracted.
Hence B = 99975,
a result in strictness applicable to the temperature 11 0 C. The hydrogen
is somewhat less compressible than according to Boyle's law, as was to be
expected from its known behaviour at pressures above atmosphere.
The Results.
After the above explanation it will suffice to record the final results of
the various sets of observations.
HYDROGEN.
Date
Source
Temperature
B
January 17 to 27, 1903
Palladium
106
99974
30 to February 10, 1903
April 9 to 24, 1903
;
106
iro
99974
99975
Mean
10*7
99974
1905] ONE ATMOSPHERE AND HALF AN ATMOSPHERE OF PRESSURE 229
In the case of hydrogen the agreement of single results is remarkably
good. This gas, as well as all the others, was carefully dried with phosphoric
anhydride.
CARBONIC OXIDE.
Date
Source
Temperature
B
May 2 to 14, 1903
Ferrocy an ide
117
100033
19 to 29, 1903
125
100024
June 17 to 29, 1903
"
145
100024
July 1 to 15, 1903
16'6
100023
Mean
138
100026
The gas was prepared from ferrocyanide of potassium and sulphuric acid*,
and purified from CO 2 by a long tube of alkali. It is barely possible that
the abnormally high number which stands first in the table may be due
to imperfect purification on that occasion ; on principle, however, it is
retained, as no suspicion suggested itself at the time.
NITROGEN.
Date
Source
Temperature
B
July 23 to August 5 1903
Chemical
16'5
100012
August 7 to 19, 1903 .
16'4
100022
October 5 to 20, 1903
154
100011
28 to November 9, 1 903
November 10 to 23, 1903
From air
132
129
100015
100016
Mean
149
100015
The "chemical" nitrogen was from potassium nitrite and ammonium
chloride. That " from air " was prepared by bubbling air through ammonia
and passing over redhot copper and sulphuric acid, with the usual pre
cautions. It contained about 1 per cent, of argon; but this could hardly
influence the observed numbers.
OXYGEN.
Date
Source
Temperature
B
November 25 to December 5, 1903
December 8 to 19, 1903
Perman
ganate
124
10'9
100034
100042
December 21, 1903, to January 5, 1904
102
100038
Mean
112
100038
See Roy. Soc. Proc. Vol. LXII. p. 204, 1897 ; Scientific Papers, Vol. rv. p. 347.
230
ON THE COMPRESSIBILITY OF GASES BETWEEN
It remains to record certain results with air (free from H 2 O and C0 2 ).
It is curious that the greatest discrepancies show themselves here.
The earlier observations, at the end of 1902, were made before the
apparatus was perfected, and gave as a mean B = T00022. Subsequently,
return was made to air.
AIR.
Date
Temperature
B
April 1 to 11, 1904
9'5
1O0035
April 12 to 26 1904
113
100016
May 14 to 26 1904
135
100018
Mean
114
100023
In partial explanation of the high number which stands first, it should
perhaps be mentioned that the set of observations in question was incomplete.
Owing to an accident, it was impossible to return from the lower pressure to
the higher pressure, as had been intended.
It may be well to repeat here that
P _ pv at  atmosphere
~ pv at 1 atmosphere'
the temperature being constant and having the values recorded in each case.
Although the accordance of results seems to surpass considerably any
thing attained in observations below atmosphere at the time this work was
undertaken, I must confess that, except in the case of hydrogen, it is not so
good as I had expected in view of the design of the apparatus and of the
care with which the observations were made. I had supposed that an error
of 3 parts in 100,000 (at the outside), corresponding to T ^ C., was as much
as was to be feared. As it is, I do not believe that the discrepancies can be
explained as due to errors of temperature, or of pressure, or of volume so far
as the readings are concerned. But it is possible that a variable contact
between mercury and glass in the lower chamber of the first manometer may
have affected the volume in an uncertain manner, though care was taken to
obviate this as far as could be. It is to be remembered, however, that,
except as to the comparison of the two manometers, all sources of error enter
independently in each set of observations ; and that a mere repetition of the
readings without a change in the gas or in the pressure (from half atmosphere
to one atmosphere, or vice versd) gave very much closer agreements.
1905] ONE ATMOSPHERE AND HALF AN ATMOSPHERE OF PRESSURE 231
The values of B here discussed are the same as those given in the
Preliminary Notice*, except that no account was there taken of the small
deviation in the ratio of volumes from 2 : 1 in consequence of the yielding to
pressure. If we measure p in atmospheres and assume, as has been usually
done, e.g., by Regnault and Van der Waals, that at small pressures' the
equation of an isothermal is
pv = PV(l+ap),
where P V is the value of the product in a state of infinite rarefaction, then
.
pv p
In applying a to correct the observed densities of gases at atmospheric
pressure, we are met with the consideration that a is itself a function of
temperature, and that the value of a. really required for our purpose is that
corresponding to C., at which temperature the weighing vessels are charged
with gas I. In the case of the principal gases a is so small that its correction
for temperature was not likely to be important for the purpose in hand, but
when we come to CO 2 and N 2 O the situation might well be altered. If we
know the pressureequation of the gas, there is no difficulty in calculating
a correction to a in terms of the critical constants. I had, in fact, calculated
such a correction for carbonic acid after Van der Waals, when I became
acquainted with the memoir of D. BerthelotJ, in which this and related
questions are admirably discussed. The " reduced " form of Van der Waals'
equation is
pressure, volume and temperature being expressed in terms of the critical
values. From this we find
I d(irv)
and the effect of a change of temperature upon the value of a is readily
deduced. Indeed, if the pressureequation and the critical values could be
thoroughly trusted, there would be no need for experiments upon the value
of a at all. The object of such experiments is to test a proposed pressure
equation, or to find materials for a new one ; but consistently with this we
may use a form, known to represent the facts approximately, to supply
a subordinate correction.
* Roy. Soc. Proc. Vol. LXXIII. p. 153, February, 1904.
t Except in the comparison of hydrogen and oxygen.
J "Surles thermometres a gaz," Travaux et Mmoires du Bureau International, tome xm.
I am indebted to the Director for an early copy of this memoir, and of that of Chappuis presently
to be referred to.
232 ON THE COMPRESSIBILITY OF GASES BETWEEN [303
A careful discussion of the available data relating to various gases has
led D. Berthelot to the conclusion that the facts at low pressure are not
to be reconciled with Van der Waals' equation, either in its original form
or as modified by Clausius (i.e., with the insertion of the absolute temperature
in the denominator of the cohesive term).
The equation which best represents the relation of d (trv)/d7r to tempera
ture is
drr
corresponding to the pressureequation for low densities
s
As an example, let us apply this formula to find for oxygen what change
must be made in o in order to pass from the temperature of the observations
(112 C.) to C. If be the value of for C., we have
6 = __?I 3 _ = ?!? = 2734JLT2 2842
"273118" 155' 155 ~ = 155"'
The factor by which the observed value of a must be multiplied is thus
'
being the same whether the pressures be reckoned in terms of the critical
pressure or in atmospheres. In the case of oxygen the factor is
568  1097 529
545  973 ~ '428
1236.
The observed value of a is  '00076, corresponding to 112 C. Hence at
C. we should have
a =  00076 x 1236 =  '00094.
It will be seen that the correction has a considerable relative effect ; but
o is so small that the calculated atomic weights are not much influenced. It
must be admitted, however, that observations for the present purpose would
be best made at C. ; to this, however, my apparatus does not lend itself.
The following table embodies the results thus obtained.
1905] ONE ATMOSPHERE AND HALF AN ATMOSPHERE OF PRESSURE 233
Gas
B
a
Temperature
a corrected to 0C.
Oxygen
100038
00076
C.
112
00094
Hydrogen
99974
+ 00052
107
+ 00053.
Nitrogen . . ....
100015
 00030
149
 00056
Carbonic oxide
Air
100026
100023
 00052
00046
138
114
00081
Carbon dioxide
Nitrous oxide . .
100279
100327
 00558
00654
150
110
 00668
 00747
The experiments on carbonic anhydride and on nitrous oxide were of
later date, having been postponed until the apparatus had been well tested
on other gases.
In both these cases it was found that the readings were less constant
than usual, signs being apparent of condensation upon the walls of the
containing vessels, or possibly upon the cement in the manometer. Under
these circumstances it seemed desirable to avoid protracted observations and
to concentrate effort upon reproducing the conditions (especially as regards
time) as closely as possible with and without the use of the large bulbs. In
this way, for example, the question of the cement is eliminated. Condensa
tion upon the walls of the large bulbs themselves, if it occurs, cannot be
eliminated from the results ; all that we can do is not unnecessarily to
increase the opportunity for it by allowing too long a time. It is certain
that, unless by chance, these results are less accurate than for the other
gases, i.e., less accurate absolutely, but the value of a is so much larger that
in a sense the loss of accuracy is less important. Two entirely independent
results for nitrous oxide agreed well. They were :
November 22, 26, 1904 . .
51003295
23, 29, 1904
51003252
Mean . .
5100327
The gas was from the same supply as had been used for density
determinations.
In applying these results to correct the ratios of densities as observed at
atmospheric pressure to what would correspond to infinite rarefaction, we
have, taking oxygen as a standard, to introduce the factor (1 + o)/(l + er ),
OQ being the value for oxygen. Taking a from the third column, which may
be considered without much error to correspond to a temperature of 13 C.
throughout, and also from the fifth column, we have :
234
ON THE COMPRESSIBILITY OF GASES BETWEEN
[303
Gas
Correcting factor for
about 13 C.
Correcting factor for
0C.
Hydrogen
1O0128
100147
Nitrogen
100046
100038
Carbonic oxide
Carbon dioxide
100024
99518
100013
!t<)42<;
Nitrous oxide
99422
99347
The double of the first number in the second column, viz., 2'00256, repre
sents, according to Avogadro's law, the volume of hydrogen which combines
with one volume of oxygen to form water, the pressure being atmospheric
and the temperature 13 C. Scott gave 200245 for 16 C. In his later
work Morley found 20027, but this appears to correspond to C. The
third column in the above table gives for this temperature 20029. The
agreement here may be regarded as very good.
In correcting the densities directly observed at C., in order to deduce
molecular weights, we must use the third column of the above table*.
Oxygen being taken as 32, the densities of the various gases at C., and
at atmospheric and very small pressures, as deduced from my own observa
tions!, are :
Gas
Atmospheric pressure
Very small pressure
H 2
20149 (16 C.)
201 73 = 2 x 10086
N 2
28005
28016 =2x14008
CO
28000
28003
N 2 d
44268
44285
44014
43996
From the researches of M. Leduc and Professor Morley it is probable that
the above numbers for hydrogen are a little, perhaps nearly one thousandth
part, too high. The correction to very small pressures has to be made in
a different manner for hydrogen and for the other gases, in consequence
of the fact that the observed ratio of densities corresponded not to C., but
to 16C.+ Now the observed values of B for hydrogen and oxygen relate
to about 11 C., so that if we correct a to C., we are, in fact, altering it in
the wrong direction. I have employed as the correcting factor to 16 C. the
value 100118.
It may be noticed that the discrepancy between my ratio of hydrogen to
oxygen and that of M. Leduc is partially explained by the fact that my
comparisons were at 16 C., and his at C.
* The correction for temperature was neglected in the Preliminary Notice,
t Roy, Soc. Proc. Vol. mi. p. 134, 1893 ; Vol. LXII. p. 204, 1897. Scientific Papers, Vol. iv.
pp. 39, 362. Also for Nitrous Oxide, Roy. Soc. Proc. Vol. LXXIV. p. 181, 1904.
J See Roy. Soc. Proc. Vol. L. p. 448, 1892 ; Scientific Papers, Vol. m. p. 533.
1905] ONE ATMOSPHERE AND HALF AN ATMOSPHERE OF PRESSURE 235
The uncorrected number for nitrogen (14*003 corresponding to O = 16)
has already been given*, and contrasted with the 14*05 obtained by Stas.
This question deserves the attention of chemists. If Avogadro's law be
strictly true, it seems impossible that the atomic weight of nitrogen can
be 1405.
The atomic weight of carbon can be derived in three ways from these
results. First from CO and O :
CO = 28003
O = 16000
C = 12003
Secondly from CO 2 and O :
C0 2 = 44014
O 2 = 32000
C = 12014
Thirdly from C0 2 and CO. This method is independent of the density
and compressibility of oxygen :
2CO = 56006
CO 2 = 44014
C = 11992
It will be seen that the number for C0 2 is too high to give the best
agreement. Were we to suppose that the true number for CO 2 was 44*004,
instead of 44'014, we should get by the second method 12*004 and by the
third 12*002, in agreement with one another and with the result of the first
method. The alteration required is less than one part in 4000, and is
probably within the limits of error for the compressibility (as reduced to
C.), and perhaps even for the density. The truth is that the second and
third methods are not very advantageous for the calculation of the atomic
weight of carbon, and would perhaps be best used conversely.
Finally the molecular weight of N 2 allows of another estimate of that
of nitrogen. Thus
N 2 O = 43*996
Q = 16000
N 2 = 27996
whence N = 13*998.
It should be remarked that these results relative to C0 2 and N 2 depend
very sensibly upon the correction of a from about 13 C. to C., and that
this depends upon the discussion of M. D. Berthelot. M. Berthelot has
himself deduced molecular weights in a similar manner, founded upon Leduc's
measures of densities.
* Rayleigh and Ramsay, Phil. Trans. A, Vol. CLXXXVI. p. 187, 1895 ; Scientific Papers, Vol. iv.
p. 133.
236 ON THE COMPBESSIBILITY OF GASES BETWEEN
It remains to refer to some memoirs which have appeared since the
greater part of the present work was finished. Foremost among these is
that of M. Chappuis*, who has investigated in a very thorough manner the
compressibilities of hydrogen, nitrogen, and carbonic anhydride at various
temperatures and at pressures in the neighbourhood of the atmospheric. For
hydrogen M. Chappuis finds at C. a = + '00057 per atmosphere, and for
nitrogen o =  00043. In the case of carbonic anhydride pv departs sensibly
from a linear function of p. M. Berthelot gives as a more accurate expression
00670
*! ^.
founded on Chappuis' measures and applicable at C. The unit of pressure
is here the atmosphere. According to this, for p = 1 we get pv = 1 '00665,
as compared with pv = 1 for v = oo , in close agreement with my value
recorded above.
A comparison of Chappuis' method and apparatus with mine may not be
without interest. On his side lay a very considerable advantage in respect
of the "nocuous space" V 3 , inasmuch as this was reduced to as little as
I'l cub. centims., whereas mine was ten times. as great. The advantage
would be important when working at temperatures other than that of the
room. Otherwise, the influence of V 3 did not appear to prejudice my results,
except in so far as V 3 might be uncertain from capillarity in the manometer ;
and this cause of error would operate equally in Chappuis' apparatus.
So far as a 2 : 1 ratio suffices, my method of varying the volume seems the
better, and, indeed, not to admit of improvement.
In the manometric arrangements it would seem that both methods are
abundantly accurate, so far as the readings are concerned. I am disposed,
however, to favour a continual verification of the vacuum by the Topler
pump, and, what is more important, a method of reading which is independent
of possible errors arising from irregular refraction at the walls of the mano
metric tubes.
It may be remarked that, with a partial exception in the case of CO 2 ,
M. Chappuis' work relates to pressures above atmosphere.
Other papers which have appeared since my Preliminary Notice are those
of M. Guye, working both alone and with the assistance of collaborators f.
Several of these relate to the atomic weight of nitrogen and insist on the
discrepancy between the number resulting from density and that of Stas.
Among the methods employed is that of decomposing nitrous oxide by an
incandescent iron wire and comparing the original volume with the residual
."Noavelles Etudes sur les Thermometres a gaz," Extrait du tome xni. det Travaux et
Mfmoiret du Bureau International des Poids et Metures.
t C. R. April 26, May 16, June 13, July 4, October 31, 1904.
1905] ONE ATMOSPHERE AND HALF AN ATMOSPHERE OF PRESSURE 237
nitrogen. In my hands* this method failed to give good results, in conse
quence, apparently, of the formation of higher oxides of nitrogen.
P.S., March 6. Some observations upon Ammonia may here be appended.
The gas was evolved (almost without warmth) from the solution in water,
and was dried by very slow passage over fragments of caustic potash. The
precautions mentioned under nitrous oxide were here followed with more
minute care. The glass surfaces were in contact with the gas for weeks,
either at half atmospheric or whole atmospheric pressure, and the observations
at full pressure were not commenced until that pressure had prevailed for
a day or more. On the reduction of pressure to the half atmospheric,
ammonia was sensibly liberated from the walls, and perhaps from the cement
in the manometer. In the observations to be compared, the same interval
was allowed to elapse between the reduction of pressure and the corresponding
readings, whether the big bulbs were in use or not. Any anomalies not
dependent upon the walls of the big bulbs themselves are thus eliminated.
In addition to commercial ammonia, a special sample prepared by Dr Scott
in accordance with Stas' directions was employed. It will be seen that there
is no certain difference between the results from the two kinds.
The departure from Boyle's law, in this case, is almost more than can be
provided for in the side apparatus. It became therefore necessary to allow
a small difference of temperature between the high and lowpressure
observations such as would somewhat prejudice the accuracy of the results,
were it possible to expect the attainment of the same high degree of accuracy
as for the less condensable gases. Under the actual circumstances the
variation of temperature was of no importance.
AMMONIA.
Date
Source
Temperature
B
December 31, 1904, January 4, 1905 ...
28, 1904, 5, 1905 ...
Commercial
103
92
100647
100631
Mean
Commercial
97
100639
January 23, 25, 1905
Scott's
9 '6
100630
24, 27, 1905
9'8
100617
Mean
Scott's
9'7
100624
Mean of all ...
97
100632
Roy, Soc. Proc. Vol. LXII. p. 204, 1897 ; Scientific Papers, Vol. iv. p. 350.
304.
ON THE PRESSURE OF GASES AND THE EQUATION
OF VIRIAL.
[Philosophical Magazine, Vol. IX. pp. 494505, 1905.]
IF m be the mass of a particle, V its velocity, p the pressure and v the
volume of the body composed of the particles, the virial equation is
i2iF' = ip + *2p*(p), (1)
where further p denotes the distance between two particles at the moment
under consideration, and <(p) the mutual force, assumed to depend upon
p only. If the mutual forces can be neglected, either because they are non
existent or for some other reason, (1) coincides with Boyle's law, since the
kinetic energy is supposed to represent temperature (T).
According to some experimenters, among whom may be especially men
tioned Ramsay and Young, the relation between pressure and temperature at
constant volume is in fact linear, or
p = T+(v)+ x (v); (2)
and it is of interest to inquire whether such a form is to be expected on
theoretical grounds, when <f> (p) can no longer be neglected. It has indeed
been maintained* that (2) is a rigorous consequence of the general laws
of thermodynamics and of the hypothesis that the forces between molecules
are functions of the distance only. The argument proceeded upon the
assumption that the distances of the particles, and therefore the mutual
forces between them, remain constant when the temperature changes, pro
vided only that the volume of the body is maintained unaltered. According
to this the virial term in (1) is a function of volume only, so that (1) reduces
to (2), with >/r(u) proportional to tr 1 . But, as Boltzmann pointed out, the
assumption is unfounded, and in fact inconsistent with the fundamental
principles of the molecular theory. The molecules are not at rest but in
motion; and when the temperature varies there is nothing to hinder the
virial from varying with it.
M. Levy, C. R. t. LXXXVII. pp. 449, 488, 554, 649, 676, 886 (1878).
1905] ON THE PRESSURE OF GASES AND THE EQUATION OF VIRIAL 239
The readiest proof of this assertion is by reference to the case where the
molecules are treated as " hard elastic spheres," that is where the force is zero
so long as p exceeds a certain value (the diameter of the spheres), and then
becomes infinite. From the researches of Van der Waals, Lorentz, and Tait
it is known that in that case
/* f ........................ (3)
where 6, denoting four times the total volume of the spheres, is supposed to
be small in relation to v. So far from the virial being necessarily inde
pendent of temperature, it is here directly proportional to temperature.
The introduction of the special value (3) into (1) gives the wellknown form
p(vb) = ^mV 2 = RT, ........................ (4)
in which b is still regarded as small in comparison with v. It is worthy of
note that this particular case, although of course sufficient to upset the
general argument that the virial is independent of temperature, nevertheless
itself conforms to (2), proportionality to T being for this purpose as good as
independence of T.
Not only is the linear relation maintained in spite of the forces of collision
of elastic spheres when no other forces operate, but it remains undisturbed
even when we introduce such forces, provided that they be of the character
considered in the theory of capillarity, that is extending to a range which
is a large multiple of molecular distances and not increasing so fast with
diminishing distance as to make the total effect sensibly dependent upon
the positions occupied by neighbours. Under these restrictions symmetry
ensures that the resultant force upon a sphere, situated in the interior and
not undergoing collision, is zero ; and the whole effect of such forces is repre
sented (Young, Laplace, Van der Waals) by an addition to the pressure
of a quantity independent of the temperature and inversely proportional to
the square of the volume. In Van der Waals' wellknown form
(5)
the relation between p and T is still linear. Even if the particles depart
from the spherical form, the virial of collisional and cohesive forces remains
a linear function of the temperature*.
The forces above considered are partly repulsive and partly attractive.
Repulsion at a certain degree of proximity seems to be demanded in order to
preserve the individuality of molecules and to prevent infinite condensation.
It will be remembered that Maxwell proposed a repulsion inversely as the
fifth power of the distance, partly as the consequence of some faulty experi
* "On the Virial of a System of Hard Colliding Bodies," Nature, XLV. pp. 8082 (1891) ;
Scientific Papers, in. p. 469.
240 ON THE PRESSURE OF GASES [304
ments upon the relation of viscosity to temperature and partly no doubt on
account of a special facility of calculation upon the basis of this law. So far
as viscosity (17) is concerned, its relation to temperature (T) when the force
of repulsion varies as p~ n is readily obtained by the method of dimensions*.
It appears that
(6)
The case of sudden collisions may be represented by taking n = oo , so
that
r>~T$; .................................... (7)
while if w = 5 y oc T. .................................... (8)
According to experiments on the more permanent gases n would vary from
68 for hydrogen to '81 for argon ; but Sutherland's lawf
( '
probably represents the facts better than (6), whatever value may be assigned
to n. According to the theory of corresponding states, C should be pro
portional to the critical temperature when we pass from one gas to another.
A similar application of the method of dimensions will give interesting
information respecting the virial, when the force of repulsion is
The virial is a definite function of N the number of molecules, m the mass of
each molecule, V the velocity of mean square on which the temperature
depends, /* the force at unit distance, and v the volume of the containing
vessel. Of these quantities the virial is of the dimensions of energy, N
has none, m is a mass simply, V is a velocity, v a volume, while p has the
dimensions
mass x (length) 71 * 1 x (time)~ 2 .
Hence if we suppose that the virial varies as v~ s , we find that it must be
proportional to
(mV*y~* ::r .^.v; ........................ (11)
or since mV* represents temperature,
8l 8
T"* 1 ./t'.tr' ............................ (12)
For example, if s = 0,
T, .............................. (13)
* Proceedings Royal Society, LXVI. p. 68 (1900) ; Scientific Papers, iv. p. 453.
t Phil. Mag. Vol. xxxvi. p. 513 (1893).
1905] AND THE EQUATION OF VIRIAL 241
whatever n may be. Hence a term in the virial equation independent of
volume must be proportional to temperature, as in (1). Again, if s = 1,
tr'.r ......................... (14)
Of this we have already had examples, both the virial terms in Van der
Waals' equation being proportional to tr 1 . The first, representing the virial
of collisional forces, corresponds in (14) to n = x , giving proportionality to T.
The second is independent of T and can be reconciled with (14) only by
supposing n = 4. It might seem that in a rare gas, whenever the virial
depends sensibly upon what occurs during the encounters of simple pairs of
molecules, there must be proportionality to v~\ so that (14) would apply.
If, as Maxwell supposed, n = 5,
2p <(p)cc V i.r*, ........................... (15)
in agreement with a result obtained by Boltzmann for this case. If we
retain n = 5, but leave the relation to v open, we get from (12)
tr'.Z T ' ......................... (16)
If we now discard the supposition that the dependence upon v follows the
law of tr*, we may interpret (16) to mean that considered as a function of v
and T, the virial is limited to the form
T.F(vT), ........................ (17)
F denoting an arbitrary function of the single variable vT*.
And more generally, whatever n may be, we find from (12) that the virial
is limited to the form
.(18)
A further generalization may be made by discarding altogether the
position that <(p) is represented by any power of p. In this case it is
convenient to write
*(p) = A*'/(p/). (19)
where a is a linear quantity. Here f itself may be supposed to be of no
dimensions, while /// has the dimensions of a force. The virial is a function
of ft, a, m, V, v; and since its dimensions are those of energy, i.e. of m V' or
T, we may write
2p<(p) = T. F(fjf, a, m, V, v),
where F is of no dimensions. It is easy to see that /*', m, and F 2 can occur
only in the combination /i'/mF 2 or ft'/T. To make this of no dimensions, we
B. v. 16
242 ON THE PRESSURE OF GASES [304
introduce the factor a. Thus F becomes a function of a, v, and p'a/T, in
which again v can occur only in the form a*/v. Accordingly
(20)
F being in general an arbitrary function of two variables.
From (20) we may fall back on (18) by the consideration that in accord
ance with (10) /*' and a can occur only in the combination p!a n .
It may be well to remark that the method of dimensions does not tell us
whether or no an available solution can be deduced from particular assump
tions. What it teaches us is the form which an available solution must
assume. For example, equation (14) gives the form of the term in the virial
proportional to w" 1 , under the law of force (10); and nothing has been said
as to any restriction upon the value of n. But it is easy to see that n must
in fact be greater than 4. Otherwise the integral representing the virial
relating to a given particle would not be convergent. We have to consider
p(j> (p) p*dp
with infinity for the upper limit, and this diverges unless n exceed 4.
It is not to be expected that any law included under (10) could represent
with completeness the mutual action of the particles of a gas. Under it no
provision can be made for repulsion at small distances and attraction at
greater ones. And when n > 4, the aggregate virial depends too much upon
the encounters which take place at exceedingly small distances.
If, as for both the virial terms in Van der W T aals' formula, there be pro
portionality to ir 1 , (20) becomes
or, if we prefer it,
......................... (22)
F in both cases denoting an arbitrary function. According to Van der Waals
F in (21) is a linear function, the constant part giving the collisional virial
and the second term the cohesional virial which is independent of T. Except
for one consideration to be mentioned presently, there would appear to be
good reason for supposing the virial of a rare gas to be proportional to v~ l ;
but on the other hand it is doubtful whether the cohesional forces are alto
gether of the kind supposed by Laplace and Van der Waals. We should
expect the cohesional virial to be more directly influenced by the approaches
of molecules during an encounter ; and on the experimental side D. Berthelot
has shown cause for preferring to that of Van der Waals the Rankine and
1905] AND THE EQUATION OF VIRIAL 243
Clausius form, in which a factor T is introduced in the denominator. The
most natural extension of the formula would be by substituting a quadratic
for a linear form of F in (21). We should then write
(23)
A, B, G being arbitrary constants; and the pressure equation, when written
after Van der Waals' manner with neglect of v~*, becomes
As has already been said, Van der Waals' form corresponds to (7=0. On the
other hand, the Rankine and Clausius form requires that B = 0, while C
remains finite. It will be evident that the two alternatives differ funda
mentally. According to the latter the cohesional terms tend to vanish when
T is sufficiently increased.
If the cohesional terms are to vanish when T is infinite, the forces con
cerned must be of an entirely different character from that contemplated in
Laplace's and Van der Waals' theory. It has been suggested by Sutherland*
that the forces may be of electric origin and in themselves (except during
actual collision) as much repulsive as attractive. This is not inconsistent
with the preponderance of attraction in the final result. " There is this
fundamental distinction in the effects of attractive and repulsive forces
whose strength decreases with increasing distance, that the attractive
forces by their own operation tend to increase themselves, while the re
pulsive forces tend to decrease themselves." The forces contemplated
by Sutherland are such as are due to electric or magnetic doublets, but a
rather simpler illustration may be arrived at by retaining the single character
of the centres of force, and supposing them to be as much positive as
negative, under the usual electrical law that similars repel while opposites
attract one another. When T is infinite, so that the paths are not influenced
by the forces, the cohesional virial will disappear, but it may become finite
as the temperature falls and room is given for the attractive forces to assert
their advantage. There is nothing in the argument upon which (21) was
founded which is interfered with by the occurrence of the two kinds of
particles, and it would seem that F must then become an even function of /*',
so that in (23) 5 = 0.
As stated, the above argument is probably not quite legitimate, inasmuch
as according to (19) a reversal of p! would imply a reversal of the collisional
forces as well as of those which operate at greater distances. The intro
duction of the two sorts of particles is not supposed to alter the repulsive
forces called into play during actual collision. I believe, however, that the
* Phil. Mag. Vol. iv. p. 625 (1902).
162
244 ON THE PRESSURE OF GASES [304
instantaneous collisional forces may be omitted from (19). The effect of the
collisions may be defined without reference to any datum having dimensions
other than a, representing the radius of a sphere. The collisions being thus,
as it were, already provided for, the argument remains that the virial must
be a definite function of N, m, V, //, a, v, of which N need not be regarded,
the force (outside actual collision) being given by (19). Equation (21) then
follows as before with its approximate form (23). If we now suppose that
the particles are repellent as much as attractive, (19) may be written
<(/>)= //(/>/); (25)
and, since odd powers of fi are now excluded, B = in (23), (24).
We have thus discovered a possible theoretical foundation for the empirical
conclusion that T should be introduced into the denominator of the cohesional
virial, and it would seem to follow conversely that, if the empirical conclusion
is correct, the forces must be intrinsically as much repellent as attractive.
This argument may be regarded as a strong confirmation of Sutherland's idea,
though a question remains as to how the attraction asserts its superiority
over repulsion.
In the above argument the particles are regarded as simple centres of
force, half of them being " positive " and half " negative." The advantage is
that the form may still be treated as spherical, so that the collisions may be
assimilated to those of " elastic spheres." But a polar constitution, such that
the positive and negative elements are combined in every particle, is certainly
more probable. This will introduce, as another linear datum, the distance
between the poles, and the collisions will admit of greater variety. Moreover,
there is now kinetic energy of rotation as well as of translation. However,
since the kinetic energies are proportional, the argument remains unaffected,
so far as it relates to the dependence of the virial of a given gas upon volume
and temperature, and the RankineClausius form (24) with B = still
obtains.
As to the preponderance of attractive over repulsive virial, I think that
the conclusion is correct, although Sutherland's argument, quoted above,
omits reference to the essential consideration of the time for which any
particular value of the virial prevails. If we fix our attention upon a pair of
particles, acting as simple centres of force, which encounter one another, the
corresponding virial varies from moment to moment, but the mean contribu
tion to the total may be represented by
f<f>(p)pdt,
the integration being taken over the whole range for which p <j>(p) is sensible.
Since only relative motion is in question, the centre of gravity of the two
1905] AND THE EQUATION OF VIRIAL 245
particles may be supposed to be at rest and the problem becomes one of
" central forces." In the usual notation we have
so that
(27)
v denoting the resultant velocity. At the upper limit dr/dt is equal to the
velocity at oo , say V, and at the lower limit dr/dt = V. Hence
s, ....................... (28)
so that the mean virial is closely connected with the " action " in the orbit.
For a simple illustration it will be more convenient to make 6 the inde
pendent variable. Thus by (26)
.r.cW ......................... (29)
Suppose for example that
P = pr 3 .................................. (30)
Then p.r.dtMe, ........................ (31)
where 6 represents twice the vectorial angle between the initial asymptote
and the apse. If h be given, a comparison between repellent and attractive
forces (fj, given in magnitude but variable in sign) shows that (31) is greater
in the case of attraction (fig. 1), so that if attractive and repellent forces
occur indifferently the average effect corresponds to attraction. In the case
of the particular law (30) we can carry out the calculation. If, as usual,
u = r~ l , the equation of the orbit is
/,: + ............................... (32)
IJL being positive in the case of attraction ; whence, if fj, be small,
u= U sin V(l  M~ 2 ) ......................... (33)
In (33) u = 0, or r = oo , when 9 = and when
= 7r + v(iM 2 ); ........................... (34)
so that from (31)
* ............... ' .......... (35)
The solutions (33), (35) hold if p be numerically less than A 2 , and (35)
246
ON THE PRESSURE OF GASES
[304
shows that when fi changes sign the virial of attraction preponderates. This
conclusion is accentuated by the consideration of what occurs if /* exceed h
nuraerically. Equations (33), (35) still hold if //, be negative, i.e. if the force
be repulsive. But when p, is positive, the form changes. Thus if yu, = h 3 ,
we have
u=Ud, (36)
and neither 8 in (31) nor the virial has a finite value. The like remains true
when p > h*.
In the above example Pr 3 remains constant, and the preponderance of
attraction over repulsion depends upon the greater vectorial angle in the
former case. If Pr*, instead of remaining constant, continually increases
with diminishing r, the preponderance of attraction follows a fortiori.
Fig. i. Fig. 2.
0, centre offeree; CD, asymptote; A, B, apses.
A particular case of (32) which arises when //, = A 1 should be singled out
for especial notice, i.e. the case of circular motion for which u constant.
The attracting particles then revolve round one another in perpetuity, and
the virial is infinite in comparison with that of an ordinary encounter. It is
this possible occurrence of reentrant orbits which causes hesitation as to the
accuracy with which we may assume the virial of a rare gas to be inversely as
the volume. It seems to be generally supposed (see, for example, Meyer's
Kinetic Theory of Oases, 4) that if a gas be rare enough no appreciable
pairing can occur. But the question is not as to the frequency with which
new pairs may form, but as to the relative number of them in existence
at any time. It is easy to recognize that the coupling or the severance of a
pair of particles cannot occur of itself, but requires always the cooperation of
a third particle. If the gas is very rare, no doubt there are few opportunities
1905] AND THE EQUATION OF VIRIAL 247
for the formation of fresh pairs, but for the same reason those already formed
have a higher degree of permanence. On the whole it would appear that the
number of pairs in existence at any moment is independent of the volume v
of a rare gas, and the same would be true of the corresponding virial. At
this rate we should have terms in the virial which by (20) come under the
form
T.F(&}.. ...(37)
It will be remarked that if these terms in the virial, independent of v, are
sensible, the density of the gas will depart from Avogadro's rule, however
greatly it may be rarefied. In the case of elastic spheres, which come into
collision when their centres approach to a certain distance, there is naturally
a limit to the magnitude of the attraction, and then pairing becomes im
possible if the velocity be sufficiently great. Any departure from Avogadro's
rule at high rarefactions would thus tend to disappear as the temperature
rises.
The behaviour of mere centres of force, which may approach one another
without limit, appears to follow a different course. Taking for example the
power law of (10), we see from (18) that for any part of the virial which is
independent of v, the function F must be constant, so that the virial is
proportional to T and independent of p.
To return to the question with which we started, there seems good reason
to doubt that the relation of pressure to temperature with volume constant
is accurately linear, even at high rarefactions. On the other hand, it is clear
that this relation is approximately satisfied ; and the natural course would
be to take it as a foundation, determining the functions ^ and ty in (2), as
well as the function of v and T jointly which may be required in supplement.
As regards the latter part of the question, a differential arrangement in which
two gases, say CO 2 and H 2 , are balanced against one another at the same
temperature, would appear to offer advantages. This is shown diagram
matically in fig. 2, where the two gasreservoirs are connected by a Utube
containing mercury. According to Boyle's law, even as modified by the
introduction of a co volume, the mercury may stand in the Utube at fixed
marks at the same level, in spite of variations of temperature affecting both
bulbs alike. And under the more general law (2) the same fixity of the
mercury thread can be attained, though now with the extremities at different
levels. With such an arrangement the departure from (2) becomes a matter
of direct observation, and so long as uniformity of temperature is secured,
a precise measurement of it, or of the total pressure, is of secondary import
ance. Useful results would probably require a total pressure of four or five
atmospheres.
305.
THE DYNAMICAL THEORY OF GASES AND OF
RADIATION.
[Nature, Vol. LXXI. p. 559; Vol. LXXII. pp. 54, 55; pp. 243, 244, 1905*.]
IN Mr Jeans' valuable work upon this subject f he attacks the celebrated
difficulty of reconciling the " law of equipartition of energy " with what
is known respecting the specific heats of gases. Considering a gas the
molecules of which radiate into empty space, he shows that in an approxi
mately steady state the energy of vibrational modes may bear a negligible
ratio to that of translational and rotational modes.
I have myself speculated in this direction; but it seems that the difficulty
revives when we consider a gas, not radiating into empty space, but bounded
by a perfectly reflecting enclosure. There is then nothing of the nature of
dissipation; and, indeed, the only effect of the appeal to the ather is to
bring in an infinitude of new modes of vibration, each of which, according
to the law, should have its full share of the total energy. I cannot give the
reference, but I believe that this view of the matter was somewhere} ex
pressed, or hinted, by Maxwell.
We know that the energy of sethereal vibrations, corresponding to a
given volume and temperature, is not infinite or even proportional to the
temperature. For some reason the higher modes fail to assert themselves.
A full comprehension here would probably carry with it a solution of the
specific heat difficulty.
I am glad to have elicited the very clear statement of his view which
Mr Jeans gives in Nature of April 27. In general outline it corresponds
* The reader interested in this subject should refer to Mr Jeans' letters, Nature, Vol. LXXI.
p. 607 ; Vol. LXXII. pp. 101, 102; pp. 293, 294.
t The Dynamical Theory of Gasei, Camb. Univ. Press, 1904.
t Mr Jeans refers to Maxwell's Collected Works, Vol. n. p. 433.
Compare "Remarks upon the Law of Complete Radiation," Phil. Mag. Vol. XLIX. p. 539,
1900. [Scientific Paper*, Vol. iv. p. 483.]
1905] THE DYNAMICAL THEORY OF GASES AND OF RADIATION 249
pretty closely with that expressed by 0. Reynolds in a British Association
discussion at Aberdeen (Nature, Vol. xxxn. p. 534, 1885). The various
modes of molecular motion are divided into two sharply separated groups.
Within one group, including the translatory modes, equipartition of energy
is supposed to establish itself within a small fraction of a second ; but
between the modes of this group and those of vibration included in the other
group, equipartition may require, Mr Jeans thinks, millions of years. Even
if minutes were substituted for years, we must admit, I think, that the law
of equipartition is reconciled with all that is absolutely proved by our ex
periments upon specific heat, which are, indeed, somewhat rough in all cases,
and especially imperfect in so far as they relate to what may happen over
long intervals of time.
As I have already suggested, it is when we extend the application of the
law of equipartition to the modes of sethereal vibration that the difficulties
thicken, and this extension we are bound to make. The first question is as
to the consequences of the law, considered to be applicable, after which, if
necessary, we may inquire whether any of these consequences can be evaded
by supposing the equipartition to require a long time for its complete
establishment. As regards the first question, two things are at once evident.
The energy in any particular mode must be proportional to 0, the absolute
temperature. And the number of modes corresponding to any finite space
occupied by the radiation, is infinite. Although this is enough to show
that the law of equipartition cannot apply in its integrity, it will be of
interest to follow out its consequences a little further. Some of them were
discussed in a former paper*, the argument of which will now be repeated
with an extension designed to determine the coefficient as well as the law
of radiation.
As an introduction, we consider the motion of a stretched string of
length I, vibrating transversely in one plane. If a be the velocity of pro
pagation, the number of subdivisions in any mode of vibration, the
frequency / is given by
/=o*/2* ..................................... (1)
A passage from any mode to the next in order involves a change of unity in
the value of , or of Zlf/a. Hence if e denote the kinetic energy of a single
mode, the law of equipartition requires that the kinetic energy corresponding
to the interval df shall be
2fe/o.d/. ................................. (2)
In terms of X the wavelength, (2) becomes
\ .................................. (3)
* "Remarks upon the Law of Complete Radiation," Phil. Mag. Vol. XLIX. p. 539, June,
1900. [Scientific Papers, Vol. iv. p. 483.]
250 THE DYNAMICAL THEORY OF OASES [305
This is for the whole length of the string. The longitudinal density of the
kinetic energy is accordingly
2e/X.d\ ..................................... (4)
In each mode the potential energy is (on the average) equal to the
kinetic, so that if we wish to reckon the whole energy, (4) must be doubled.
Another doubling ensues when we abandon the restriction to one plane of
vibration ; and finally for the total energy corresponding to the interval from
X to X + d\ we have
8e/\*.d\ ..................................... (5)
When we proceed to three dimensions, and consider the vibrations
within a cube of side I, subdivisions may occur in three directions. In
place of (1)
/a/2JV(F + Vm ........................... (6)
where , 17, may assume any integral values. The next step is to
ascertain what is the number of modes which corresponds to an assigned
variation of f.
If the integral values of f , 77, f be regarded as the coordinates of a point,
the whole system of points constitutes a cubic array of volumedensity unity.
If R be the distance of any point from the origin,
E 2 = f +7? 2 + 2 ;
and the number of points between R and R + dR, equal to the included
volume, is
Hence the number of modes corresponding to df is
47r(2l/ay/*df,
or in terms of X
47r.8/ s .X<dX .................................. (7)
If e be the kinetic energy in each mode, then the kinetic energy corre
sponding to d\ and to unit of volume is
32 . TT . e . \~*d\ ............................... (8)
Since, as in the case of the string, we are dealing with transverse vibrations,
and since the whole energy is the double of the kinetic energy, we have
finally
128 . TT . e . X*dX .............................. (9)
as the total energy of radiation per unit of volume corresponding to the
interval from X to X + d\, and in (9) e is proportional to the absolute
temperature 6.
Apart from the numerical coefficient, this is the formula which I gave
in the paper referred to as probably representing the truth when X is large,
in place of the quite different form then generally accepted. The suggestion
1905] AND OF RADIATION 251
was soon confirmed by Rubens and Kurlbaum, and a little later Planck
(Drude Ann. Vol. IV. p. 553, 1901) put forward his theoretical formula,
which seems to agree very well with the experimental facts. This contains
two constants, h and k, besides c, the velocity of light. In terms of \ it is v
reducing when X is great to
Ed\ = 8irke\ 4 d\ ........................... (11)
in agreement with (9). E d\ here denotes the volumedensity of the energy
of radiation corresponding to dX.
A very remarkable feature in Planck's work is the connection which he
finds between radiation and molecular constants. If N be the number of
gaseous molecules in a cubic centimetre at C. and under a pressure of one
atmosphere,
1013 x 10*
273JV
Though I failed to notice it in the earlier paper, it is evident that (9)
leads to a similar connection. For e, representing the kinetic energy of a
single mode at temperature 0, may be identified with onethird of the
average kinetic energy of a gaseous molecule at that temperature. In the
virial equation, if N be the total number of molecules,
so that
e=pv/2N. ................................. (13)
If we apply this to one cubic centimetre of a gas under standard conditions,
N has the meaning above specified, v = l, and p = l'Q 13x10" C.G.S. Ac
cordingly, at C.
e= 1013
and at
1013 x 10" x
Introducing this into (9), we get as the number of ergs per cubic centimetre
of radiation
64.7r.r013.10 6 .0.dX
273.#.\<
6 being measured in centigrade degrees. This result is eight times as large
as that found by Planck. If we retain the estimate of radiation used in his
calculations, we should deduce a value of N eight times as great as his, and
probably greater than can be accepted. [See below.]
A critical comparison of the two processes would be of interest, but not
having succeeded in following Planck's reasoning I am unable to undertake
252 THE DYNAMICAL THEORY OF GASES [305
it. As applying to all wavelengths, his formula would have the greater
value if satisfactorily established. On the other hand, the reasoning which
leads to (15) is very simple, and this formula appears to me to be a necessary
consequence of the law of equipartition as laid down by Boltzmann and
Maxwell. My difficulty is to understand how another process, also based
upon Boltzmann's ideas, can lead to a different result.
According to (15), if it were applicable to all wavelengths, the total
energy of radiation at a given temperature would be infinite, and this is an
inevitable consequence of applying the law of equipartition to a uniform
structureless medium. If we were dealing with elastic solid balls colliding
with one another and with the walls of a containing vessel of similar
constitution, energy, initially wholly translational, would be slowly converted
into vibrational forms of continually higher and higher degrees of subdivision.
If the solid were structureless, this process would have no limit ; but on an
atomic theory a limit might be reached when the subdivisions no longer
included more than a single molecule. The energy, originally mechanical,
would then have become entirely thermal.
Can we escape from the difficulties, into which we have been led, by
appealing to the slowness with which equipartition may establish itself?
According to this view, the energy of radiation within an enclosure at given
temperature would, indeed, increase without limit, but the rate of increase
after a short time would be very slow. If a small aperture is suddenly
made, the escaping radiation depends at first upon how long the enclosure
has been complete. In this case we lose the advantage formerly available
of dividing the modes into two sharply separated groups. Here, on the
contrary, we have always to consider vibrations of such wavelengths as to
bear an intermediate character. The kind of radiation escaping from a
small perforation must depend upon the size of the perforation.
Again, does the postulated slowness of transformation really obtain ? Red
light falling upon the blackened face of a thermopile is absorbed, and the
instrument rapidly indicates a rise of temperature. Vibrational energy is
readily converted into translational energy. Why, then, does the thermopile
not itself shine in the dark ?
It seems to me that we must admit the failure of the law of equipartition
in these extreme cases. If this is so, it is obviously of great importance to
ascertain the reason. I have on a former occasion (Phil. Mag. Vol. XLIX.
p. 118, 1900)* expressed my dissatisfaction with the way in which great
potential energy is dealt with in the general theory leading to the law of
equipartition.
* [This Collection, Vol. iv. p. 451.]
1905] AND OF RADIATION 253
In Nature, May 18, I gave a calculation of the coefficient of complete
radiation at a given absolute temperature for waves of great length on
principles laid down in 1900, and it appeared that the result was eight times
as great as that deduced from Planck's formula for this case. In connection
with similar work of his own Mr Jeans (Phil. Mag. July [1905]) has just
pointed out that I have introduced a redundant factor 8 by counting
negative as well as positive values of my integers , 77, ,
I hasten to admit the justice of this correction. But while the precise
agreement of results in the case of very long waves is satisfactory so far
as it goes, it does not satisfy the wish expressed in my former letter for
a comparison of processes. In the application to waves that are not long,
there must be some limitation on the principle of equipartition. Is there
any affinity in this respect between the ideas of Prof. Planck and those
of Mr Jeans ?
[1911. Since the date of these letters further valuable work has been
done by Planck, Jeans, Lorentz, Larmor, Einstein and others. But I suppose
the question can hardly yet be considered settled.]
306.
AN OPTICAL PARADOX.
[Philosophical Magazine, Vol. ix. pp. 779781, 1905.]
CONSIDER the following combination : A pointsource A of approximately
homogeneous light (X) is focused by the lens LL upon the objectglass of a
telescope T. In its turn the telescope is focused upon L. According to
geometrical optics the margin of the lens L should be seen sharp by an eye
applied to the telescope ; but when we consider the limitation of aperture at
the objectglass of the telescope, we come to the conclusion that the definition
must be very bad. The image of A at G constitutes the usual diffraction
pattern of which most of the light is concentrated in the central disc. The
diameter of this disc is of the order \ . LC/LL. If this be regarded as the
effective aperture of T, the angular resolving power will be found by dividing
\ by the above quantity, giving LL/LC ; so that the entire angular magnitude
of the lens LL is on the limits of resolving power.
If this be admitted, we may consider next the effect of enlarging the
source A, hitherto supposed to be infinitely small. If the process be carried
far enough, the objectglass of T will become filled with light, and we may
expect the natural resolving power to be recovered. But here we must
distinguish. If the enlarged source at A be a selfluminous body, such as
a piece of whitehot metal or the carbon of an electric arc, no such conclusion
will follow. There is no phaserelation between the lights which act at
different parts of the objectglass, and therefore no possibility of bringing
into play the interferences upon which the advantage of a large aperture
depends. It appears, therefore, that however large the selfluminous source
at A may be, the definition is not improved, but remains at the miserably
low level already specified. If, however, the source at A be not a real one,
but merely an aperture through which light from real sources passes, the case
may be altered.
1905] AN OPTICAL PARADOX 255
Returning to the extended selfluminous source, we see that the inefficiency
depends upon the action of the lens L. If the glass be removed from its seat,
so that A is no longer focused upon the objectglass, the definition must
recover.
I do not know how far the above reasoning will seem plausible to the
reader, but I may confess that I was at first puzzled by it. I doubt whether
any experimenter would willingly accept the suggested conclusion, though he
might be unable to point out a weak place in the argument. He would prob
ably wish to try the experiment ; and this is easily done. The lens L may be
the collimatinglens of an ordinary spectroscope whose slit is backed by a
flame. The telescope is removed from its usual place to a distance of say
10 feet and is focused upon L. The slit is at the same time focused upon the
objectglass of the telescope. Although the image of the slit is very narrow,
the definition of L as seen in the telescope does not appear to suffer, the
vertical parts of the circular edge (parallel to the slit) being as well defined
as the horizontal parts. If, however, at the objectglass a material screen be
interposed provided with a slit through which the image of the first slit can
pass, the definition at the expected places falls off greatly, even although a
considerable margin be allowed in the width of the second slit.
This experiment gives the clue to the weak place in the theoretical
argument. It is true that the greater part of the light ultimately reaching
the eye passes through a very small area of the objectglass; but it does not
follow that the remainder may be blocked out without prejudice to the
definition of the boundary of the field. In fact, a closer theoretical dis
cussion of the diffraction phenomena leads to conclusions in harmony with
experiment.
In the case of a pointsource and the complete circular aperture LL, the
question turns upon the integral
J , J l being the Bessel's functions usually so denoted. The integral passes
from to 1/yS, as a passes through the value $*.
If the aperture of LL be reduced to a narrow annulus, the integral to be
considered is
o
This assumes an infinite value when a =
If the apertures be rectangular, the integrals take still simpler forms.
* A theorem attributed to Weber. See Gray and Matthews' Bessel's Functions, p. 228.
t See Theory of Sound, 203, equations (14), (16).
307.
THE PROBLEM OF THE RANDOM WALK.
[Nature, Vol. LXXII. p. 318, 1905.]
THIS problem, proposed by Prof. Karl Pearson in the current number of
Nature*, is the same as that of the composition of n isoperiodic vibrations of
unit amplitude and of phases distributed at random, considered in Philosophical
Magazine, x. p. 73, 1880; XLVII. p. 246, 1889 (Scientific Papers, I. p. 491 ; iv.
p. 370). If n be very great, the probability sought is
Probably methods similar to those employed in the papers referred to
would avail for the development of an approximate expression applicable
when n is only moderately great.
* [Vol. LXXII. p. 294. " A man starts from a point and walks I yards in a straight line ;
he then turns through any angle whatever and walks another I yards in a second straight line.
He repeats this process rj times. I require the probability that after these n stretches he is at a
distance between r and r + Sr from his startingpoint 0. The problem is one of considerable
interest, but I have only succeeded in obtaining an integrated solution for two stretches. I think,
however, that a solution ought to be found, if only in the form of a series in powers of 1/n, when
n is large."]
308.
ON THE INFLUENCE OF COLLISIONS AND OF THE MOTION
OF MOLECULES IN THE LINE OF SIGHT, UPON THE
CONSTITUTION OF A SPECTRUM LINE.
[Proceedings of the Royal Society, Vol LXXVI. (A), pp. 440444, 1905.]
APART from the above and other causes of disturbance, a line in the
spectrum of a radiating gas would be infinitely narrow. A good many years
ago*, in connection with some estimates by Ebert, I investigated the widening
of a line in consequence of the motion of molecules in the line of sight,
taking as a basis Maxwell's wellknown law respecting the distribution of
velocities among colliding molecules, and I calculated the number of inter
ferencebands to be expected, upon a certain supposition as to the degree of
contrast between dark and bright parts necessary for visibility. In this
investigation no regard was paid to the collisions; the vibrations issuing
from each molecule being supposed to be maintained with complete regularity
for an indefinite time.
Although little is known with certainty respecting the genesis of radiation,
it has long been thought that collisions act as another source of disturbance.
The vibrations of a molecule are supposed to remain undisturbed while a free
path is described, but to be liable to sudden and arbitrary alteration of phase
and amplitude when another molecule is encountered. A limitation in the
number of vibrations executed with regularity necessarily implies a certain
indeterminateness in the frequency, that is a dilatation of the spectrum line.
In its nature this effect is independent of the Doppler effect for example, it
will be diminished relatively to the latter if the molecules are smaller ; but
the problem naturally arises of calculating the conjoint action of both causes
upon the constitution of a spectrum line. This is the question considered by
Mr C. Godfrey in an interesting paper f, upon which it is the principal object
* Phil. Mag. Vol. xxvii. p. 298, 1889 ; Scientific Papers, Vol. in. p. 258.
t "On the Application of Fourier's Double Integrals to Optical Problems," Phil. Trans. A,
VoL cxcv. p. 329, 1899.
B.V. 17
258 INFLUENCE OF MOLECULAR COLLISIONS, ETC., [308
of the present note to comment. The formula? at which he arrives are some
what complicated, and they are discussed only in the case in which the
density of the gas is reduced without limit. According to my view this
should cause the influence of the collisions to disappear, so that the results
should coincide with those already referred to where the collisions were
disregarded from the outset. Nevertheless, the results of the two calculations
differ by 10 per cent., that of Mr Godfrey giving a narrower spectrum line
than the other.
The difference of 10 per cent, is not of much importance in itself, but a
discrepancy of this kind involves a subject in a cloud of doubt, which it
is desirable, if possible, to dissipate. Mr Godfrey himself characterises the
discrepancy as paradoxical, and advances some considerations towards the
elucidation of it. I have a strong feeling, which I think I expressed at the
time, that the 10percent. correction is inadmissible, and that there should be
no ambiguity or discontinuity, in: passing to the limit of free paths infinitely
long. In connection with some other work I have recently resumed the
consideration of the question, and I am disposed to think that Mr Godfrey's
calculation involves an error respecting the way in which the various free
paths are averaged.
The first question is as to the character of the spectrum line corresponding
to a regular vibration which extends over a finite interval of time. As the
energy lying between the limits and n + dn of frequency (or rather inverse
wavelength), Mr Godfrey finds from Fourier's theorem
,
dn >
r denoting the finite length of the train of waves, and n being measured from
that value which would be dominant if r were infinitely long. For the total
energy of all wavelengths we have
* dn==7r3r (2)
That the total energy should be proportional to r is what we would expect.
The maximum coefficient in (1) occurs, of course, when n = 0, and is propor
tional to r 8 ; once proportional to r on account of the greater total energy
as given in (2), and again on account of the greater condensation of the
spectrum as r increases. Expression (1) may be taken to represent the
spectrum of the radiation from a single molecule which describes in a given
direction and with a given velocity a free path proportional to r. If there
be N independent molecules answering to this description, N may be intro
duced as a factor into (1). From this expression Mr Godfrey proceeds to
investigate the spectrum corresponding to the aggregate radiation of the gas*,
:: M
1905] ON THE CONSTITUTION OF A SPECTRUM LINE 259
integrating first for the different lengths (r) of parallel free paths described
with constant velocity, and afterwards for the various component velocities
across and in the line of sight, the latter giving rise to the Doppler effect.
It is with the first of these integrations that I am more particularly
concerned.
In order to effect it, we need to know the probabilities of the various
lengths of free path described with given velocity. " Now, Tait has shown
that, of all atoms moving with velocity v, a fraction e~S ft penetrates unchecked
to distance p where f is [a function of v and of the permanent data of the
gas]. From this we see that, of molecules moving with velocity v, a fraction
fe'^dp have free paths between p and p 4 dp. Now, such a molecule will
emit an undisturbed train of waves of length between r and r + dr, where
r = p . V/v, and V is the velocity of light. Hence, of all molecules moving
with velocity v, a fraction (fv/V)e~ v S r/v will give free paths between r and
r + dr. Returning to the expression (1) for the energy of a single train of
length r, we see that with the aggregates of molecules now under considera
tion (definite thwart and lineofsight velocities) we have for n a proportion
of energy
S i n2 H7rr . dr
A_
or, on effecting the integration,
27T 2
The next steps are integrations over the various velocities, but it is not
necessary to follow them here in detail, inasmuch as the objection which I
have to take arises already. It appears to me that what we are concerned
with is not the momentary distribution of free paths among the molecules
which are 'describing them, but rather the statistics of the various free paths
(described with velocity v) which occur in a relatively long time. During this
time various free paths occur with frequencies dependent on the lengths.
Fix the attention on two of these, one long and one short. They present
themselves in certain relative numbers, or say in a certain proportion, and it
is with this proportion that we have to do. The other procedure takes, as it
were, an instantaneous view of the system and, surveying the molecules,
inquires what proportions of them are pursuing free paths of the two lengths
under contemplation. It is not difficult to recognize that this is a different
question. Of the paths which are described in a given period of time, an
instantaneous survey is more likely to hit upon a long one than upon a short
one. Thus Mr Godfrey's integration favours unduly the long paths.
* Mr Godfrey's expression (iii) differs somewhat from (3). A 4^ appears to have been
temporarily dropped, but this is not material for my present purpose.
172
260 INFLUENCE OF MOLECULAR COLLISIONS, ETC.
The above consideration indicates that we ought to divide by r previously
to integration, that is, evaluate
If we write
in place of (3).
It must be remarked, however, that an overvaluation of long paths
relatively to shorter ones which all correspond to the same velocity would
not of itself explain the 10percent. discrepancy; for, when the gas is
infinitely rare, all the paths must be considered to be infinitely long, and
then the proportion of relatively longer and shorter paths becomes a matter
of indifference. In fact (3) should give the correct result in the limit (/= 0),
even though it be of erroneous form as respects n, provided a suitable function
of f and v be introduced as a factor. If we integrate (3) as it stands with
respect to n between the limits  oo and + oo , we obtain
(6)
But this should certainly be independent of/. I think that if we introduce
the factor vf into (3), Mr Godfrey's analysis would then lead to the same
result as is obtained by neglecting the influence of collisions ab initio.
It may be convenient to recite the constitution and visibility of a
spectrum line according to the simple theory, where the Doppler effect is
alone regarded. If % be the velocity of a molecule in the line of sight, the
number of molecules whose velocities in this direction lie between and
% + d% is, by Maxwell's theory,
According to Doppler's principle the reciprocal wavelength of the light
received from these molecules is changed from A" 1 , corresponding to = 0,
to A 1 (l +/y)f V being the velocity of light. If a; denote the variation of
reciprocal wavelength, x = \ l j/V, and the distribution of light in the
dilated spectrum line may be taken to be
................................. (8)
1905] ON THE CONSTITUTION OF A SPECTRUM LINE 261
When this light forms interferencebands with relative retardation D, the
" visibility " accorded to Michelson's reckoning is expressed by
cos (27,Da;) dx  r
that is e * F * A2 (9)
If v be the velocity of mean square, on which the pressure of the gas
depends,
In terms of v the exponent in (8) is
..(11)
and that in (9) is  (12)
309.
ON THE MOMENTUM AND PRESSURE OF GASEOUS VIBRA
TIONS, AND ON THE CONNEXION WITH THE VIRIAL
THEOREM.
[Philosophical Magazine, Vol. x. pp. 364374, 1905.]
IN a paper on the Pressure of Vibrations (Phil. Mag. in. p. 338, 1902 ;
Scientific Papers, v. p. 41) I considered the case of a gas obeying Boyle's
law and vibrating within a cylinder in one dimension. It appeared that in
consequence of the vibrations a piston closing the cylinder is subject to an
additional pressure whose amount is measured by the volumedensity of the
total energy of vibration. More recently, in an interesting paper (Phil. Mag.
ix. p. 393, 1905) Prof. Poynting has treated certain aspects of the question,
especially the momentum associated with the propagation of progressive
waves. Thus prompted, I have returned to the consideration of the subject,
and have arrived at some more general results, which however do not
in all respects fulfil the anticipations of Prof. Poynting. I commence with
a calculation similar to that before given, but applicable to a gas in which
the pressure is any arbitrary function of the density.
By the general hydrodynamical equation (Theory of Sound, 253 a),
/**>
where p denotes the pressure, p the density, < the velocitypotential, and U
the resultant velocity at any point. If we integrate over a long period of
time, < disappears, and we see that
(2)
retains a constant value at all points of the cylinder. The value at the
piston is accordingly the same as the mean value taken over the length of
the cylinder.
1!)05] ON THE MOMENTUM AND PRESSURE OF GASEOUS VIBRATIONS 263
If p\, p\ denote the pressure and density at the piston, and p , p the
pressure and density that would prevail throughout were there no vibrations,
we have
P=f(p)=f(p + pp*)> ........................ (3)
and approximately
,  ,, ............. (4)
For the mean value of TB at the piston we have only to write p x for p in
(4) and integrate with respect to t. And at the piston /"=0.
For the mean of the whole length I of the cylinder (parallel to x), we have
to integrate with respect to x as well as with respect to t. And in the
integration with respect to # the first term of (4) disappears, inasmuch as the
mean density remains the same as if there were no vibrations. Accordingly
the terms on the right being of the second order in the quantities which
express the vibration.
Again, J( PI  Po ) dt = j {/( Pl )  / ( Po )} dt
so that by (5) j ( Pl p ) dt = f // ?*
+ (^W*^
The three integrals on the right in (6) are related in a way which we may
deduce from the theory of infinitely small vibrations. If the velocity of
propagation of such vibrations be denoted by a, then f'(p ) = a a . By the
usual theory we have
TT _d<f> PP D _ 1 d<p m
' " '
If we suppose that the cylinder is closed at x = and at x = I, a normal
vibration is expressed by
STTX sirat /ox
9 = cos T . cos j , ........................... (o)
264 ON THE MOMENTUM AND [309
where is any integer, giving
the integrations with respect to x in (9) being taken from to /, that is over
the length of the cylinder.
The same conclusions (9) follow in the general case where <f> is expressed
by a sum of terms derived from (8) by attributing an integral value to s.
The latter part expresses the equality of the mean potential and kinetic
energies.
Introducing the relations (9) into (6), so as to express the mean pressure
upon the piston in terms of the mean kinetic energy, we get as the final
formula
Among special cases let us first take that of Boyle's law, where p = a?p,
so that
/'0>o) = a 2 ,
We have at once
The expression on the right represents double the volumedensity of the
kinetic energy, or the volumedensity of the whole energy, and we recover
the result of the former investigation.
According to the adiabatic law
p/p = (plp)*; .............................. (12)
so that /'(*) = 77> .f"(P) = Poy(fY p r l) ................ (13)
Hence from (10)
* ................ (14)
The mean pressure upon the piston is now $ (7 + 1) of the volumedensity of
the total energy. We fall back on Boyle's law by taking 7 = 1.
It appears therefore that the result is altered when Boyle's law is departed
from. Still more striking is the alteration when we take the case treated in
Theory of Sound, 250 of the law of pressure
p = Const. a'pf/p ............................ (15)
According to this
.................. (16)
and (10) gives j( pl p )dt = ............................... (17)
1905] PRESSURE OF GASEOUS VIBRATIONS 265
The law of pressure (15) is that under which waves of finite condensation can
be propagated without change of type.
In (17) the mean additional pressure vanishes, and the question arises
whether it can be negative. It would appear so. If, for example,
p = Const ', ........................... (18)
and topta fa ................... (19)
I now pass on to the question of the momentum of a progressive train of
waves. This question is connected with that already considered; for, as
Prof. Poynting explains, if the reflexion of a train of waves exercises a
pressure upon the reflector, it can only be because the train of waves itself
involves momentum. From this argument we may infer already that
momentum is not a necessary accompaniment of a train of waves. If the
law were that of (15), no pressure would be exercised in reflexion. But it
may be convenient to give a direct calculation of the momentum.
For this purpose we must know the relation which obtains in a progressive
wave between the forward particle velocity u (not distinguished in one
dimensional motion from U) and the condensation (p po)/p , usually denoted
by s. When the disturbance is infinitely small, this relation is well known to
be u = as, in the case of a positive wave. Thus
<u:s=</(dp/dp) ............................... (20)
The following is the method adopted in Theory of Sound, 351 : ' ; If the
above solution be violated at any point a wave will emerge, travelling in the
negative direction. Let us now picture to ourselves the case of a positive
progressive wave in which the changes of velocity and density are very
gradual but become important by accumulation, and let us inquire what
conditions must be satisfied in order to prevent the formation of a negative
wave. It is clear that the answer to the question whether, or not, a negative
wave will be generated at any point will depend upon the state of things in
the immediate neighbourhood of the point, and not upon the state of things
at a distance from it, and will therefore be determined by the criterion
applicable to small disturbances. In applying this criterion we are to con
sider the velocities and condensations not absolutely, but relatively, to those
prevailing in the neighbouring parts of the medium, so that the form of (20)
proper for the present purpose is
266 ON THE MOMENTUM AND [309
which is the relation between u and p necessary for a positive progressive
wave. Equation (22) was obtained analytically by Earnshaw (Phil. Trans.
1859, p. 146).
In the case of Boyle's law, ^(dp/dp) is constant, and the relation between
velocity and density, given first, I believe, by Helmholtz, is
w = alog(p//3 ),
if p be the density corresponding to u = 0."
In our previous notation
dp/dp =f (p) = a 2 +/" ( Po ) . (p  po),
a being the velocity of infinitely small waves, equal to V{/'(po)} ; and
by (22>
* a ^ + S( 2* S)^^ ? (23)
the first term giving the usual approximate formula.
The momentum, reckoned per unit area of cross section,
f [f  \
= I pudx = po I ( 1 4  J udx.
Introducing the value of u from (23) and assuming that the mean density
is unaltered by the vibrations, we get
(24)
or, if we prefer it, g  Po ^ Po) + 1 1 [ u?dx (25)
The total energy of the length considered is p I u*dx ; and the result may
be thus stated
momentum =  ^^* + ^ I x total energy (26)
This may be compared with (10). If we suppose the long cylinder of length
I to be occupied by a train of progressive waves moving towards the piston,
the integrated pressure upon the piston during a time t, equal to I/a, should
be equal to twice the momentum of the whole initial motion. The two
formulae are thus in accordance, and it is unnecessary to discuss (26) at
length. It may suffice to call attention to Boyle's law, where /" (/> ) = 0, and
to the law of pressure (15) under which progressive waves have no momentum.
It would seem that pressure and momentum are here associated with the
tendency of waves to alter their form as they proceed on their course.
The above reasoning is perhaps as simple as could be expected ; but an
argument to be given later, relating to the kinetic theory of gases, led me to
1905] PRESSURE OF GASEOUS VIBRATIONS 267
recognize, what is indeed tolerably obvious when once remarked, that there
is here a close relation with the virial theorem of Clausius. If x,y,z be
the coordinates ; v x , v y , v z the component velocities of a material particle of
mass m, then
with two similar equations, X being the impressed force in the direction of x
operative upon m. If the motion be what is called stationary, and if we
understand the symbols to represent always the mean values with respect to
time, the last term disappears, and
(27)
The mean kinetic energy of the system relative to any direction is equal to
the virial relative to the same direction.
Let us apply (27) to our problem of the onedimensional motion of a gas
within a cylinder provided with closed ends. As in other applications of
the virial theorem, the forces X are divided into two groups, internal and
external. The latter reduces to the forces between the ends (pistons) and
the gas. If p 1 be the pressure on the pistons it will be the same on the
average at both ends the external virial is per unit of area %pj simply.
As regards the internal virial, I do not remember to have seen its value
stated, probably because in hydrodynamics the mechanical properties of a
fluid are not usually traced to forces acting between the particles. There
can be no doubt, however, what the value is. If we suppose that the whole
mass of gas in (27) is at rest, the lefthand member vanishes, so that the sum
of the internal and external virial must vanish. Under a uniform pressure
p , the internal virial is therefore ^p l. In an actual gas the virial for any
part can depend only on the local density, so that whether the gas be in
motion or not, the value of the internal virial is
*/>*
(28)
JO
Hence (27) gives
ri n
kinetic energy = ^pil^\ pdx = ^(p 1 p )l^ (pp )dx. (29)
Jo Jo
If the gas be subject to Boyle's law, pressure is proportional to density,
and the last term in (29) disappears. The additional pressure on the ends
(Pi ~P) is thus equal to twice the density of the kinetic energy.
In general, pp =a 2 (p />) + \f" (p ) . (p  p ),
and j\p p )dx = \f" (/>) j(p  p Y dx.
268 ON THE MOMENTUM AND [309
If we introduce expressly the integration with respect to t already implied,
(29) gives
regard being paid to (9). Equation (10) is thus derived very simply from
the virial.
In all that precedes, the motion of the gas has been in one dimension,
and even when we supposed the gas to be confined in a cylinder, we were
able to avoid the consideration of lateral pressures upon the walls of the
cylinder by applying the virial equation in its onedimensional form. We
now pass on to the case of three dimensions, and the first question which
arises is as to the value of the virial. In place of (27) we have now
VS,mU* = &(Xic+7y + Zz), .................... (30)
U being the resultant velocity, Y, Z impressed forces parallel to the axes of y
and z. Let us first apply this to a gas at rest under pressure p . The total
virial, represented by the righthand member of (30), is now zero ; that is,
the internal and external virial balance one another. As is well known and
as we may verify at once by considering the case of a rectangular chamber,
the external virial is $p v, v denoting the volume of gas. The internal virial
is accordingly $p v ; and from this we may infer that whether the pressure
be uniform or not, the internal virial is expressed by
ifjfpdvdyd* ............................ (31)
The difference between the internal virial of the gas in motion and in
equilibrium is
(31*)
According to the law of Boyle, (31*) must vanish, since the mean density
of the whole mass cannot be altered. The internal virial is therefore the
same whether the gas be at rest or in motion.
A question arises here as to whether a particular law of pressure may not
be fundamentally, inconsistent with the statical Boscovitchian theory of the
constitution of a gas upon which the application of the virial theorem is
based. If, indeed, we assume Boyle's law in its integrity, the inconsistency
does exist. For Maxwell has shown (Maxwell's Scientific Papers, vol. n. p. 422)
that on a statical theory Boyle's law involves between the molecules of a gas a
repulsion inversely as the distance. This makes the internal virial for any
pair of molecules independent of their mutual distance, and thus the virial
1905] PRESSURE OF GASEOUS VIBRATIONS 269
for the whole mass independent of the distribution of the parts. But such
an explanation of Boyle's law violates the principle upon which (31) was
deduced, making the pressure dependent upon the total quantity of the mass
and not merely upon the local density; from which Maxwell concluded that
all statical theories are to be rejected. It is to be remarked, however, that
our calculations involve the law of pressure only as far as the term involving
the square of the variation of density, and that a law agreeing with Boyle's
to this degree of approximation may perhaps not be inconsistent with a
statical Boscovitchian theory*.
Passing over this point, we find in general from (30)
(32)
whenever the character of the motion is such that the mean pressure (p^ is
the same at all points of the walls of the chamber. Further, as before,
ffj(P p }dxdydz=^f" ( Po ) fjj( P  ptfdxdydz,
and finally, regard being paid to (9) as extended to three dimensions,
( Pl  Po )v= (jJL + fcQ0L>) x total energy ............. (33)
In the case of Boyle's law/"= 0, and we see that the mean pressure upon
the walls of the chamber is measured by onethird of the volumedensity of
the total energy.
For the adiabatic law (12), (13) gives
x total energy. ............... (34)
In the case of certain gases called monatomic, 7= If, and (34) becomes
(Pi/V)v = f x total energy ...................... (35)
Thirdly, in the case of the law (15) for the relation between pressure and
density,
(Pipo)v = %x total energy, ..................... (36)
the mean pressure upon the walls being less than if there were no motion.
So far we have treated the question on the usual hydrodynamical basis,
reckoning the energy of compression or rarefaction as potential. It was,
* I think the difficulty may be turned by supposing the force, inversely as the distance, to
operate only between particles whose mutual distance is small, and that outside a certain small
distance the force is zero. All that is necessary is that a pair of particles once within the range
of the force should always remain within it a condition easily satisfied so long as small dis
turbances alone are considered.
270 ON THE MOMENTUM AND [300
however, on the lines of the kinetic theory that I first applied the virial
theorem to the question of the pressure of vibrations. In the form of this theory
which regards the collisions of molecules as instantaneous, there is practically
no potential, but only kinetic, energy. And if the gas be monatomic, the
whole of this energy is translational. If V be the resultant velocity of the
molecule whose mass is m, the virial equation gives
%p l v = &mV* t (37)
fr denoting, as before, the pressure upon the walls, assumed to be the same
over the whole area. If necessary, p l and 2mV* are to be averaged with
respect to time.
It is usually to a gas in equilibrium that (37) is applied, but this
restriction is not necessary. Whether there be vibrations or not, p l is equal
to  of the volumedensity of the whole energy of the molecules. Consider a
given chamber whose walls are perfectly reflecting, and let it be occupied by
a gas in equilibrium. The pressure is given by (37). Suppose now that
additional energy (which can only be kinetic) is communicated. We learn
from (37) that the additional pressure is measured by of the volumedensity
of the additional energy, whether this additional energy be in the form of
heat, equally or unequally distributed, or whether it take the form of
mechanical vibrations, i.e. of coordinated velocities and density differences.
Under the influence of heatconduction and viscosity the mechanical vibra
tions gradually die down, but the pressure undergoes no change.
The above is the case of the adiabatic law with 7 = 1 already considered
in (35), and a comparison of the two methods of treatment, in one of which
potential energy plays a large part, while in the other all the energy is
regarded as kinetic, suggests interesting reflexions as to what is really
involved in the distinction of the two kinds of energy.
If we abandon the restriction to monatomic molecules, the question
naturally becomes more complicated. We have first to consider in what
form the virial equation should be stated. In the case of a diatomic molecule
we have, in the first instance, not only the kinetic energy of the molecule as
a whole, but also the kinetic energy of rotation, and in addition the internal
virial of the force by which the union of the two atoms is maintained. It is
easy to see, however, that the two latter terms balance one another, so that
we are left with the kinetic energy of the molecule as a whole. For general
purposes a theorem is required for which I have not met a complete state
ment. For any part of a wider system for which we wish to form the virial
equation, we may omit the kinetic energy of the motion relative to the centre
of gravity of the part, if at the same time we omit the virial of the internal
forces operative in this part and treat the forces acting from outside upon the
part, whether from the remainder of the system or wholly from outside, a
1905] PRESSURE OF GASEOUS VIBRATIONS 271
acting at the centre of gravity of the part. In applying (37) to a gas
regarded as composed of molecules, we are therefore to include on the right
only the kinetic energy of translation of the molecules. If a gas originally at
rest be set into vibration, we have
I (Pi P) v = additional energy of translation (38)
The pressure p l does not now, as in the case of monatomic gases, remain
constant. Under the influence of viscosity and heatconduction, part of the
energy at first translational becomes converted into other forms.
A complete discussion here would carry us into the inner shrine of the
kinetic theory. We will only pursue the subject so far as to <x>nsider briefly
the case of rigid molecules for which the energy is still entirely kinetic
partly that of the translatory motion of the molecules as wholes and partly
rotatory. Of the additional energy E representing the vibrations, half may
be regarded as wholly translational. Of the other half, the fraction which is
translational is 3/ra, where ra is the whole number of modes. The transla
tional part of E is therefore ^Z7(l+3/ra); so that
(39)
If m = 3, as for monatomic molecules, we recover the former result;
otherwise p l p is less. In terms of 7 we have
7=l + 2/m, (40)
and accordingly (p 1 p )v = E(^ ^^J, (41)
in agreement with (34) where what was there called the total energy is now
regarded as the additional energy of vibration. In the case of a diatomic
gas, m = 5, 7 = If.
310.
THE ORIGIN OF THE PRISMATIC COLOURS.
[Philosophical Magazine, Vol. x. pp. 401407, 1905.]
THE fact that by the aid of a spectroscope interferences may be observed
with light originally white used to be regarded as a proof of the existence
of periodicities in the original radiation ; but it seems now to be generally
agreed that these periodicities are due to the spectroscope. When a pulse
strikes a grating, it is obvious that the periodicity and its variation in
different directions are the work of the grating. The assertion that Newton's
experiments prove the colours to be already existent in white light, is usually
made in too unqualified a form.
When a prism, which has no periodicities of figure, is substituted for a
grating, the modus operandi is much less obvious. This question has been
especially considered by Schuster (Phil. Mag. xxxvu. p. 509, 1894 ; vn.
p. 1, 1904), and quite recently Ames has given an " Elementary Discussion
of the Action of a Prism upon White Light " (Astrophysical Journal,
July 1905). The aim of the present note is merely to illustrate the matter
further.
I commence by remarking that, so far as I see, there is nothing faulty
or specially obscure in the traditional treatment founded upon the con
sideration of simple, and accordingly infinite, trains of waves. By Fourier's
theorem any arbitrary disturbance may be thus compounded; and the
method suffices to answer any question that may be raised, so long at least
as we are content to take for granted the character of the dispersive
medium the relation of velocity to wavelength without enquiring further
as to its constitution. For example, we find the resolvingpower of a prism
to be given by
\ T dp
d\~ T d\> (1)
in which X denotes the wavelength in vacuo, T the "thickness" of the
prism, p. the refractive index, and d\ the smallest difference of wavelength
1905] THE ORIGIN OF THE PRISMATIC COLOURS 273
that can be resolved. A comparison with the corresponding formula for a
grating shows that (1) gives the number of waves (X) which travel in the
prescribed direction as the result of the action of the prism upon an in
cident pulse.
But, although reasoning on the above lines may be quite conclusive,
a desire is naturally felt for a better understanding of the genesis of the
sequence of waves, which seems often to be regarded as paradoxical.
Probably I have been less sensible of this difficulty from my familiarity
with the analogous phenomena described by Scott Russel and Kelvin, of
which I have given a calculation*. " When a small obstacle, such as
fishingline, 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 wavepattern, fixed relatively to the obstacle.
On the upstream side the wavelength 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 wavelength, of those parts of the wavepattern where the
fronts are oblique to the direction of motion. If the angle between this
direction and the normal to the wavefront be called 6, the velocity of
propagation of the waves must be equal to v cos 0, where v represents the
velocity of the water relatively to the (fixed) obstacle." In the laboratory
the experiment may be made upon water contained in a large spongebath
and mounted upon a revolving turntable. The fishingline is represented
by the impact of a small jet of wind. In this phenomenon the action of
a prism is somewhat closely imitated. Not only are there sequences of
waves, unrepresented (as would appear) either in the structure of the medium
or in the character of the force, but the wavelength and velocity are variable
according to the direction considered.
For the purposes of Scott Russel's phenomenon the localized pressure is
regarded as permanent ; but here it will be more instructive if we suppose
it applied for a finite time only. Although the method is general, we may
fix our ideas upon deep water, subject to gravity (cohesion neglected), upon
which operates a pressure localized in a line and moving transversely with
velocity V. In the general twodimensional problem thus presented, the
effect of the travelling pressure is insignificant unless V is a possible
wave velocity ; but where this condition is satisfied, a corresponding train
of waves is generated. In the case of deep water under gravity the
* " The Form of Standing Waves on the Surface of Running Water," Proc. Lond. Math. Soc.
Vol. xv. p. 69 (1883) ; Scientific Papers, Vol. n. p. 258.
B. V. 18
274 THE ORIGIN OF THE PRISMATIC COLOURS [310
condition is always satisfied, for the wavevelocities vary from zero to in
finity.
The limitation to a wavetrain of velocity V is complete only when the
time of application of the pressure is infinitely extended. Otherwise,
besides the train of velocity V we have to deal with other trains, of velo
cities differing so little from V that during the time in question they
remain sensibly in step with the first. As is known*, the behaviour of
such aggregates is largely a matter of the groupvelocity U, whose value is
given by
oo
k being proportional to the reciprocal of the wavelength in the medium. In
the particular case of deepwater waves U= %V.
From this point of view it is easy to recognize that the total length of
the train of waves generated in time t' is + ( V U) t'. If T be the periodic
time of these waves, the wavelength in the medium is FT, and the number
of waves is therefore
But for our present purpose of establishing an analogy with prisms and their
resolvingpower, what we are concerned with is not the number of waves
at any time in the dispersive medium itself, but rather the number after
emergence of the train into a medium which is nondispersive ; and here
a curious modification ensues. During the emergence the relative motion
of the waves and of the group still continues, and thus we have to introduce
the factor V/U, obtaining for the number N of waves outside
If X be the distance through which the pressure travels, X = Vt' ; and
if F be the (constant) velocity outside and X the wavelength outside,
X = F O T. Thus
AT
To introduce optical notation, let fi= F /F, so that p, is the refractive
index. In terms of /u
Kxf (6)
u a\
so that finally
* See, for example, Nature, Vol. xxv. p. 51 (1881) ; Scientific Papers, Vol. i. p. 540.
1905] THE ORIGIN OF THE PRISMATIC COLOURS 275
in close correspondence with (1). A very simple formula thus expresses the
number of waves (after emergence) generated by the travel of the pressure
over a distance X of a dispersive medium.
The above calculation has the advantage of being clear of the complication
due to obliquity; but a very little modification will adapt it to the case of
a prism, especially if we suppose that the waves considered are emergent
at the second face of the prism without refraction.
In the figure, A G represents an incident plane pulse
whose trace runs along the first face of the prism
from A to B. AF, BE is the direction of propa
gation of the refracted waves under consideration, to
which the second face of the prism is perpendicular.
As before, if r be the period, V the wave velocity of
the waves propagated in direction BE, U the corre F
spending groupvelocity, if the time of travel of the pulse from A to B, the
number of waves within the medium is
F U t'
~^F~ r'
giving on emergence the number of waves expressed in (4). If F be the
velocity in vacuum, T = X/F , and
BC_AD
~ V ~ V '
so that
t' _AD FQ
r~ \ V
Thus, as in (5), (6), (7),
F V \AD dp
in agreement with (1).
Although the process is less easy to follow, the construction of a train of
waves from an incident pulse is as definite in the case of a prism as is that
of a grating; and its essential features are presented to the eye in Scott
Russel's phenomenon.
The above treatment suffices for a general view, but it may be instructive
to give an analytical statement ; and this I am the more inclined to do as
affording an opportunity of calling attention to a rather neglected paper by
Lord Kelvin entitled "On the Waves produced by a Single Impulse in
Water of any Depth, or in a Dispersive Medium"*. When we know the
* Proc. Boy. Soc. Vol. XLII. p. 80 (1887).
182
276 THE ORIGIN OF THE PRISMATIC COLOURS [310
effect of an impulse, that of a uniform force applied for a finite time can be
deduced by integration. It may be convenient to recite the leading steps
of Kelvin's investigation.
Let f(k) denote the velocity of propagation corresponding to wavelength
(in the medium) 27T/&. The FourierCauchyPoisson synthesis gives
u=t"dkcosk[xtf(k)] (9)
for the effect at place and time (x, t) of an infinitely intense disturbance
at place and time (0, 0). When x tf(k) is very large, the parts of the
integral (9) which lie on the two sides of a small range, AC a to K+OL,
vanish by annulling interference ; K being a value, or the value, of k,
which makes
(10)
or x =*{/() + /'(*)} =Ut, ........................ (11)
U being the group velocity. By Taylor's theorem when k K is very
small,
k [x  tf(k)} = *tf(*) + & (k  )* {  /"(*)  2/V) } .
Using this in (9) and integrating with the aid of
/+ /+>
d<r cos o 2 = I do sin a 2
00 J 00
we find as an approximate value
V(2,r). cos {*</>) + JTT}
V*.V{ */"(*) 2 /'(*)}'
As a particular case, for deepwater gravity waves
/(*)  vW*), /(*) =  to**" 1 ,  kf"(k)  2/'(&) = tf AT*,
and finally with use of (11)
...as)*
This gives the effect of the impulse at (0, 0). If the impulse be at of, t',
we are to write x a/ for x and t t' for t. For our purpose of finding the
effect of a travelling force, we are to make x' = Vt' and integrate with respect
to If from to t', t' being the duration of the force. The integral will depend
mainly upon the part where
(ttj
xVt"
An almost equally simple formula applies when more generally /(fc) oc A".
1905] THE ORIGIN OF THE PRISMATIC COLOURS 277
under the cosine, is stationary. This occurs when
(14)
and then
9(ttJ g(Vtx) ,
4(*FrT ~~F^~ ......................
Omitting the variation of the other factors as less important, we see that,
when sensible, the effect is proportional to
representing simple waves of velocity V. But this is limited to such values
of x and t as make t' in (14) lie between and if. Thus if t be given, the
range of x is from ^Vt to ^Vt + ^Vt'; so that the train of waves covers
a length \Vt ', agreeing with the general value given before, since here
U=$V. If, as would be more convenient in order to find the length of the
train after emergence into a nondispersive medium, we regard x as given,
we find that t ranges from 2a;/F to 2#/FM'.
I have taken the particular case first, as the reasoning is rather simpler
when we have, as in (13), an explicit expression in terms of x and t*. In
general K cannot be eliminated between (11) and (12), and we must proceed
rather differently. The question is when will
k[xtf(k)] .............................. (17)
with
x=t\f(k} + kf'(k)} ....................... (18)
be stationary with respect to t', x  Vt' being substituted for x and t t' for t
in (17), (18) ? Now
of which the second term on the right vanishes by (18). The variation of
(17) vanishes when F =/(&). Accordingly
k[xVt'(tt')f(k)\
with
is stationary with respect to t ', if F =/(&), and then assumes the form
k[xVt].
Here t' must lie between and t'. Thus if t be constant, x has
a range
* For an admirable discussion of the general problem of deepwater wavea arising from a
localized disturbance, see Lamb, Proc. Lond. Math. Soc. Vol. n. p. 371 (1904).
278 THE ORIGIN OF THE PRISMATIC COLOURS [310
And if a; be given, t has a range
 (19)
These are the limits over which the waves of velocity V extend. And
(19) shows that the number of waves which pass a fixed point, either within
the dispersive medium or on emergence from it, has the expression
H_ U V
r
where r is the periodic time, in agreement with (4).
311.
ON THE CONSTITUTION OF NATURAL RADIATION.
[Philosophical Magazine, Vol. xi. pp. 123 127, 1906.]
THE expression of Prof. Larmor's views in his paper* "On the Con
stitution of Natural Radiation" is very welcome. Although it may be true
that there has been no direct contradiction, public and private communications
have given me an uneasy feeling that our views are not wholly in harmony;
nor is this impression even now removed. It may conduce to a better
understanding of some of these important and difficult questions if without
dogmatism I endeavour to define more clearly the position which I am
disposed to favour on one or two of the matters concerned.
On p. 580, in comparing white light and Rontgen radiation, Prof. Larmor
writes : " Both kinds of disturbance are resolvable by Fourier's principle
into trains of simple waves. But if we consider the constituent train having
wavelength variable between A, and \ + 8\, i.e. varying irregularly from part
to part of the train within these limits, a difference exists between the two
cases. In the case of the white light the vibrationcurve of this approxi
mately simple train is in appearance steady; it is a curve of practically
constant amplitude, but of wavelength slightly erratic within the limits 8\
and therefore of phase at each point entirely erratic. In the Fourier analysis
of the Rontgen radiation the amplitude is not regular, but on the contrary
may be as erratic as the phase." This raises the question as to the general
character of the resultant of a large number of simple trains of approximately
equal wavelength. In what manner will the resultant amplitude and phase
vary? In several papers f I have considered particular cases of approximately
simple waves, showing how they may be resolved into absolutely simple trains
of approximately equal wavelengths. But now the question presents itself in
* Phil. Mag. Vol. x. p. 574 (1905).
t See especially Phil. Mag. Vol. L. p. 135 (1900). [Scientific Papers, Vol. iv. p. 486.]
280 ON THE CONSTITUTION OF NATURAL RADIATION [311
the converse form. What are we to expect from the composition of simple
trains, severally represented by
ch cos {(71 I &ij) t + e,}, (1 )
where Sn^ is small, while the amplitude a, and the initial phase e, vary from
one train to another ?
In virtue of the smallness of &jj we may appropriately regard (1) as
a vibration of speed n and of phase e t + SnJ, variable therefore with the time.
The amplitude and phase may be represented in the usual way by the polar
coordinates of a point; and the point representing (1) accordingly lies on
the circle of radius j and revolves uniformly with small angular velocity.
For the present at any rate I suppose that the amplitudes a,, a,, &c. are all
equal (1), in which case the points lie all upon the same circle. The radius
from the centre to any of the points P upon the circumference is a vector
fully representative of the vibration, and the resultant of the vectors repre
sents the resultant of the vibrations.
After the lapse of a time t the points have moved from their initial
positions P to other positions Q, and the aggregate of the vectors OP is
replaced by the aggregate of OQ. The difference is the aggregate of PQ.
Now we suppose that t is so related to the greatest &n that all the arcs PQ
are small fractions of the quadrant, and the question before us is the amount
of the difference between the resultants of the OP's and the OQ's, i.e. of the
PQ's. There are certain cases where we can say at once that the difference
of resultants is small, small that is relatively to the whole. This happens
when all the P's are rather close together, i.e. when the component vibrations
have initially nearly the same phase. It is then certain that at the end
of the time t the amplitude and phase are but little altered from what they
were at the beginning. Over this range the vibration is approximately
simple, and the range is inversely as the greatest departure from the mean
frequency n.
But in general the distribution of initial phases e causes the resultant to
be much less than if the phases were in agreement, and it may even happen
that the initial resultant is zero. At the end of the time t the resultant
will probably not be zero, so that in this case the change is relatively large.
The proposition that small changes in the phases of the components can
lead only to relatively small changes in the resultant is thus not universally
true; and we must inquire further as to the conditions under which the
conclusion is probable.
The most important case for our purpose is when the initial phases are
distributed at random, as they would presumably be when Rontgen radiation
1906] ON THE CONSTITUTION OF NATURAL RADIATION 281
is concerned. If the components are very numerous (and of equal amplitude
unity), the problem is one which I have considered on former occasions*.
It appears that the probability of a resultant amplitude lying between
r and r + dr is
e" m rdr, (2)
m
where m is the number of components. Or the probability of an amplitude
exceeding r is e^ 2 ^. The mean intensity (when the phases are redistributed
at random a great many times) is m, corresponding to the amplitude *Jm.
When r is great compared with \Jm, the probability of an amplitude
exceeding r becomes vanishingly small. When on the other hand r is small,
the probability of a resultant less than r is approximately r^/m. It appears
that the chance of the resultant lying outside the range from say *Jm to
2\/w is comparatively small.
We have next to consider the resultant of the components PQ. Here
again the phases are distributed in all directions. The amplitudes, however,
are no longer equal, but they are small relatively to unity. Although the
contrary is not impossible, it would seem that in all probability the resultant
amplitude of the PQ's is small in comparison with that of the OP's, from
which it follows that, exceptional cases apart, the amplitude and phase of the
resultant remain but little changed at the end of a time t, such that the
changes of phase of the individual components are small.
From the above discussion I am disposed to infer that a Fourier element
of radiation necessarily possesses in large degree the characteristic which
(if I rightly understand him) Prof. Larmor associates with white light in
contrast to Rontgen radiation. Of course, after the lapse of a sufficient
time the final phases of the components lose all simple relation to the initial
phases. The final phase of the resultant is then without relation to the
initial phase, and the amplitudes may differ finitely, but in all probability
within somewhat restricted limits. From this variation it seems to me white
light cannot be exempt.
In the above and, so far as I remember, in what I have written previously,
the question is purely kinematical. In saying that Fourier's theorem is
competent to answer any question that may be raised respecting the action of
a dispersive medium, I take for granted that the law of dispersion is given
in its entirety. I quite admit that if there are any wavelengths for which
the behaviour of the medium is unknown, a corresponding uncertainty must
attach to the fate of any aggregate in which these are included. Doubtless
* Phil. Mag. Vol. x. p. 73 (1880) ; Scientific Papers, Vol. i. p. 491 ; Theory of Sound, 2nd ed.
Vol. i. 42 a.
282 ON THE CONSTITUTION OF NATURAL RADIATION [311
a complete statement of the law of dispersion may involve the case of wave
lengths for which the medium is not transparent.
As regards the passage quoted from Sir G. Stokes, his object was, I think,
to explain the absence of refraction when Rontgen rays traverse matter.
Taking light of ordinary and absolutely definite wavelength incident upon
transparent matter, he contemplates the lapse of 10,000 periods before
harmony is established between the aetherial and molecular vibrations, that
is, as I understand it, before regular refraction is possible. At this rate the
light from a soda flame would be incapable of regular refraction, for the
vibrations are certainly not regular for more than 500 periods. Indeed
Stokes's argument appears better adapted to prove that Rontgen rays could
not traverse material media at all in a regular manner, than that they would
do so without change of wavevelocity.
I must confess that I have never fully understood Stokes's position in
this matter. A medium is nonrefractive and nearly transparent for the
pulses constituting Rontgen rays*. What reception would it give to simple
waves of half wavelength equal to the thickness of the pulses? I should
suppose that it would be nonrefractive and transparent for these also, but
Stokes's argument seems to imply the contrary. The paradox would then
have to be met that the medium treats simple waves less simply than
compound ones.
* [1911. The question whether Rontgen radiation is really of this character at all seems still
to be an open one.]
312.
ON AN INSTRUMENT FOR COMPOUNDING VIBRATIONS, WITH
APPLICATION TO THE DRAWING OF CURVES SUCH AS
MIGHT REPRESENT WHITE LIGHT.
[Philosophical Magazine, Vol. xi. pp. 127130, 1906.]
IN discussions respecting the character of the curve by which the
vibrations of white light may be expressed, I have often felt the want of
some ready, even if rough, method of compounding several prescribed simple
harmonic motions. Any number of points on the resultant can of course
always be calculated and laid down as ordinates ; but the labour involved in
this process is considerable. The arrangement about to be described was
exhibited early in the year during lectures at the Royal Institution. As it is
inexpensive to construct and easily visible to an audience, I have thought
that such a description might be useful, accompanied with a few specimens of
curves actually drawn with its aid.
A wooden batten, say 1 inch square and 5 feet long, is so mounted hori
zontally as to be capable of movement only along its length. For this
purpose it suffices to connect two points near the two ends, each by means of
two thin metallic wires, with four points symmetrically situated in the roof
overhead. This mounting, involving four constraints only, allows also of a
rotatory or rolling motion, which could be excluded, if necessary, by means of
a fifth wire attached to a lateral arm. In practice, however, this provision
was not used or needed. The movement of the batten along its length is
controlled by a piece of springsteel against which the pointed extremity
of the batten is held by rubber bands. Any force acting in the direction of
the length of the batten produces a displacement proportional to the force*.
The tracing point, by which the movements are recorded, is at the other end,
* In strictness this presupposes the fulfilment of a condition involving the period of the force
and that of free vibration under the influence of the spring, which it is scarcely necessary to
enter upon.
284 ON AN INSTRUMENT FOR COMPOUNDING VIBRATIONS [312
as nearly as possible in the line joining the two points of attachment of the
four suspending wires.
The longitudinal forces are due to the vibrations of pendulums hanging
from horizontal crosspieces attached to the batten at their centres. The
two ends of a wire or cord are attached to the extremities of a crosspiece,
the bob of the pendulum being a mass of lead (perhaps half a pound) carried
at the middle of the cord. When set swinging the movements of the
pendulums are thus parallel to the batten and tend to displace it along its
length. In my apparatus the length of the longest pendulum is 3 feet.
Under the influence of one pendulum the tracing point describes a small
simple harmonic motion along the length of the batten. In order to draw
a curve of sines the smoked glass destined to receive the record should move
vertically in its own plane. I found it more convenient and sufficient for
my purpose to substitute a movement of rotation. A disk (like the faceplate
of a lathe) revolves freely in a vertical plane round a horizontal axis. To
this disk a piece of smoked glass is cemented and the tracing is taken
near the circumference, the axis of rotation being at the same level as the
tracing point, so that the movement of vibration is radial.
The disk must be made to revolve slowly and with uniform angular
velocity. To effect this I employed a sandclock, a device which works better
than would be expected *. The sand, carefully sifted and dried, is contained
in a vertical metal tube of about 1 inch diameter, and escapes below through
a small aperture of size to be determined by trial. On the sand rests a
weight, of such diameter as to fit the tube easily; and this in its descent
rotates the disk by means of a thread, of which the free part is vertical
while the remainder engages a circumferential groove. The descent of the
weight is practically independent of the quantity of sand remaining at any
timef. It is scarcely necessary to say that the revolving parts must be so
weighted as to keep the thread tight.
The advantages of the apparatus depend of course upon the facility with
which a number of vibratory movements can be combined J. It is as easy to
record the effect of a number of pendulums as of a single one, the contri
bution in each case being proportional to the amplitude of vibration. In my
instrument there are six pendulums, the shortest of such length as to vibrate
about twice as quickly as the longest. The frequencies are in fact somewhat
as the numbers 5, 6, 7, 8, 9, 10. No precise adjustment was attempted, the
object being in fact rather to avoid anything specially simple.
The lengths of the pendulums were chosen so as to afford an illustration
of the vibrations constituting white light. Of course a complete physical
* It was used by H. Draper to drive an equatorially mounted telescope.
t See Note at end of paper.
t The principle of mechanical addition is employed in an instrument devised by Michelson.
1906] ON AN INSTRUMENT FOR COMPOUNDING VIBRATIONS 285
representation of light from the sun or from the electric arc would need
a much larger range of frequency. But we may suppose this light filtered
through media capable of sensibly absorbing the ultrared and ultraviolet,
while still remaining white so far as the eye could tell, even with the aid
of a prism. The range of an octave, for which provision is made, then amply
suffices.
The number of pendulums may seem, and perhaps is, rather small. The
frequency, e.g. 7, given by one of the pendulums must be taken to represent
a range from 6^ to 7, with an error therefore up to 1 in 14. Such an error
will be serious after 7 vibrations, but not so for 3 or 4 vibrations. Hence if
we limit ourselves to sequences of 3 or 4 waves, the representation is about
good enough.
Connected with the above is the question what amplitudes of vibration
are to be assigned to the various pendulums. It would not be difficult to
give effect to an assigned law of spectrum intensity whether suggested by
theory or found in observation. It is to be remembered, however, that such
laws relate to averages, and do not give the relative amplitudes at any
particular time, which will indeed vary fortuitously over a rather large range.
I thought it therefore unnecessary to be very particular in this respect.
The vibrations of the shorter pendulums die down more rapidly than the
slower ones. By giving the former an advantage at starting a somewhat
wide range is covered.
The tracings presented no general features that might not have been
anticipated. A few specimens are reproduced one showing the operation of
the longest and shortest pendulums alone, the others the effect of all the
pendulums.
Note on the Principle of the SandClock.
The difficulty of propelling a column of sand, occupying a tube, by forces
pushing at one end is well known; but I do not remember to have seen any
discussion of the question on mechanical principles. A similar phenomenon
occurs in the storage of grain, the weight of which, when contained in tall
bins, is found to be taken mainly on the sides and but little on the bottom of
the bin*.
The unexpectedness of these effects depends upon a half unconscious
comparison with fluids which in a state of rest are exempt from friction. In
the present case, when the sand is moving, the tangential force at the wall is
reckoned at p times the normal force. We may suppose, as a rough approxi
mation, that there is something like a fluid pressure p. If a be the radius of
* I. Roberts, Proc. Roy. Soc. Vol. xxxvi. p. 226, 1884. "In any cell which has parallel
sides, the pressure of wheat upon the bottom ceases when it is charged up to twice the diameter
of the inscribed circle."
286
ON AN INSTRUMENT FOR COMPOUNDING VIBRATIONS
[312
the tube and dx an element of length along the axis, the tangential force
acting upon the surface Ziradx is pp. 'lira. dx. This is to be equated
to the difference of the forces upon the two faces of the slice, viz. rra^dp.
Accordingly
p
or
the pressure diminishing as a increases. Hence a powerful pressure at x =
is unable to overcome a very feeble one acting in the opposite direction at a
section many diameters away. The case is similar to that of a rope coiled
round a post, as used to check the motion of steamers coming up to a pier.
As regards numbers, it will not be out of the way to suppose p=^.
When x = 10a, p/p = e~* = 14.
Fig. 1.
Fig. 2.
313.
ON ELECTRICAL VIBRATIONS AND THE CONSTITUTION
OF THE ATOM.
[Philosophical Magazine, Vol. xi. pp. 117123, 1906.]
IN illustration of the view, suggested by Lord Kelvin, that an atom may
be represented by a number of negative electrons, or negatively charged
corpuscles, enclosed in a sphere of uniform positive electrification, Prof.
J. J. Thomson has given some valuable calculations* of the stability of
a ring of such electrons, uniformly spaced, and either at rest or revolving
about a central axis. The corpuscles are supposed to repel one another
according to the law of inverse square of distance and to be endowed with
inertia, which may, however, be the inertia of aBther in the immediate
neighbourhood of each corpuscle. The effect of the sphere of positive
electrification is merely to produce a field of force directly as the distance
from the centre of the sphere. The artificiality of this hypothesis is partly
justified by the necessity, in order to meet the facts, of introducing from the
beginning some essential difference, other than of mere sign, between positive
and negative.
Some of the most interesting of Prof. Thomson's results depend essentially
upon the finiteness of the number of electrons ; but since the experimental
evidence requires that in any case the number should be very large, I have
thought it worth while to consider what becomes of the theory when the
number is infinite. The cloud of electrons may then be assimilated to a fluid
whose properties, however, must differ in many respects from those with
which we are most familiar. We suppose that the whole quantities of
positive and negative are equal. The difference between them is that the
positive is constrained to remain undisplaced, while the negative is free to
move. In equilibrium the negative distributes itself with uniformity
throughout the sphere occupied by the positive, so that the total density is
everywhere zero. There is then no force at any point ; but if the negative
* Phil. Mag. Vol. vn. p. 237 (1904).
288 ON ELECTRICAL VIBRATIONS AND [313
be displaced, a force is usually called into existence. We may denote the
density of the negative at any time and place by p, that of the positive and
of the negative, when in equilibrium, being p . The repulsion between two
elements of negative pdV, p'dV at distance r is denoted by
ry.r^.pdV.p'dV ............................... (1)
The negative fluid is supposed to move without circulation, so that
a velocitypotential (<) exists ; and the first question which presents itself, is
as to whether there is " condensation." If this be denoted by s, the equation
of continuity is, as usual *,
.......................... ........ (2)
Again, since there is no outstanding pressure to be taken into account,
the dynamical equation assumes the form
where R is the potential of the attractive and repulsive forces. Eliminating
</>, we get
~ .................................. <*>
In equilibrium R is zero, and the actual value depends upon the dis
placements, which are supposed to be small. By Poisson's formula
VlR=47T7/> s, ...................... ........... (5)
so that
2
............................ (6)
This applies to the interior of the sphere ; and it appears that any
departure from a uniform distribution brings into play forces giving stability,
and further that the times of oscillation are the same whatever be the
character of the disturbance. It is worthy of note that the constant (ypo) of
itself determines a time.
In considering the significance of the vibrations expressed by (6), we
must remember that when s is uniform no external forces having a potential
are capable of disturbing the uniformity.
We now pass on to vibrations not involving a variable s, that is of such
a kind that the fluid behaves as if incompressible. An irrotational dis
placement now requires that some of the negative fluid should traverse the
surface of the positive sphere (a). In the interior VlR = 0.
To represent simple vibrations we suppose that <f>, &c. are proportional to
e* 1 . By (3) V*</> = ; and we take (at any rate for trial)
4> = e*'rS n , .............................. (7)
* Theory of Sound, 244.
1906] THE CONSTITUTION OF THE ATOM 289
where S n is a spherical surface harmonic of the nth order. The velocity
across the surface of the sphere at r = a is
and thus the quantity of fluid which has passed the element of area dor at
time t is
.................. (8)
The next step is to form the expression for R, the potential of all the
forces. In equilibrium the positive and negative densities everywhere
neutralize one another, and thus in the displaced condition R may be
regarded as due to the surface distribution (8). By a wellknown theorem
in Attractions we have
_ .n
But by (3) this is equal to d<f>/dt, or tpe^r n n . The recovery of r n S n
proves that the form assumed is correct ; and we find further that
This formula for the frequencies of vibration gives rise to two remarks.
The frequency depends upon the density p , but not upon the radius (a) of
the sphere. Again, as n increases, the pitch rises indeed, but approaches
a finite limit given by p* = S^yp^. The approach to a finite limit as we
advance along the series is characteristic of the series of spectrumlines found
for hydrogen and the alkali metals, but in other respects the analogy fails.
It is p*, rather than p, which is simply expressed; and if we ignore this
consideration and take the square root, supposing n large, we find
p oc 1  l/2n,
whereas according to observation n 2 should replace n. Further, it is to be
remarked that we have found only one series of frequencies. The different
kinds of harmonics which are all of one order n do not give rise to different
frequencies. Probably the simplicity of this result would be departed from
if the number of electrons was treated merely as great but not infinite."
The principles which have led us to (10) seem to have affinity rather
with the older views as to the behaviour of electricity upon a conductor than
with those which we associate with the name of Maxwell. It is true that
the vibrations above considered would be subject to dissipation in consequence
of radiation, and that this dissipation would be very rapid, at any rate in the
case of n equal to unity*. But this hardly explains the difference between
the two views.
* In this case we should have to consider how the positive sphere is to be held at rest.
R. V. 19
290 ON ELECTRICAL VIBRATIONS AND [313
[191 1. Some paragraphs dealing with the question of electrical vibrations
outside a conducting sphere (J. J. Thomson, Proc. Lond. Math. Soc. Vol. xv.
p. 197, 1884; Recent Researches, 312, 1893), or of sonorous vibrations
outside a rigid and fixed sphere, are omitted as involving a misconception.
The matter had already been satisfactorily treated by Lamb (Proc. Lond.
Math. Soc. Vol. xxxn. p. 208, 1900) and by Love (Ibid. Vol. n. p. 88, 1904).]
In the calculation of frequencies given above for a cloud of electrons the
undisturbed condition is one of equilibrium, and the frequencies of radiation
are those of vibration about this condition of equilibrium. Almost every
theory of this kind is open to the objection that I put forward some years
ago*, viz. that p 2 , and not p, is given in the first instance. It is difficult to
explain on this basis the simple expressions found for p, and the constant
differences manifested in the formulae of Rydberg and of Kayser and Runge.
There are, of course, particular cases where the square root can be taken
without complication, and Ritzf has derived a differential equation leading
to a formula of this description and capable of being identified with that of
Rydberg. Apart from the question whether it corresponds with anything
mechanically possible, this theory has too artificial an appearance to inspire
much confidence.
A partial escape from these difficulties might be found in regarding
actual spectrum lines as due to difference tones arising from primaries of
much higher pitch, a suggestion already put forward in a somewhat different
form by Julius.
In recent years theories of atomic structure have found favour in which
the electrons are regarded as describing orbits, probably with great rapidity.
If the electrons are sufficiently numerous, there may be an approach to
steady motion. In case of disturbance, oscillations about this steady motion
may ensue, and these oscillations are regarded as the origin of luminous
waves of the same frequency. But in view of the discrete character of
electrons such a motion can never be fully steady, and the system must tend
to radiate even when undisturbed \. In particular cases, such as some
considered by Prof. Thomson, the radiation in the undisturbed state may be
very feeble. After disturbance oscillations about the normal motion will
ensue, but it does not follow that the frequencies of these oscillations will be
manifested in the spectrum of the radiation. The spectrum may rather be
due to the upsetting of the balance by which before disturbance radiation
was prevented, and the frequencies will correspond (with modification) rather
to the original distribution of electrons than to the oscillations. For example,
if four equally spaced electrons revolve in a ring, the radiation is feeble and
* Phil. Mag. xuv. p. 362, 1897 ; Scientific Papert, iv. p. 845.
t Drude, Ann. Bd. xn. p. 264, 1903.
J Confer Larmor, Matter and Mther.
1906] THE CONSTITUTION OF THE ATOM 291
its frequency is four times that of revolution. If the disposition of equal
spacing be disturbed, there must be a tendency to recovery and to oscillations
about this disposition. These oscillations may be extremely slow; but
nevertheless frequencies will enter into the radiation once, twice, and thrice
as great as that of revolution, and with intensities which may be much
greater than the original radiation of fourfold frequency.
An apparently formidable difficulty, emphasised by Jeans, stands in the
way of all theories of this character. How can the atom have the definiteness
which the spectroscope demands ? It would seem that variations must exist
in (say) hydrogen atoms which would be fatal to the sharpness of the observed
radiation ; and indeed the gradual change of an atom is directly contemplated
in view of the phenomena of radioactivity. It seems an absolute necessity
that the large majority of hydrogen atoms should be alike in a very high
degree. Either the number undergoing change must be very small or else
the changes must be sudden, so that at any time only a few deviate from one
or more definite conditions.
It is possible, however, that the conditions of stability or of exemption
from radiation may after all really demand this definiteness, notwithstanding
that in the comparatively simple cases treated by Thomson the angular
velocity is open to variation. According to this view the frequencies observed
in the spectrum may not be frequencies of disturbance or of oscillations in
the ordinary sense at all, but rather form an essential part of the original
constitution of the atom as determined by conditions of stability.
192
314.
ON THE PRODUCTION OF VIBRATIONS BY FORCES OF RELA
TIVELY LONG DURATION, WITH APPLICATION TO THE
THEORY OF COLLISIONS.
[Philosophical Magazine, Vol. XL pp. 283291, 1906.]
THE problem of the collision of elastic solid bodies has been treated
theoretically in two distinct cases. The first is that of the longitudinal
impact of elongated bars, which for simplicity may be supposed to be of the
same material and thickness. Saint Venant* showed that, except when the
lengths are equal, a considerable fraction of the original energy takes the
form of vibrations in the longer bar, so that the translational velocities after
impact are less than those calculated by Newton for bodies which he called
perfectly elastic. It will be understood that in Saint Venant's theory the
material is regarded as perfectly elastic, and that the total mechanical energy
is conserved. The duration of the impact is equal to the period of the slowest
vibration of the longer bar.
The experiments of Voigtf, undertaken to test this theory, have led to
the conclusion that it is inapplicable when the bars differ markedly in length.
The observations agree much more nearly with the Newtonian law, in which
all the energy remains translational. Further, Hamburger}: found that the
duration of impact was much greater than according to theory, though
it diminished somewhat as the relative velocity increased. I do not think
that these discrepancies need cause surprise when we bear in mind that the
theory presupposes a condition of affairs impossible to realise in practice.
Thus it is assumed that the pressure during collision is uniform over the
whole of the contiguous faces. But, however accurately the faces may be
prepared, the pressure, at any rate in its earlier and later stages, must
certainly be local and be connected with the approach by a law altogether
* Liouville's Journal, xn. (1867). See also Love's Treatise on the Theory of Elasticity,
Vol. 11. p. 137 (1893).
t Wied. Ann. xn. (1883).
J Wied. Ann. xxvm. (1886).
1906] ON THE PRODUCTION OF VIBRATIONS 293
different from that assumed in the calculation. Since the region of first
contact would yield with relative ease, we may expect a prolongation of
the impact, and in consequence, as we shall see more in detail presently,
a diminished development of vibrations. Possibly with higher velocities
and longer bars a nearer approach might be attained to the theoretical
conditions.
But it is with Hertz's* solution, under certain conditions, of the problem
of impinging curved bodies with which I am now more concerned. He
commences with the purely statical problem of contact under pressure.
Thus if two equal spheres of similar material be pressed together with a
given force P , the surfaces of contact are moulded to a plane; and it is
required to find the radius of the circle of contact, and more especially the
distance (a) through which the centres (or other points remote from the place
of contact) approach one another. It appears that the relation between
P and a is simply
Po = M, ................................. (1)
where & 2 depends only on the forms and materials of the two bodies. In the
particular case abovementioned,
where r is the radius of the spheres, E Young's modulus, and a Poisson's
ratio.
In applying this result to impacts Hertz proceeds: "It follows both from
existing observations and from the results of the following considerations,
that the time of impact, i.e. the time during which the impinging bodies
remain in contact, is very small in absolute value; yet it is very large com
pared with the time taken by waves of elastic deformation in the bodies in
question to traverse distances of the order of magnitude of that part of their
surfaces which is common to the two bodies when in closest contact, and
which we shall call the surface of impact. It follows that the elastic state
of the two bodies near the point of impact during the whole duration of
impact is very nearly the same as the state of equilibrium which would be
produced by the total pressure subsisting at any instant between the two
bodies, supposing it to act for a long time. If, then, we determine the
pressure between the two bodies by means of the relation which we previously
found to hold between this pressure and the distance of approach along the
common normal of two bodies at rest, and also throughout the volume of
each body make use of the equations of motion of elastic solids, we can trace
the progress of the phenomenon very exactly. We cannot in this way expect
* Journal filr reine und angeicandte Mathematik, xcn. p. 156 (1881); Hertz's Miscettaneout
Papers, English edition, p. 146. A good account is given by Love, loc. cit.
294 ON THE PRODUCTION OF VIBRATIONS [314
to obtain general laws; but we may obtain a number of such if we make the
further assumption that the time of impact is also large compared with the
time taken by elastic waves to traverse the impinging bodies from end to end.
When this condition is fulfilled, all parts of the impinging bodies, except
those infinitely close to the point of impact, will move as parts of rigid
bodies ; we shall show from our results that the condition in question may be
realised in the case of actual bodies." The abovementioned condition may
in fact always be satisfied by taking the relative velocity of impact to be
sufficiently small.
The solution of the problem, thus limited, is now easily found. For the
case of two spheres the relative acceleration a is connected with P by the
equation
P =a/&i, ................................. (3)
where ki
and m,, m? are the masses of the spheres. Eliminating P between (1) and
(3), we get
a + ^2^ = 0, ................................. (4)
and on integration as the equation of energy
ddo + $Mia f 0, ........................... (5)
a being the relative velocity before impact.
" The greatest compression takes place when a vanishes, and if ^ be the
value of a at this instant
Before the instant of greatest compression the quantity a increases from zero
to a maximum a 1( and d diminishes from a maximum do to zero. After the
instant of greatest compression a diminishes from i to zero and d increases
to d,,. The bodies then separate, and the velocity with which they rebound
is equal to that with which they approach. This result is in accord with
Newton's Theory. It might have been predicted from the character of the
fundamental assumptions."
" The duration of the impact is
2 f ai da = ??? p ^
** V(0* $fcl&2^) " ^ \/(l 
where a, is given by (6).
1906] BY FORCES OF RELATIVELY LONG DURATION 295
The duration of impact, therefore, varies inversely as the fifth root of the
initial relative velocity*."
So long as the condition is satisfied that the duration of the impact is
very long in comparison with the free periods, vibrations will not be excited
in a sensible degree, the energy remains translational, and Newton's laws
find application. It would be of great interest if we could enfranchise our
selves from this restriction. It is hardly to be expected that a complete
solution of the problem will prove feasible, but I have thought that it
would be worth while to inquire into the circumstances of the first appear
ance of sensible vibrations. We should then be in a better position to
appreciate at least the range over which Newton's laws may be expected
to hold.
In the case of spheres the vibrations to be considered are those of the
"second class" investigated by Lambf. They involve spherical harmonic
functions of the various orders, limited in the present case to the zonal kind.
But for each order there are an infinite number of modes corresponding to
greater or less degrees of subdivision along the radius. The first appearance
of vibrations will be confined to those of longest period, of which the most
important is of the second order. In this mode the sphere vibrates sym
metrically with respect both to a polar axis and to the equatorial plane, the
greatest compression along the axis synchronizing with the greatest expansion
at the equator. In what follows we shall denote by <j, <> 2 , &c. the radial
displacement at the pole (point of contact) corresponding to the several
modes, the first ^ being appropriated to that mode in which the sphere
moves as a rigid body (spherical harmonic of order 1), the next <> 2 to the
mode of the second order above described which gives the principal vibration.
Since there is no force of restitution corresponding to fa, the equation
for it takes the simple form
oA = Po> (8)
P being as before the total pressure between the spheres at any time, and
Oj a coefficient of inertia in this case the simple mass of a sphere. On the
other hand, the equations for < a &c. are of the form
o,&+c^ = P* (9)
c 2 &c. being coefficients of stability to be treated as large. This form applies
to all the lower modes, for which the force of collision operating at any
moment may be treated as a whole. By equation (1) of Hertz's theory
P = & 2 a 2 , but now that we are admitting the possibility of vibrations a
must be reckoned no longer from the centre, but from a point which is at
* Love, toe. cit. p. 154.
t Proc. Lond. Math. Soc. Vol. arm. p. 189 (1882).
ON THE PRODUCTION OF VIBRATIONS [314
once near the surface and yet distant from it by an amount large in com
parison with the diameter of the circle of contact. We may write
a = fa + fa + ..., .............................. (10)
inclusion being made of the coordinates of the lower modes only. The sum
of all the coordinates would be zero, since (in the case of equal spheres) the
pole does not move. Thus
P a =k(2fc + 2fc + ...)* ...................... (11)
In the first approximation c 2 &c. are regarded as infinite, so that fa &c.
vanish. P reduces to & 2 (2fa)%, and so from (8)
OifckWO 1 , .............................. (12)
the solution of which gives fa as in Hertz's theory. If P be regarded as a
known function of the time, fa is determined by (9); but it may be well at
this stage to ascertain how far P is modified in a second approximation.
Retaining for brevity fa only, we have approximately fa = P /c 2 . Hence
.................. (13)
and we infer that P is changed by a term of the order cr 1 
We will now pass on to consider the general problem of a vibrator whose
natural vibrations are very rapid in comparison with the force which operates.
We write (9) in the form
$ + w'< = P /a 2 = <l>, ........................... (14)
where n^ = c^a^, and is to be treated as very great. If < and <j> vanish
when t = Q, the solution of (14) is*
If the force operates only between t = Q and t = r and we require the
value of ^> at a time t greater than T, that is after the operation of the
force has ceased, we may write
1 f r
< =  8uin(tt')3?t=t'dt' ................... (16)
nJ Q
If T be infinitely small, the force reduces to an impulse, and we get
4> = n l smnt.fdt; ........................ (17)
but it is the other extreme which concerns us at present.
In many cases, especially when O = at the limits, we may advantageously
integrate (16) by parts. Thus
t')d1f. ...(18)
* Theory of Sound, 66.
1906] BY FORCES OF RELATIVELY LONG DURATION 297
Again
0^ ...(19)
and so on if required. In this way we obtain a series proceeding by descending
powers of n, and thus presumably advantageous when n is great.
As an example, let <i> = t, so that d*<&fdt 2 = 0. The force rises from zero
at t = to a greatest value at t = r and then suddenly drops to zero. From
(18), (19) we find
= n~*r cos n (t T) + n~ 3 sin n (t r) n~ s sin nt .......... (20)
Again, take the parabolic law
= trt*, ................................. (21)
so that <f> = at both limits,
dto/dt = r2t, d^fdt* =  2.
From (18), (19)
<f> =  rn~ 3 sin n (t r) rn~ z sin nt + 2n,~ 4 cos n (t T) 2nr 4 cos nt
=  2n 3 rcos %nr . sin n(t T) + 4n 4 sin \m . sin n (t  T). . . .(22)
If 3> and its differential coefficients up to a high order are continuous
within the range of integration and vanish at the limits, the leading term in
the development of (16) is of high inverse power in n. An extreme case of
this kind is considered by Mr Jeans*, who takes
In this case the solution involves the factor e~ ne , smaller when n is great
than any inverse power of n. But the force is not here limited to a finite
range of time.
iUMjir.fl)
The application of these results to the problem of the collision of equal
elastic spheres is not quite so straightforward as had been expected. In (9),
a denoting, as in (4), (5), (6), (7), the approach of the spheres. The terminal
values of a and of a 2 are zero. Again,
_ ( a f) _ 3 a l _ t (24)
dt dt '
* Dynamical Theory of Gases, Cambridge, 1904, 241. I should perhaps mention that moat
of the results of the present paper were obtained before I was acquainted with Mr Jeans' work.
298 ON THE PRODUCTION OF VIBRATIONS [314
so that d$>ldt vanishes at the limits of the range. But
use being made of (4), (5); and the first part of this becomes infinite at the
limits where a = 0.
Equations (18), (19) give
t')~dt', .................. (26)
and in this we have now to consider the two parts
fsinn(tt')oL'^dt' and fsmn(t
For the second we get on integration by parts, since a? vanishes at both
limits,
which is of order n~ l , or less. In the first part the relation between dt' and
do. is, as in (7),
......................... (27)
If we exclude the terminal parts of the range, the integral would be of
order n~ l , or less, so that it is only the terminal parts that contribute to the
leading term. For the beginning we see from (27), or independently, that
= <M, .................................... (28)
nearly, so that for this terminal region
^ IM o~*f ^fcosnt'd(nt') r8mnt'd(nt')}
sin n (t t) a * dt = r \ sm nt  r   cos nt  V } .
n* I J n* J n* >
When we suppose n very great, the limits of integration may be identified
with zero and infinity; and further by a known theorem
fsinxdx f cosxdx
. = I   / = V(i^)
Jo *Jx Jo *Jx
Thus, so far as it depends upon the early part of the collision,
...................... (29)
4 a 2 n
There will be a similar term due to the end of the collision, derivable
from (29) by replacing nt with n (t T).
If, as I think must be the case, (29) gives the leading term in the expres
sion for a vibration, the next question is as to the order of magnitude of the
1906] BY FORCES OF RELATIVELY LONG DURATION 299
corresponding energy in comparison with the energy before collision, viz.
Mass x d 2 , or
The maximum kinetic energy of the vibration is given by
32 ojw 8 '
and the ratio (R) of this to the energy before collision is
27 frd. = 3 Q
64 a 2 n<X 32 (1  o 2 ) 2 a 2 n 5 pr 2 '
if we introduce the value of k^ from (2).
The precise value of a,, would have to be calculated from Lamb's theory.
It is easy to see that it is decidedly smaller than, but of the same order of
magnitude as, the mass of the sphere, viz. f Trpr 3 .
The precise value of R would depend also upon <r, but for our purpose it
will suffice to make a = . Thus we take
According to Lamb's calculation * for the principal vibration of the second
order in spherical harmonics (a = ^)
so that approximately R = ^? *
50
< 32 >
In (33) ^/(E/p) is the velocity of longitudinal vibrations along a bar of
the material in question, and the comparison is between this velocity and
the velocity of approach before collision. In steel the velocity of longi
tudinal vibrations is about 500,000 cms. per second, or about 16 times that
of sound in air. It will be seen that in most cases of collision R is an
exceedingly small ratio.
The general result of our calculation is to show that Hertz's theory of
collisions has a wider application than might have been supposed, and that
under ordinary conditions vibrations should not be generated in appreciable
degree. So far as this conclusion holds, the energy of colliding spheres
remains translational, and the velocities after impact are governed by
Newton's laws, as deducible from the principles of energy and momentum.
* Loc. cit. p. 206.
315.
ON THE DILATATIONAL STABILITY OF THE EARTH.
[Proceedings of the Royal Society, A. Vol. LXXVII. pp. 486499, 1906.]
THE theory of elastic solids usually proceeds upon the assumption that
the body is initially in a state of ease, free from stress and strain. Displace
ments from this condition, due to given forces, or vibrations about it, are then
investigated, and they are subject to the limitation that Hooke's law shall be
applicable throughout and that the strain shall everywhere be small. When
we come to the case of the earth, supposed to be displaced from a state of
ease by the mutual gravitation of its parts, these limits are transgressed ; and
several writers* who have adopted this point of view have indicated the
obstacles which inevitably present themselves. In his interesting paperf
Professor Jeans, in order to attain mathematical definiteness, goes the length
of introducing forces to counteract the selfgravitation : " That is to say, we
must artificially annul gravitation in the equilibrium configuration, so that
this equilibrium configuration may be completely unstressed, and each
element of matter be in its normal state." How wide a departure from
actuality is here implied will be understood if we reflect that under such
forces the interior of the earth would probably be as mobile as water.
It appears to me that a satisfactory treatment of these problems must
start from the condition of the earth as actually stressed by its selfgravita
tion, and that the difficulties to be faced in following such a course may not
be so great as has been supposed. The stress, which is so enormous as to
transcend all ordinary experience, is of the nature of a purely hydrostatic
pressure, and as to this surely there can be no serious difficulty. After great
compression the response to further compressing stress is admittedly less
than at first, but there is no reason to doubt that the reaction is purely
elastic and that the material preserves its integrity. At this point it may be
well to remark, in passing, upon the confusion often met with in geological
See, for example, Love, Theory of Elasticity, 127; Chree, Phil. Mag. Vol. xxrn. p. 233,
1891 ; Jeans, Phil. Tram. A. Vol. cci. p. 157, 1903.
t Loc. cit. p. 161.
1906] ON THE DILATATIONAL STABILITY OF THE EARTH 301
and engineering writings arising from the failure to distinguish between
a onedimensional and a threedimensional, or hydrostatic, pressure. When
rock or cast iron is said to be crushed by such and such a pressure, it is the
former kind of pressure which is, or ought to be, meant. There is no evidence
of crushing under purely hydrostatic pressure, however great.
Not only is the integrity of a body unimpaired by hydrostatic pressure,
but there is reason to think that the superaddition of such a pressure may
preserve a body from rupture under stresses that would otherwise be fatal.
FitzGerald raises this question in a review* of Hertz's Miscellaneous Papers.
He writes : " In considering the cracking of a material like glass, Hertz seems
to think its cracking will depend only on the tension; that it will crack
where the tension exceeds a certain limit. He does not seem to consider
whether it might not crack by shearing with hardly any tension. It is
doubtful whether a material in which there were sufficient general compres
sion to prevent any tension at all, would crack. Rocks seem capable of being
bent about and distorted to almost any extent without cracking, and this
might very well be expected if they were at a sufficient depth under other
rocks to prevent their parts being under tension. It is an interesting
question whether a piece of glass could be bent without breaking if it were
strained at the bottom of a sufficiently deep ocean. On the other hand,
there seems very little doubt that the parts of a body might slide past one
another under the action of a shear, and would certainly crack unless there
were a sufficiently great compressional stress to prevent the crack ; and that
consequently a body might crack, even though the tensions were not by
themselves sufficiently great to cause separation, arid might crack where the
shear was greatest, and not where the tensions were greatest."
When we reflect that pieces of lead may be made to unite under pressure
when the surfaces are clean, and upon what is implied when insufficiently
lubricated journals, or slabs of glass under polish, seize, we may well doubt
whether it is possible to disintegrate a material at all when subjected to
enormous hydrostatic pressure. In the words of Dr Chreef : " The conditions
under which the deepseated materials of the earth exist are fundamentally
different from those we are familiar with at the surface. The enormous
pressure, and the presumably high temperature, very likely combine to
produce a state to which the terms solid, viscous, liquid, as we understand
them, are alike inapplicable."
A study of the mechanical operations of coining and of stamping (in
recent years, I believe, much developed) would probably throw light upon
this question. We know that rod or tube may be " squirted " from hot (but
solid) lead. Is the obstacle to a similar treatment of harder material purely
* Nature, November 5, 1896 ; Scientific Writings, p. 433.
t Phil. Mag. Vol. XLIII. p. 173, 1897.
302 ON THE DILATATION AL STABILITY OF THE EARTH [315
practical ? In the laboratory I have experimented upon jellies of various
degrees of stiffness, on the principle of suiting the material to the appliances
rather than the appliances to the material. In the simplest arrangement
a leaden bullet is imbedded in jelly contained in a strong glass tube which
the bullet somewhat nearly fits. Although the tube stand vertical for
several days, there is no appreciable descent. But if by numerous longi
tudinal impacts against a suitable pad the inertia of the bullet be brought
into play, movements through several inches may be obtained. Here,
although the deformations are very violent, there is no rupture visible, either
before or behind the bullet.
When an elastic body is slightly displaced from the condition of ease, the
potential energy ( V) is expressed by terms involving the squares and products
of the displacements. If, however, we suppose given finite forces to be
constantly imposed, so that the initial condition is one of strain, the case
is somewhat, though not essentially, altered. It may be convenient to make
a statement, once for all, in terms of generalised coordinates. If under the
action of the forces 4>, , . . . the coordinates assume the values <, 6, ... we
have in terms of the potential energy of strain V,
If the forces permanently imposed be distinguished by the suffix (0), they
are connected with the corresponding values of the coordinates, <f> , etc., by
the equations
This strained condition is now to be regarded as initial, and displacements
from it are denoted by ascribing to the coordinates the slightly altered
values < + 8<, Q + 80, etc. For the potential energy of strain we have
which is of the first order of the small quantities S<j>, etc. But V V is not
now the whole potential energy. In addition to the potential energy of
strain we have to include that of the steadily imposed forces, represented by
the terms
<I> 8<*>@ 80 ............................... (4)
The whole potential energy is thus
regard being paid to (2). The total potential energy, as given by (5), is now
1906] ON THE DILATATIONAL STABILITY OF THE EARTH 303
of the second order in 8$, &c., as is obviously required by the circumstance
that the strained condition </> , &c., is one of equilibrium under the proposed
forces. The coefficients of stability are d?V/d<f><?, d?V/d<f> Q d0 , &c., and they
may differ finitely from the values which obtained previously to the applica
tion of the forces <I> , &c.
As an example having an immediate bearing upon the matter in hand, let
us consider the case of a uniform nongravitating body originally in a state
of ease. If a small hydrostatic pressure <1> act upon it, the volume changes
proportionally, and the ratio gives the " compressibility " of the body in this
condition. Under the action of a finite pressure <I> the volume may be
greatly altered, especially if the body be gaseous, but the new condition is
still one of equilibrium and may be regarded as initial. The compressibility
now may be quite different from before, but it may be treated in the same
way as depending upon the small change of volume S< accompanying the
imposition of a small additional pressure S3>.
To those who, while accepting the usual elastic theory for bodies in a state
of ease, repudiate the application to bodies subject to great hydrostatic
pressure, I would suggest that liquids and solids, as we know them, are not
really free from stress. In virtue of cohesional forces, there is every reason
to believe, the interior of a drop of water is under pressure not insignificant
even in comparison with those prevailing inside the earth, and the same may
be said of a piece of steel.
The conclusion that I draw is that the usual equations may be applied to
matter in a state of stress, provided that we allow for altered values of the
elasticities. In general, these elasticities will not only vary from point to
point, but be seolotropic in character. If, however, we suppose that the body
is naturally isotropic, and that the imposed stress is everywhere merely a
hydrostatic pressure, so that by pure expansion a state of ease could be
attained, the case is much simpler and probably suffices for an approximate
view of the condition of the earth. But although the initial state is one free
of shear, we are not to conclude that the rigidity is the same as it would be
without the imposed pressure. On the contrary, there is much reason to
think that the rigidity would be increased. If there is any analogy to be
found in a pile of mutually repellant hard spheres, it will follow that
an infinite pressure will entail infinite rigidity as well as infinite incom
pressibility.
In the original draft of this paper I had supposed that it would be possible
upon these lines to find another and a more practical basis for Professor
Jeans' analysis. A correspondence with Professor Love* has, however, con
vinced me that this hope is destined to disappointment, and the remainder of
* To whom I am indebted also for other corrections.
304 ON THE DILATATIONAL STABILITY OF THE EARTH [315
the paper loses accordingly much of the interest which at first I felt for it.
In Professor Jeans' theory, if A be the dilatation, so that the altered density
is p (1 A), U the radial outward displacement, E the potential of a volume
distribution of density pA, and a surfacedistribution of density pU, the
displacements , 77, are subject to
w
and two similar equations relating to 17, In (6) \ and p. are the elastic
constants of Lame's notation, and they relate to displacements from the
compressed initial condition. From equations (6) we obtain, as usual,
pVff; ..................... (7)
and by Poisson's equation V*E = 4nryp&, .............................. (8)
7 being the constant of gravitation. Thus
(9)
which is Professor Jeans' equation*.
The solution of these equations is developed by Professor Jeans with the
view of determining at what point instability sets in. Attention is con
centrated mainly upon the solution of (9) expressed by a spherical function of
order one, as being that which bears upon the question of the evolution of the
moon.
I had intended merely to indicate a somewhat simpler treatment, following
more closely the notation and method of Lamb's memoir, " On the Vibrations
of an Elastic Sphere "f ; but as the results so obtained do not agree with those
of Jeans, it appears necessary to set forth the argument in fuller detail, so as
to facilitate criticism.
If in (9) we assume that A is proportional to cospt, we get
(V 2 + A 2 ) A = 0, ...... ........................ (10)
*> ........................... <">
and the solution of (10), subject to the condition of finiteness at the centre, is
A=(Ar)*.7 n + i (Ar).S n .cosp*, .................. (12)
J being the symbol of Bessel's functions, and S n a spherical surface function
of order n. As is well known, J n+ , is expressible in finite terms ; in the case
ofn = > (hr)^J^(hr) may be replaced in (12) by sin hr/hr, a constant factor
being disregarded.
Loc. eit. p. 162.
t Proceedingi London Mathematical Society, Vol. xm. p. 192, 1882.
1906] ON THE DILATATION AL STABILITY OF THE EARTH 305
Before going further it may be well to consider the particular case of a
fluid for which p, = 0. Here the solution for A already given suffices to solve
the problem, and the condition of no pressure at the surface (r = a) gives at
once
J n + ^(ha) = Q, .............................. (13)
which with (11) determines p in terms 7, p, a and the elastic constant X. The
criterion of stability follows by setting p = 0. In the case of n = 0, where the
displacements are symmetrical, ha = mtr, m being an integer ; and we see
that equilibrium is unstable for symmetrical displacements if
a*p*y>lir\ ............................... (14)
In general, by (8) and (10)
(15)
so that #A + , ........................... (16)
where e satisfies throughout the sphere
V 2 e = ..................................... (17)
Substituting the value of E in (6), we get with regard to (11)
where & 2 =p 2 />//* ..................................... (19)
Equation (18) and its companions may be treated as in Lamb's classical
paper. A solution is
1 c?A 1 de /orix
fwa; + i>S' etc .......................... (20)
where A satisfies (10). In virtue of (17) these values satisfy the relation
and the solution may be completed by the addition of terms u, v, w, satisfying
(V 2 + & 2 ) u = 0, etc., as well as the relation
du dv dw _
B+^S"*
Professor Lamb gives the general values of u, v, w. For our present
purpose, and with limitation to one order of spherical harmonics, it suffices to
take
and two similar equations, where <j> n is a solid harmonic of degree n ;
r = ^(a? + y 2 + z*) ; and >^ n is defined by the equation
& B 4
R. v. 20
306 ON THE DILATATIONAL STABILITY OF THE EARTH [315
Save as to a constant multiplier ^>/r n (^)is identical with B~^J n+ ^(6), as
employed in (12). ^ n is thus associated with solid in place of surface
harmonics. The function possesses the following properties
(23)
(25)
A formula in spherical harmonics frequently required is
The term of the nth order in A is thus
A n = Vr u (Ar).o) M .............................. (27)
and corresponding thereto
, = _!^ 1^
A 2 <& V <& +
where M is defined as above, and e n , as well as <f> n and o> u , is a solid harmonic
of degree n.
The formation of the boundary conditions to be satisfied at the free surface
of the sphere (r = a) proceeds almost exactly as in Lamb's investigation
(p. 199), the only difference arising from the fact that h has now a different
value. The first of the three symmetrical surface conditions may be written
(29)
The terms in (29) depending on the parts of 77, f which involve A n are
found to be
..
........................... (30)
where A n = "y "^7' ?^*W<*); (31)
In like manner Lamb finds for the terms in (29) arising from u, v, w,
n d<b n . r. d <t> n
/QQ v
(33)
where (7. =  j ^ + n (ka)  2 (n  1) ^ (te) } ; (34)
1906] ON THE DILATATIONAL STABILITY OF THE EARTH 307
We have now further an additional part arising from e n , which, it should
be observed, makes no contribution to A n . In this
T
so that the additional part is
The two most important cases where n = and n = 1 are also especially
simple, in that (36) disappears. It will be convenient to consider them first.
When n = 0, u, v, w vanish : also, since e is constant, (28) reduces to
t 1 dAo 1 dAo ., 1 dA
= /T^' ^/T*^' ?= /T*^' ......... (37)
where A is proportional to ^ (hr). The motion is everywhere purely radial.
Exactly as in Lamb's investigation of vibrations without gravity, the expression
(30) reduces to
D d &>
^7'
where &> is a constant, so that the surface conditions yield simply B = 0, or
from (32)
Writing 6 for Aa, and for ^ and i/r : (=  ^B~ 1 ^} their values, we get
......................... (39)
Except for a slight difference of notation, this is the same as Lamb's
equation, and his results are therefore available. They are expressed by
means of Poisson's elastic constant <r, and they exhibit ha/7r as dependent
on <r and on the order of the root. To adapt them it is only necessary to
remember that A. 2 , as given by (11), has here a different value from that which
obtains when there is no gravitation (7 = 0). On the other hand, although
7 be finite, hajir may still be equated to TJr, where T^ is the time occupied
by a plane wave of longitudinal vibration in traversing a space equal to the
diameter of the sphere, and r denotes the time of complete oscillation. The
following are the smallest values of ha/Tr corresponding to selected values of
<r, as given by Lamb :
06626 08160
08500 08733
For example, if <r= \ (Poisson's value), the criterion of stability is
(081 60) a .
202
308 ON THE D1LATATIONAL STABILITY OF THE EARTH [315
If & = , the material is incompressible, and motion of the kind now under
contemplation is excluded.
When n = 1, (36) again vanishes, though for a different reason from before*.
The form of the solution is accordingly the same as if there were no gravita
tion. We have from (31), (32)
i (ha) \ + fy a^ (ha)
~
a .
3 "' ~~JT ~~3~ 5
and from (34), (35) omitting some common factors which have no effect,
(42)
a? ( 6 ) a 3 f )
A = sr I "^i (& a ) + rr~ n ka&i (ka) \ = ^ { Jr. (&a) $"*K (^'a) K (43)
2 I T ^a 2 j 2 I T J
The surface conditions (29) are of the form
(I (O\ m^ CL Cl)i >^ ( '^P\ r\ v (Pi
and, as Professor Lamb shows, they require that
It follows that <j and &>, must be of the same form, and also that
B l IA l = D l jC lt (45)
in which the values A lt B lt C lt D l are to be substituted from (40), (41), (42),
(43). We find
or if, in accordance with (23), we replace i^ 2 (6) by b6~ 1 ^ (d),
' .
Except for the different value of h, this agrees with Professor Lamb's
equation (87).
In equation (46) there is no limitation upon the value of/). If to find the
criterion of stability we put p or k equal to zero,
*, (Aw) fi (*)!,
and the equation reduces to
*.<*) l<X 4 + 2 A) fl(Att) ...................... (48 )
The equation may also be written in terms of the Bessel functions. The
relation between J and ^ is
Jj (a;) x V(i = # f (#) = sin ,
/2 (a;) x \/(i 7r#) = i^ 2 ^! (^) = or 1 sin #  cos x,
, (a?) = (3ar  1) sin  Sa; 1 cos a; ;
* The terms de^dx, etc., in (28) denote in this case a uniform displacement, as of a rigid
body, and naturally contribute nothing to the surface condition.
1906] ON THE DILATATIONAL STABILITY OF THE EARTH 309
so that in terms of J^s (48) becomes
X + 2it , 20 r
J ., {x) = jr  /, (ac) ; (49)
or, if we introduce the circular functions,
a?tana?a = ^ ~~^T (50)
Unfortunately (49) does not agree with the result given by Professor
Jeans. In his notation, when n = 1,
y, (from 59) =  f ^ , y, (from 60) = + ^ ^ ;
and (54), (57), (58) give
156 r
The comparison of processes is rendered difficult by the occurrence of
several errors (possibly misprints) in Professor Jeans' paper. Thus (33) does
not seem correct, and (41), (42) do not follow from (38), (39). Starting
from Professor Jeans' equations just mentioned and making use of his (30),
(43), (44), (45), (48), (49), I have obtained a result in harmony with my
equation (49).
From (50) it is easy to calculate the value of X//A corresponding to any
value of x. When /z = 0, tan x = x, of which the first root is
# = 143037T = 4493.
This gives an angle of 77 (+ 180). Calculating for angles of 60, 50 C ,
40, we find
v/*
oo
4493
3840
4189
2056 1221
4014 3840
It seems that the value of x is not very sensitive to variation of /*, and for
such values of the ratio of X.//A as are likely to occur, especially under high
pressure, we might almost content ourselves with the fluid solution (ft = 0).
The simplicity of the cases so far considered, viz., n = and n = 1, depends
upon our having escaped the necessity of determining the value of e n . For
values of n greater than unity this function remains in the equations, which
now demand a more elaborate treatment. From (20), (21)
f 1<^ + 1* + M> (52)
/i 2 dx p dx
in which the second term becomes infinite when p = 0. In order to balance
310 ON THE DILATATIONAL STABILITY OF THE EARTH [315
this, <t> n in (21) must be made infinite of the order p~* or Ar*. Thus writing
> n = <j>n, we have
!*+  p <** , Mi
"*" ~ '
. 2n + 1 da;
n _ r>+' f /,V' I ^ _<>
n+12n + i.2n + 3( 2 . 2 + 5 " \ dx i*"" '
Thus, as in the theory of differential equations with equal roots, we have
when' = 0,
den _<yn,P_
dx " l "i
with two similar equations, f n denoting again a solid harmonic of degree n.
From (52), (53) we find for the radial displacement
rr af + yiy+* _ 1 ^An , n& , P ^^gn
r h* dr r * 2.2?i + 3
at the surface, where r = a.
The boundary condition (29) requires a parallel treatment. The terms
depending on A n remain as in (30), (31), (32). From (34), (35), we get as
appropriate for the present purpose
D n
_
n + 1 .2/1 + 1 . 2n + 3 '
Equations (33), (36) now give, 4> being written as before for frfa
dx j 2n +
which, when fc 2 is made to vanish, is to be replaced by
r< (56)
This is additional to (30). The equations to be satisfied at the surface
are thus
.(57)
.(58)
1906] ON THE DILATATIONAL STABILITY OF THE EARTH 311
When n= 1,f n disappears, and the final condition is found by eliminating
the ratio <a n : e n from (57), (58). This would conduct us again to the results
already arrived at for that case. In general we require another equation
connecting e n with w n and / .
For this purpose we must recur to the definition (16) of e n and of E.
A calculation is made by Jeans on the basis of the expression (12) of A
by means of Bessel's functions. We have at the surface
. (59)*
In order to express this in our present notation (by means of ^rs), we see
by comparison of (12) and (27) that
J n + .(lia).Sn = (h<rf*n(ha).a> n ................... (60)
so that with use of (24)
Eliminating U between this and (54), we find
f = pa?e n ( (2n + l)ji _JL _ \
Jn " p \ n(\+2fj<,)hW 2.2w + 3j
to n ( 2n + 1 . h?a? \ ,_ .
F  S* + *  , * ' (62)
The substitution for/ ft in (57) now gives
2w2
pa?e n \ (2n
2n2
This equation and (58) determine two values of o> n :e n ,and the elimination
of this ratio gives the required final result. We will write (63) for brevity as
Fa*a> n + G(ptf/ fJ i)e n = 0, ........................ (64)
where by (31), and reduction with the aid of (25),
j* 2/i2 n
* In Professor Jeans' equations (5), (23) the sign of U is positive, but this appears to be
an error.
312 ON THE DILATATION AL STABILITY OF THE EARTH [315
Similarly, if (58) be written
(69)
and the final result is FKGH=Q, .............................. (70)
given the ratio i/r n+1 (ha) : >/r n (ha) in terms of n, ha, and (A, + 2/i)//z.
In applying results of calculation based upon the assumption of a uniform
compressibility to the case of the earth where the variation is likely to be
very considerable, we must have regard to the character of the function (12)
by which the dilatation is expressed. When n = I or a greater number, (12)
vanishes at the centre and (when /* = 0) at the surface. The values to be
ascribed to the elasticities are those proper to an intermediate position, such
as halfway between the centre and the surface. For a more complete
treatment we might calculate the balance of the elastic and gravitational
potential energies on the basis of a displacement still following the same law
as has been found to apply to a uniform sphere. In accordance with a general
principle the result, so calculated, will be correct as far as the first powers of
the variations from uniformity.
Another question, interesting to geologists, upon which our results have a
bearing is as to the effect of denudation in altering the surface level. The
immediate effect of the removal of material is, of course, to lower the level ;
but if the material removed is heavy and the substratum very compressible,
the springing up of the foundation may more than neutralize the first effect,
and leave the new surface higher than the old one. So far as I am aware
discussions have been based upon the elastic quality merely of the interior
without regard to selfgravitation ; but, as is easy to see, if the condition be
one approaching instability, the effect of a pressure applied to the surface may
be immensely increased.
316.
SOME MEASUREMENTS OF WAVELENGTHS WITH A
MODIFIED APPARATUS.
[Philosophical Magazine, Vol. xi. pp. 685703, 1906.]
As the result of discussions held during the last three or four years, it
seems to be pretty generally agreed that the use of the diffractiongrating in
fundamental work must be limited to interpolation between standard wave
lengths determined by other means. Even under the advantageous con
ditions rendered possible by Rowland's invention of the concave grating,
allowing collim'ators and objectglasses to be dispensed with, the accuracy
attained in comparisons of considerably differing wavelengths is found to
fall short of what had been hoped. I think that this disappointment is partly
the result of exaggerated expectations, against which in 1888* I gave what
was intended to be a warning. Quite recently, Michelsonf has shown in
detail how particular errors of ruling may interfere with results obtained by
the method of coincidences ; but we must admit that the discrepancies found
by Kayser + in experiments specially designed to test this question, are greater
than would have been anticipated.
Under these circumstances, attention has naturally been directed to inter
ference methods, and especially to that so skilfully worked out by Fabry and
Perot. In using an accepted phrase it may be well to say definitely that
these methods have no more claim to the title than has the method which
employs the grating. The difference between the grating and the parallel
plates of Fabry and Perot is not that the latter depends more upon inter
ference than the former, but that in virtue of simplicity the parallel plates
allow of a more accurate construction. In Fabry and Perot's work the wave
lengths are directly compared with the green and red of cadmium ; and they
have obtained numbers, apparently of great accuracy, for artificial lights from
vacuumtubes containing various substances, e.g. mercury, for numerous
* Wave Theory, Enc. Brit. ; Scientific Papers, in. p. Ill, footnote.
t Astrophysical Journal, xvm. p. 278 (1903).
t Zeitschrift fiir wiss. Photographic, Bd. n. p. 49 (1904).
314 SOME MEASUREMENTS OF WAVELENGTHS [316
lines from an iron arc, and also for various rays of the solar spectrum.
While, so far as I can judge, there has been every disposition to receive with
favour work which not only bears the marks of care but is explained with
great discrimination, it must still be felt that, in accordance with an almost
universal rule, confirmation by other hands is necessary to complete satis
faction. It was with this feeling that about a year ago I commenced some
observations of which I now present a preliminary account. I was not
without hope that I might be able to introduce some variations which would
turn out to be improvements, and which would, at any rate, promote the
independence of my results.
In this method the interference rings utilized are of the kind first
observed by Haidinger, dependent upon obliquity. Their theory is con
tained in the usual formulae for the reflexion and transmission of parallel
light by a " thin plate." Thus, if \ be the wavelength of monochromatic
light, K = 27T/X, 8 the retardation, e the reflecting power of the surface, we
have, in the usual notation for the intensity of reflected light*,
and S = 2/ucosa', (2)
where t denotes the thickness of the plate, /j, the refractive index, and a' the
obliquity of the rays within the plate.
Another form of (1) is
1 i .L I 1 **)* M
R = + 4* gin* (**) "
and from this we see that if e = 1 absolutely,
l/R = R = l
for all values of 8. If e = I very nearly, R = 1 nearly for all values of 8 for
which sin(*8) is not very small. In the light reflected from an extended
source, the ground will be of the full brightness corresponding to the source,
but it will be traversed by narrow dark lines. By transmitted light the
ground, corresponding to general values of the obliquity, will be dark, but
will be interrupted by narrow bright rings whose position is determined by
sin(*S) = 0. In permitting for certain directions a complete transmission
in spite of a high reflecting power (e) of the surfaces, the plate acts the part
of a resonator.
There is no transparent material for which, unless at high obliquity,
e approaches unity. In Fabry and Perot's apparatus the reflexions at nearly
perpendicular incidence are enhanced by lightly silvering the surfaces. In
this way the advantage of narrowing the bright rings is attained in great
* See, for example, Wave Theory, Enc. Brit.; Scientific Paper*, HL pp. 64, 65.
1906] WITH A MODIFIED APPARATUS 315
measure without too great a sacrifice of light. The plate in the optical sense
is one of air, and is bounded by plates of glass whose inner silvered surfaces
are accurately flat and parallel*. The outer surfaces need only ordinary
flatness, and it is best that they be not quite parallel to the inner ones.
It will be seen that the optical parts are themselves of extreme sim
plicity ; but they require accuracy of construction and adjustment, and the
demand in these respects is the more severe the further the ideal is pursued of
narrowing the rings by increase of reflecting power. Two forms of mounting
are employed. In one instrument, called the interferometer, the distance
between the surfaces the thickness of the plate is adjustable over a wide
range. In its complete development this instrument is elaborate and costly.
The actual measurements of wavelengths by Fabry and Perot were for the
most part effected by another form of instrument called an etalon or inter
ferencegauge. The thickness of the optical plate is here fixed ; the glasses
are held up to metal knobs, acting as distancepieces, by adjustable springs,
and the final adjustment to parallelism is effected by regulating the pressure
exerted by these springs.
The theory of the comparison of wavelengths by means of this apparatus
is very simple, and it may be well to give it, following closely the statement
of Fabry and Perotf. Consider first the cadmium radiation \. It gives a
system of rings. Let P be the ordinal number of one of these rings, for
example the first counting from the centre. This integer is supposed known.
The order of interference at the centre will be p = P + e. We have to
determine this number e, lying ordinarily between and 1. The diameter
of the ring under consideration increases with e ; so that a measure of the
diameter allows us to determine the latter. Let e\ be the thickness of the
plate of air. The order of interference at the centre is p = 2e/\. This
corresponds to normal passage. At an obliquity i the order of interference
is pcosi. Thus if x be the angular diameter of the ring P, pcos ^x =P',
or since as is small,
In like manner, from observations upon another radiation X' to be compared
with \ we have
whence if e be treated as an absolute constant,
X/ P '!+?$).
The ratio X/X' is thus determined as a function of the angular diameters
x, x and of the integers P, P'.
* The most important requirement is the equidistance of the surfaces, and would not be
inconsistent with equal and opposite finite curvatures.
+ Ann. de Chimie, xxv. p. 110 (1902). A good account is given in Baly's Spectroscopy.
Now with an altered meaning.
316 SOME MEASUREMENTS OF WAVELENGTHS [3.16
One of the principal variations in my procedure relates to the manner in
which P is determined. MM. Fabry and Perot* say: "L'e"talon, une fois
rg!6, est mesure* en fonction des longueurs d'onde du cadmium, par les
m^thodes que nous avons pre'ce'dement de"crites ; 1'emploi de interfe"rometre
est ncessaire pour cela." I wished to dispense with the sliding interfero
meter, and there is no real difficulty in determining P without it. For this
purpose we use a modified form of (4), viz.:
a?
expressing P'/P as a function of A/A/, regarded as known, and of the
diameters. To test a proposed (integral) value of P, we calculate P' from (5).
If the result deviates from an integer by more than a small amount (depending
upon the accuracy of the observations), the proposed value of P is to be
rejected. In this way, by a process of exclusion the true value is ultimately
arrived at.
The details of the best course will depend somewhat upon circumstances.
It will usually be convenient to take first a ratio of wavelengths not
differing much from unity. Thus in my actual operations the mechanical
measure of the distance between the plates was 4'766 mm., and the first
optical observations calculated related to the two yellow lines of mercury.
The ratio of wavelengths, according to the measurements of Fabry and
Perot, is 1 '003650; giving after correction for the measured diameters
1003641 as the ratio P'/P. From the mechanical measure we find as a
rough value of P, P = 16460. Calculating from this, we get P t = 16519'92,
not sufficiently close to an integer. Adding 22 to P we find as corresponding
values
P= 16482, P' = 1654200,
giving P as closely as it can be found from these observations. This makes
the value of P for the cadmiumred ring observed at the same time about
14824, and this should not be in error by more than 30.
Having obtained an approximate value of P for the cadmium red, we may
now conveniently form a table, of which the first column contains all the so
far admissible (say 60) integral values of P. The other columns contain the
results by calculation from (5) of comparisons between other radiations and
the cadmium red. The second and third columns, for example, may relate
to cadmium green and cadmium blue. These almost suffice to fix the value
of P, but any lingering doubt will be removed by additional columns relating
to mercury green and mercury yellow (more refrangible). An extract from
the table (p. 317) may make the matter clearer.
Inasmuch as the ratio of cadmium red to cadmium green is 12659650,
very nearly 5 : 4, only every fourth number for red is admissible on this
* Loc. cit. p. 112.
1906]
WITil A MODIFIED APPARATUS
317
ground alone. If we consider a number such as 14803 not excluded by the
comparison with cadmium green, we see that while it would pass the mercury
green test, it is condemned by the cadmium blue and still more by the mercury
yellow test. The only possible value of P is found to be 14814.
The criticism may probably suggest itself that, although other values of
P may be excluded, the agreement of the row containing 14814 with integers
is none too good. It is to be remembered that these observations were of a
preliminary character, and were taken without the full precautions with
regard to temperature afterwards found to be necessary. The formula at the
basis of the calculation assumes that e, the thickness of the plate, is constant,
Cd
red
Cd
green
Cd
blue
Hg
green
Hg
yellow
14788
Q
1872103
1983604
1743524
14790
1
1984007
2
1872609
3
4
1984409.
5
1872989
6
1873115
7
1984812
9
1873495
1985080
14800
1985214
1
2
3
1874001
1985616
1745901
1651868
4
5
6
1986019
7
1874508
8
1986287
9
14810
1874888
1
1875014
1986690
1746236
3
4
1875394
1987092
1746590
1653096
5
but in fact it changes with temperature. On this account alone erroneous
results will be obtained unless the observations are well alternated, so as to
eliminate such effects. The numbers finally arrived at, in substitution for
the row in the table, are
14814, 1875395, 1987095, 1746597, 1653100.
The deviations from integers still outstanding have their origin in a
complication which must be admitted to be a drawback to the method and
might at first sight be estimated even more seriously. The optical thickness
318 SOME MEASUREMENTS OF WAVELENGTHS [316
e of the plate, on which everything depends, is not really constant, as has
been assumed, when we pass from one part of the spectrum to another some
what distant from the first. The question is discussed by Fabry and Perot.
If, to take account of this factor, we denote the thicknesses for the two wave
lengths by e x , e A > we have
;/ _v X
p ~ 6* X"
and accordingly in place of (5)
P' X
But although I was prepared to find the calculated values of P differing
somewhat from integers, I was disturbed by the amount and at first by the
direction of the difference. For in their paper of 1899* Fabry and Perot
remark : "La surface optique du me*tal pour la radiation rouge est, par suite,
situe un peu plus profonde'ment dans le metal que celle de la lumiere verte,
et a une distance de 4 fi/j," At this rate e>, (red) would exceed e\~ (green),
and the introduction of the new factor in (6) would increase, and not remove,
the discrepancy. It would seem, however, that the passage above quoted is
in error and inconsistent with the discussion given in the later paperf, itself
indeed embarrassed by several misprints}.
The amount of the correction required to bring the number for cadmium
green up to an integer about 2 parts in a million is 2 times as great as
one would expect from Fabry and Perot's indications . As to this, it may be
observed that the wavelengths employed in the calculation of the cadmium
radiations are those of Michelson, and were obtained by a method free of the
complication now under discussion. If these are correct, as there is no
reason to doubt, and if there is no mistake in the identification of the
ring and there can be none here it follows that the change of optical
thickness in passing from red to green is determined by the numbers given
and may be used to correct ratios of wavelengths not previously known with
precision.
If we wish to make the results of the present method entirely inde
pendent, we must obtain material from observation sufficient to allow the
variation of thickness with wavelength to be eliminated, that is, we must
use the same silvered plates at two different distances. In Fabry and Perot's
Ann. de Chimie, xvi. p. 311.
t Loc. tit. pp. 120124.
J Of these it may be worth while to note that the sign of 66 up on p. 123, line 5 should
apparently be  instead of + .
It is known that the effect depends upon the thickness of the silver films ; perhaps also
upon the process used in silvering and upon the condition of the surfaces in other respects.
Surfaces that have stood some time in air are almost certain to be contaminated with layers of
volatile greasy matter.
1906] WITH A MODIFIED APPARATUS 319
work the sliding interferometer was employed ; the silvered surfaces were
brought to very small distances, and the coincidences of two band systems,
e.g. cadmium red and cadmium green, were observed, the telescope being
focussed upon the plate, and not as before for infinity. It appears that ex
cellent results were obtained in this way, affording material for eliminating
the complication due to change of optical thickness.
It is rather simpler in principle, and has the incidental advantage of
allowing the sliding interferometer to be dispensed with, if we follow the
same method for the small as for the greater distance. If the calculation be
conducted on the same lines as before by means of (5), we ought to obtain
the same fractional part again in the value of P', e.g. '95 for cadmium green
referred to cadmium red. For, as we see from (6), the proportional error in
P'/P as calculated from (5) is (e^ e^je^ In the second set of operations,
writing 77 for e, we find as the proportional error (^x i7x)/^x, in which
77 v 77 A = g v e\', so that the proportional errors are as T/A : e\ , or inversely as
P or P'. Thus the absolute error in P', as calculated from (5), is unaffected
by the change of e to ??. If the fractional part is not recovered, within the
limits of error, it is a proof that the assumed ratio of wavelengths calls for
correction, and the discrepancy gives the means for effecting such correction.
The above procedure is the natural one, when it is a question of identi
fying a ring or of confirming ratios of wavelengths already presumably
determined with full accuracy; but when the object is to find more accurately
wavelengths only roughly known, it has an air of indirectness. Otherwise,
we have as before,
Ze^ = p\, 2e\> = p"\,' ;
and again for a smaller interval between the surfaces,
2T7A = 7rX., 277 A = 7r'X/.
Hence
2 (e,  T;,) = (p  TT) \, 2 (e v  ^) = (p' 
and e,, 77* = en  i;v, so that
Hence p, IT, p', TT' are the ordinal numbers at the centre. They are to be
deduced, as before, from the integral numbers proper to the rings actually
observed and from the measured angular diameters of these rings.
It is obvious that p and TT must not be nearly equal. If p be the larger
number corresponding to the greater interval, TT should not exceed \p. On
the other hand, too great a reduction of TT would lead to difficulties on account
of the increased angular diameter of the rings. Perhaps it was for this
reason that Fabry and Perot adopted an altered course. In my experiments
320 SOME MEASUREMENTS OF WAVELENGTHS [316
the longer interval was, as already mentioned, about 5 mm., and the shorter
interval was about 1 mm., so that the angular diameter of the rings was
rather more than doubled in the latter case.
The facility with which angular diameters larger than usual could be
observed is due, in part at any rate, to the special construction of my
apparatus. MM. Fabry and Perot employed a fixed interferencegauge and
a fixed telescope, measuring the diameters of the rings by an eyepiece
micrometer. There are, I think, some advantages in a modified arrangement,
whereby it becomes possible to refer the rings to a wire fixed in the optic
axis of the telescope. To this end the wire is made vertical, and the rings
are brought to coincidence with it by a rotation of the gauge, which is
mounted upon a turntable giving movement round a vertical axis. The
middle plane of the gauge is vertical and adjusted so as to include the axis
of rotation. In this way of working the reference wire is backed always by
the same light, whether opposite sides of one ring or of different rings are
under observation. It is perhaps a more important advantage that the same
part of the objectglass is always in use, and to a better approximation the
same parts of the plates of the gauge. The diaphragm which limits the
latter should be as close to the plates as possible (or to their image near the
eye), but when the multiple reflexions are taken into account it is impossible
to secure that exactly the same part should always be in action.
The revolving turntable carried with it a thick strip of plateglass upon
which was scratched a radial line. The point observed described a circle of
10 inches radius, and the rotation was measured by means of a travelling
microscope reading to *001 inch. The angles involved are sufficiently small
to allow the diameter of a ring to be taken as proportional to the difference
of readings at the microscope.
As regards the gauge itself, the plates are by Brashear. For the mounting
of the 5 mm. gauge, which is of brass, I am indebted to my son Mr R. J.
Strutt. The 1 mm. gauge is of iron and was made by my assistant Mr Enock.
They are much after the design of Fabry and Perot. For the final adjust
ment to parallelism the eye is moved in various directions across the line of
vision so as to bring different parts of the plates into action, and for this
purpose it may be desirable to increase the aperture. A dilatation of the
rings means that the corresponding parts of the plates need approximation
by additional pressure. The aperture employed in the actual measurements
was of about 9 mm. diameter.
The (achromatic) objectglass of the telescope is of 15 inches focus. In
rigid connexion with it is the vertical reference wire accurately adjusted to
focus, and close to the wire a small frame suitable for carrying the horizontal
slits (cut out of thin sheet zinc) necessary for the isolation of the various
1906] WITH A MODIFIED APPARATUS 321
colours*. The eyepiece is a single lens of 5 inches focus, mounted indepen
dently, so that it can be readjusted without fear of disturbing the object
glass and reference wire. The change of position required for the best
seeing in passing from red to blue or even from red to green is so great as
to occasion surprise that good results can be attained in the absence of such
a pro vision f.
The separation of the colours was usually effected by direct vision prisms
held between the eyepiece and the eye. Of these two were available. The
larger containing (in all) three prisms was usually the more convenient, but
sometimes a smaller and more dispersive combination containing five prisms
was preferred. It is better to use more dispersion than unduly to narrow
the slit. The refracting edges of the prisms are, of course, horizontal.
In order to secure that the proper parts of the ring systems should be
visible, the axis of the telescope was adjusted in the vertical plane with
substitution for the slit of a horizontal wire coincident with the middle line
of the former.
The advantage of this arrangement is that the ring systems (or at least
so much of them as is necessary) of the various radiations emitted by one
source of light are all in view at the same time.
In some cases, directvision prisms held between the 5inch eyelens and
the eye do not suffice. The soda lines, for example, require a high dispersion.
Even the yellow lines of mercury, which are about three times as far apart
as the soda lines, could not be fully separated by the prisms already spoken
of. Here a good deal depends upon chance. If the rings of one mercury
system happen to bisect approximately those of the other system, both can
be measured in the interferometergauge, and the only question which
remains open is the distinction of the two systems. For this purpose a prism
of moderate power, by which one system is lifted a little relatively to the
other, suffices. If, however, the two ring systems chance to be nearly in co
incidence, a much more powerful dispersion is required in order to measure
them separately.
In such cases recourse was had to a special directvision prism of glass
and bisulphide of carbon through which a selected ray of the spectrum passes
without refraction at all at any of the surfaces]:. In this instrument the
upper edge of the beam traverses 20 inches of glass and the lower edge
20 inches of bisulphide of carbon. This prism cannot be inserted between
the eyepiece already described and the eye, which latter must be placed at
the image of the objectglass. Additional lenses are therefore required.
* Fabry and Perot, C. E. March 27, 1904.
t Especially in using the method of coincidences. I ought perhaps to mention that my eyes
have now very little power of accommodation.
t Nature, LX, p. 64 (1899) ; Scientific Papers, Vol. iv. p. 394.
R. v. 21
322 SOME MEASUREMENTS OF WAVELENGTHS [316
These are merely ordinary spectaclelenses and constitute a telescope of unit
magnifying power. A more precise description is postponed, as I am not
sure that I have as yet hit upon the best arrangement. It may suffice to
say that with this instrument rings formed of spectral rays even closer than
the soda lines could be readily separated, and that without too great a con
traction of the slit limiting the visible portion of the rings.
The source of light, sometimes very small, was focussed upon the dia
phragm at the gauge, and it is necessary that the aperture be completely filled
with light. This gives the ratio between the distances of the lens from the
source (u) and from the gauge (v). Again, the angular diameter of the field
of light, which must not be too small, fixes the ratio of the aperture of the
lens to v; so that only the absolute scale of the three quantities is left
open. It is desirable that the lens be achromatic. I have used a oneinch
lens from a small operaglass, and this worked well with u = 2 inches and
w = 4 feet
As sources of light in experiments involving high interference, vacuum
tubes are by far the most convenient, and their introduction is one of the
many services which Optics owes to Prof. Michelson. At the head of the
list stands the helium tube, both on account of its not requiring to be heated
and also of the brilliancy of the yellow radiation. Hitherto, however, the
wavelengths have not been measured with the highest accuracy. The tube
that I have employed was made some years ago by my son and had already
seen a good deal of service in experiments designed to answer the question :
"Is Rotatory Polarization influenced by the Earth's Motion?*" From the
overpowering brilliancy of the yellow line, it may be inferred that the
pressure is not very low. Mercury too, for which the principal wavelengths
have been determined with great accuracy by Fabry and Perot, is convenient
as requiring only a very moderate heating; and cadmium, in spite of the
higher temperature demanded, is indispensable. Not only is the cadmium
red by general consent the ultimate standard, but a comparison of the red
and green ring systems, even without a prism, gives rapid information as to
the condition of the gauge, slightly variable from time to time on account of
temperature and of necessary readjustments. Thus, in most of my observa
tions, the red ring under measurement was in very approximate coincidence
with a green ring. If, owing to rise of temperature, this ring had so far
expanded as to make it advisable to substitute the next interior one, there
could still be no uncertainty as to the order (one higher) of the ring actually
under observation.
As cadmium tubes appear to have been found troublesome, it may be
well to describe a simple construction specially adapted to private workers
whose skill in glassblowing is limited. It was thought that alloying and
Phil Mag. Vol. iv. p. 215 (1902) [Scientific Papers, Vol. Y. p. 58].
1906] WITH A MODIFIED APPARATUS 323
consequent expansion of platinum sealings was a likely source of difficulty,
and these were accordingly dispensed with. The diagram exhibits half the
complete tube. The working capillary A, the enlargement BD, and the
lateral tube C for attachment to the pump are much as usual. But the
enlargement is continued by a second capillary DE, perhaps 1 mm. in
diameter and 15 cm. long, through which passes with approximate fit a
straight aluminium wire, serving as electrode. The airtight joint at E
between the wire and the glass is made with sealingwax. The length DE
must be sufficient to allow E to remain cool, although D, enclosed in a
copper case, is hot enough to keep the cadmium vapour uncondensed. The
lateral tube C projects from the case, and the cadmium condensed in it may
need to be driven back from time to time by temporary application of the
flame of a spiritlamp or bunsenburner.
This construction, used with cadmium, mercury, and thallium, has so far
answered my expectations. Cadmium tubes, apart from failures by cracking,
are said often to deteriorate rapidly. My experience did not contradict this ;
for after four or five evenings' work the red radiation, which at first had been
very brilliant, was no longer serviceable, although the green did not seem
to have suffered much. At this stage the tube was reexhausted and then
appeared to behave differently, the red radiation being much better main
tained. One must suppose that something deleterious had been emitted and
been pumped away. There is much in the behaviour of vacuumtubes which
at present defies explanation.
To excite the electric discharge a large Ruhmkorff, actuated by five small
storagecells, was usually employed. Sometimes, especially in the comparison
of the cadmium radiations, an alternate current was substituted ; but there
was no perceptible difference in the measurements. In this case a trans
former of home construction was fed from a De Meritens magneto machine.
The radiations from zinc (and occasionally from cadmium) were obtained
by an arrangement similar to Fabry and Perot's "trembler"*. The behaviour
was very capricious. Sometimes, even when actuated by five secondary cells
only, the zinc rings were magnificent; but the deterioration was usually
rapid as the zinc points lost their metallic surfaces. This change appears to
* C. R. 130. p. 406 (1900).
212
324
SOME MEASUREMENTS OF WAVELENGTHS
[316
be independent of oxidation. When the current was from a dynamo giving
about 80 volts, the apparatus was less troublesome, but even then required
careful management. The fineness of the points needs to be accommodated
to the current employed.
As an example of the observations and calculations therefrom, I will
take a series of Dec. 20, 1905, relative to the three radiations from the
cadmium vacuumtube. In this series the temperature conditions were more
favourable than usual.
Cadmium (5 mm. gauge).
Bed
Green
Blue
Right
Left
Right
Left
Right
Left
398
221
401
217
406
213
399
221
402
217
404
211
399
220
402
216
406
210
3987
2207
4017
2167
4053
2113
Diff. = 1780
Diff. = 1850
Diff. = 1940
The numbers entered are the actual readings of the microscope in inches
for settings on the right and left sides of the rings. Each horizontal row
constitutes really a complete set. In order to eliminate temperature effects
as far as possible, the readings are taken in a certain sequence. Thus in the
first row the sequence was Red (R), Green (R), Blue (R), Blue (L), Green (L),
Red (L). The differences, representing the diameters of the rings, are thus
appropriate to the middle of the time occupied. If, as happened here and
usually, the temperature was rising, so that the rings dilated, the first
reading ('398) on the red is too small, but the error is compensated in the
last reading ('221), which is equally too small. As a matter of convenience
the next row would be taken in the reverse order, beginning with a repetition
of Red (L), and so on.
Since the radius of the circle described by the point of observation is
10 inches, the angular diameters (x) of the rings are as follows:
Red
Green
Blue
X
O1780
01850
O1940
3?
10~ 4 x3168
10* x 3422
10~ 4 x 3764
i**
Diff . ..
10* x 3960
10 4 x 4277
10* x O317
10 x 4705
10~ 4 x 0745
1906]
WITH A MODIFIED APPARATUS
325
The calculation now proceeds by means of (5). If P refer to cadmium
red and P' to green, we have with Michelson's values of the wavelengths :
^ = 12659650 (1  00000317) = 12659610,
which with P = 14814 gives
P'= 14814 + 3939945 = 18753945.
In like manner for the blue referred to the red,
P
whence
= 13413733 (1  00000745) = 13413633,
P / = 14814 + 5056955 = 19870955.
The wavelengths of the various radiations from a single source can thus
be compared with great ease, and but little fear of temperature error. A
set of observations from which this error is practically eliminated can be
made in a short time and a few repetitions give all the security necessary.
But the situation is not so favourable when we compare radiations from
different sources. More time is occupied and there is corresponding oppor
tunity for temperature change. It is necessary to alternate the observations,
taking the first source twice and the second once, or preferably the first
three times and the second twice. Even with this precaution I believe that
temperature change was the principal source of error in the results of a single
evening's work.
In the observations with an interval of one millimetre between the
silvered surfaces, the influence of temperature is of course much less per
ceptible. For a similar reason the identification of the rings is a much
easier matter. I will give as a specimen a series of operations (Feb. 9) in
which helium was compared with cadmium. The first and third sets, each
containing a repetition, related to cadmium ; the second set (twice repeated)
related to helium. Only the mean diameter for each set is here recorded :
Cadmium.
Red
Green
Blue
I
749
636
641
Ill
752
643
643
Mean
750
639
642
326
SOME MEASUREMENTS OF WAVELENGTHS
Helium.
[316
Bed
7065
Bed
6678
Yellow
5876
Green
5016
Green
4922
Blue
4713
Violet
4472
II
704
673
721
630
684
665
637
The first question is as to the ratio P'/P derived by (5) from these
numbers for cadmium. The integral value of P for the cadmium red ring
was 3328. From this we find
P' cad. green = 4213946,
F cad. blue =4464935,
on the basis of Michelson's wavelengths. For the green the fractional part
is practically identical with that deduced above from one set of observations
with the 5 mm. gauge. In the case of the blue the fractional part is now
distinctly lower.
The above are the results of work on single evenings. On the mean of
all the comparisons with the two intervals there resulted :
Cadmium.
5 HUM.
1 mm.
Red
14814
3328
Green
1875395
421395
Blue
1987095
446494
As already explained the agreement of the fractional parts constitutes a
complete verification of Michelson's ratios of wavelengths, accurate to one
part in 2 millions in the case of red and blue and to a still closer accuracy
in the case of red and green. And it appears further that the phasechanges,
upon which depend the deviations from integers, are decidedly greater than
in the examples recorded by Fabry and Perot.
The above results for cadmium suffice to indicate what deviations from
integral values are to be expected when any radiation is compared with
cadmium red assumed integral. In so far as the expected fractional parts
appear in the results, so far are the ratios of wavelengths assumed in the
calculation verified. The following are the wavelengths, reckoned in air at
1906] WITH A MODIFIED APPARATUS 327
15 C. and 760 mm. pressure, whose ratios to cadmium red have been verified
by my observations to about one part in a million :
Cadmium
Mercury
Zinc
Soda
64384722
50858240
4799911
5790659
5769598
5460742
4358343
6362345
4810535
4722164
4680138
5895932
5889965
Michelson.
Fabry and Perot.
Fabry and Perot.
I Fabry
and Perot.
I have spoken of an agreement to about 1 part in a million. In several
cases the confirmation was decidedly closer. In one only, that of zinc red,
did there appear an indication of a disagreement rather outside the limits
of error. My observations would point to a wavelength about 1 millionth
part greater than that of Fabry and Perot ; but in view of the difficulty of
observations with the trembler, I am not disposed to insist upon it. The
soda observations were on light from a cadmium vacuumtube in which soda
accidentally presented itself. The numbers quoted from Fabry and Perot
relate to a sodaflame.
As an example in which the ratios of wavelengths were less accurately
known beforehand, I will give some details relative to helium, beginning
with observations of Feb. 9 by the 1 mm. gauge, already referred to. The
Table I. annexed gives, in the second column the wavelengths of the various
helium lines recorded by Runge*, in the fifth the same reduced to Michelson's
scale as employed by Fabry and Perot. The third column gives the correc
tions for obliquity as calculated from the observations with the 1 mm.
apparatus already recorded, the fourth the differences from the corresponding
quantity for cadmium red. Taking, for example, the helium ray of longest
wavelength in comparison with cadmium red, we get by (5)
F = 3328 r (1 + '000084) = 303303.
These numbers should be integers, were the wavelengths accurate, and
were there no phase change. On account of the phase change as determined
from the cadmium observations, the fractional parts should be those entered
in the 7th column. The differences are trifling, except in the cases of 5016
* Astrophysical Journal, January 1896.
SOME MEASUREMENTS OF WAVELENGTHS
[31G
il !
(N rt O ip O5 O
O 00 C *C ** W i
QD l^ 1^ (H CM iH
O CD OC O C5 1 s *
I CD iC ^ rj*
. CO t^ r O5 ift <t
: ^r co oo r o <N CD
: ib oo >b >b (N r^H
O CD 00 O O5 1^ ^
r CD >o o ** Tt ^<
Sa
f
.5$2gS
: o oo o o 11 co fi
8
X
5
II
o'a
j 1 +t^+ : :
IT
in <M i i oo >o o ^ :
s
osooagsgs : :
1 1
3
o r^r os CD o
05050C30 ?P . .
<
<
H
x r? ^' 5: ~ ' " ' '
COiOOOCOCO(MrH
X
llllll
Qf^ c s ^ r
1906] WITH A MODIFIED APPARATUS 329
and 4713. The proportional corrections by which the \'s of column 5 are to
be increased are set out in column 8 expressed in millionths ; but of course
an accuracy of 4 or 5 millionths is hardly to be expected in results from a
single set of observations with the 1 mm. gauge.
In the observations (Table II.) with the 5 mm. gauge the comparisons were
with the cadmium green, for which P is assumed to be 18753'95, correspond
ing to 1481400 for cadmium red. The numbers given embody the results
of three days' observations, but they do not include the wavelengths 4713,
4472. The procedure is the same as for Table I. If the observations with
the 5 mm. gauge stood alone, we should be in doubt whether P for 5016
should be 19016'95, or 1901595. The results with the 1 mm. gauge show
that the latter alternative must be chosen. Except in this respect, the
5 mm. results are independent; and they are of course to be preferred as
presumably more accurate. The final numbers for helium are therefore those
given in column 6 of Table II.
The only further remark that I will make is that the observations on the
helium yellow (5876) are not improbably somewhat embarrassed by a com
panion of feeble luminosity which could not be separated. In the 5 mm.
apparatus the two components would be nearly but not quite in coincidence.
[1911. See further Phil. Mag. Vol. xv. p. 548, 1908; This Collection, Vol. v. Art. 327.]
317.
ON THE EXPERIMENTAL DETERMINATION OF THE RATIO
OF THE ELECTRICAL UNITS.
[Philosophical Magazine, Vol. XH. pp. 97108, 1906.]
AN able discussion of the principal determinations of the above quantity,
usually denoted by v, has been given in the Reports of the Paris Physical
Congress (1900) by H. Abraham himself a contributor to the series. This
ground it is unnecessary to retraverse, but I desire to place on record one or
two suggestions which have occurred to me but which I may probably have
no opportunity of myself putting into practice.
The most approved methods involve the construction either of a condenser
or of an electrometer, of which in the first case the capacity, and in the
second the potential, can be calculated in electrostatic measure. The first
method, on the whole, offers the greatest advantages, and I preferred it when
(about 1882, and with the advice of Prof. Stuart) the Cambridge condenser
was designed*. In this method two currents are compared by a galvano
meter. The first is that due to a given electromotive force in a resistance
whose value is known in electromagnetic measure. The second is the inter
mittent current due to the same electromotive force charging n times per
second a condenser whose capacity is known from the data of construction
in electrostatic measure. The comparison may be conducted by the aid of
Wheatstone's bridge.
There are, however, one or two matters as to which doubts may arise.
Thus it is essential that the commutator by whose action the condenser is
periodically charged and discharged, should introduce no electromotive force
on its own account. A more serious doubt hangs over the behaviour of the
galvanometer. It is assumed that this instrument indicates exactly the
mean current, whether the current be steady or intermittent. The principal
error to be feared, arising from a somewhat oblique position of the needle
* For description see J. J. Thomson, Phil. Trans. 1883, p. 711 ; Thomson and Searle, Phil.
Trans. 1890, p. 586.
1906] THE RATIO OF THE ELECTRICAL UNITS 331
and its temporary magnetization under the condensed charging currents,
would be eliminated by reversing the battery. But is it certain that the
axial magnetization remains constant, even when this axis is strictly perpen
dicular to the magnetic forces due to the currents*?
Another question relates to the leads connecting the condenser with the
remainder of the apparatus. These must themselves have capacity, and the
effect is easily allowed forf if the capacity is definite. It is here that a
doubt arises. Consider for example the coaxial cylinders of the Cambridge
condenser. When the condenser is to be in action, a leading wire is brought
into contact with the inner surface of the inner cylinder. A rupture of this
contact throws the condenser out of action ; but whatever be done with the
end of the lead, its electrical situation is not the same as before. It is only
in very special cases, if at all, that capacities can be added by simply making
contacts.
Passing on to the condenser itself, we may notice that in almost all cases
it has been necessary to provide a guardring, on the principle first intro
duced by Lord Kelvin. This leads to complications, though perhaps not
very serious ones. Thomson and Searle have shown how to allow for the
guardring in the calculation of electrostatic capacity, and how to connect it
with the bridge in the electromagnetic measurement. It is a further slight
complication that the potential is not quite the same for the guardring and
for the main part of the condenser.
It has occurred to me that a condenser, not very different from the
Cambridge one, may be so arranged as not only to dispense with the guard
ring, but also to eliminate all questions connected with the capacity of the
leads. The principle is that of the variable condenser described in Maxwell's
Electricity, 127, and further considered below. There are three outer
A, G, D and two inner cylinders B, F, the components of two pairs being of
the same length ; r and the outer surfaces of the inner cylinders and the
inner surfaces of the outer cylinders being accurately worked to the same
diameters. One pair A and B are mounted coaxially upon an insulating
base and remain undisturbed. The other parts are movable and allow
of the formation of two condensers. In the first of these the third outer
cylinder D is mounted upon A so that the inner surfaces correspond. Upon
the accurately worked top of B is placed a disk G of the same diameter, and
D is also closed above by a plate E. The leads make contact with the
cylinders A , B at their bases. Of this condenser and its leads the capacity
is unknown.
* It is possible that the difficulty arising from the uncertain behaviour of steel magnets might
be obviated by the use of a galvanometer of the socalled d'Arsonval type. The string galvano
meter of Einthoven (Drude, Ann. xn. p. 1059, 1903) would appear to be specially suitable.
t Thomson and Searle, loc. cit.
332
ON THE EXPERIMENTAL DETERMINATION OF
[317
In the second arrangement the long pair of cylinders F, G are inter
polated, G resting directly upon A and F upon B, while G is removed so as
to close F in place of B. The third outer cylinder D with its cover E now
rests upon G instead of A. In this way we obtain a second condenser.
Although its capacity is unknown, the increase of capacity is accurately that
of the intermediate cylinders F, G considered as forming parts of infinitely
prolonged wholes. That is, if I be the length, b the larger and a the smaller
radius, the increase of capacity is 1 = log (6/a).
The circumstance that in this method the smaller capacity is much
greater than that of the leads alone is scarcely an objection. In the approxi
mate formula the electromagnetic capacity is proportional to the resistance
Fig. 1.
of the opposite member of the Wheatstone quadrilateral, so that it is merely
with the difference of resistances needed in this branch that we are mainly
concerned. The resistance that must be added as we pass from one condenser
to the other can be determined with full accuracy.
The length I and the smaller diameter 2a are readily measured. The
inner diameter 26 of the outer cylinder is less easily dealt with ; and even if
the error were no greater than for 2a, it would be seriously multiplied in
log (6/a), which is approximately proportional to (6  a). In the Cambridge
apparatus the interval between the cylinders was intended to be found by
gauging the space with water, and the process is described by Thomson and
1906] THE RATIO OF THE ELECTRICAL UNITS 333
Searle (p. 600). If this plan be adopted, there is no need to measure b other
wise. If v be the included volume,
v = Trl (b 2  a 2 ),
I
and
or approximately C = .*
It is to be remarked that by this method we determine what we really
require, i.e. the mean value of b a.
In carrying out the necessary measurements there should be no difficulty
over I or a. The evaluation of v is more troublesome, and the principal
uncertainty would seem to arise out of the possible presence of airbubbles.
Thomson and Searle used a vacuum towards the later stages of the filling.
Perhaps it would be an improvement to have a vacuum (? from carbonic acid)
from the first, and to introduce the previously boiled water from below. It
would be possible, though probably more elaborate, to determine v without
water by the behaviour of included air.
The investigation of the formula for the electromagnetic measure of the
capacity as derived from observations with Wheatstone's bridge is given in
Maxwell's Electricity, 775, 776, but so succinctly that the full bearing of it
may easily be misunderstood. Thus Thomson f speaks of it as "only an
approximation," and substitutes a fuller treatment. After pointing out that
in simple circuit the combination of commutator (period T) and condenser
(capacity C) is equivalent to a resistance R, where R = T/C$, Maxwell pro
ceeds to consider the bridge arrangement. " Let us suppose that ... a zero
deflexion of the galvanometer has been obtained, first with the condenser
and commutator, and then with a coil of resistance R! in its place, then the
quantity T T [2] C will be measured by the resistance of the circuit of which
the coil R l forms part, and which is completed by the remainder of the
conducting system including the battery. Hence the resistance R, which
we have to calculate, is equal to R l} that of the resistancecoil, together with
R 2 , the resistance of the remainder of the system (including the battery),
the extremities of the resistancecoil being taken as the electrodes of the
system."
* In the Cambridge condenser 1=61 cm., 2a=23 cm., and 26  2a = Tl cm. I do not know
that these dimensions are susceptible of much improvement.
t Phil. Trans. 1883, p. 708.
+ Maxwell has 2(7 in place of C, inasmuch as he supposes the charge of the condenser to be
reversed instead of merely annulled.
334 ON THE EXPERIMENTAL DETERMINATION OF [317
" Using the notation of Art. 347 [see figure], and supposing the condenser
and commutator substituted for the conductor AC in Wheatstone's Bridge,
Fig. 2.
and the galvanometer inserted in OA, and that the deflexion of the galvano
meter is zero, then we know that the resistance of a coil, which placed in
AC would give a zero deflexion, is
(3)
The other part of the resistance, R z , is that of the system of conductors
AO, 00, AB, BC, and OB, the points A and C being considered as the elec
trodes. Hence
=
" In this expression a denotes the internal resistance of the battery and
its connexions, the value of which cannot be determined with certainty ; but
by making it small compared with the other resistances, this uncertainty will
only slightly affect the value of R^.
"The value of the capacity of the condenser in electromagnetic
measure is
.(5)"
 [2] (#, + ,) '
Apart from the difference of notation, (5) is the same as the formula
arrived at by Prof. Thomson. Maxwell's idea would appear to have been
that it makes no difference to the galvanometer in OA whether in AC we
have the resistance /,, which gives the ordinary balance, or the commutator
and condenser, provided that the condition be satisfied that the same integral
current passes from A to C in both cases f. In considering the fulfilment of
this condition we must remember that the difference of potential (A C) at
A and C under the steady current is not the same as that (A' C') to which
* In Maxwell's statement a and a are interchanged in the first term of the denominator,
t E.g. in the case of steady currents the introduction of an electromotive force into AC has
no effect, provided the resistance of that branch be so altered as to satisfy the above condition.
1906] THE RATIO OF THE ELECTRICAL UNITS 335
the condenser is charged. The latter corresponds to the rupture of AC, so
that no current there passes. The condition may be expressed :
.,,. AC
and what we have further to consider is the relation between A' G' and
AC.
Let E' be the electromotive force which must act in R^ in order to stop
the current in it. Then E' = A' C'. From another point of view the zero
current in AC may be regarded as the resultant effect of two independent
electromotive forces E, E' acting in the system composed of R l and the other
resistances. Thus
E' AG
so that
And /c.j/c.vyxujr _ * } simply.
But although the condenser method may be the best, it is not so perfect
but that a desire remains to see results so obtained confirmed otherwise.
The construction of an absolute electrometer is beset with difficulties, some
of which have been remarked upon by M. Abraham. In point of theory the
Fig. 5.
best arrangement is that described by Maxwell (Electricity, 127, 1873) in
which (fig. 5) an inner cylinder C moves coaxially in the interior of fixed
coaxial cylinders A, B. It will suffice to suppose that C and A are at
potential zero while B is at electrostatic potential B.
" The capacities of the parts of the cylinders near the [gap] and near the
ends of the inner cylinder will not be affected by the [motion] provided a
considerable length of the inner cylinder enters each of the hollow cylinders.
Near the ends of the hollow cylinders, and near the ends of the inner
cylinder*, there will be distributions of electricity which we are not yet able
to calculate, but the distribution near the [gap] will not be altered by the
motion of the inner cylinder provided neither of its ends comes near the
[gap], and the distributions at the ends of the inner cylinder will move with
* A solution of this problem for the case of two dimensions has been given by Prof. J. J. Thomson
(Recent Researches, 237, 1893).
336 ON THE EXPERIMENTAL DETERMINATION OF [317
it, so that the only effect of the motion will be to increase or diminish the
length of those parts of the inner cylinder where the distribution is similar
to that on an infinite cylinder."
Thus if a be the radius of the inner cylinder and b of the outer, the force
with which the former is drawn into B is
log(6/a)'
It appears that F depends only on the ratio of the diameters of the cylinders
B, C. Suppose for example that 2a = 2, 26 = 4 (perhaps in inches), then
log (b/a) = log 2 = 69. If the potential B correspond to 2000 volts,
2xlO"2xlO"20
v 3 x 10' 3
and F= 16 dynes, or mgs. weight. This is rather small; but since F <x B*,
we get 64 mgs. for 4000 volts, and 144 mgs. for 6000 volts.
As regards the effect of errors in the fundamental measurements 6, a, we
have if y = log (b/a),
dy _db
^
'
or with the above proportions
dy = d(26) _ d(2a)
y~'7(26)
The outer cylinder is the more difficult to measure, but a given absolute error
in it is less important. If we suppose 26 = 4 inches, and d (26) = T75 ^ inch,
and the proportional error in y is halved when we pass to that of B.
It should be borne in mind that what we have to do with here is not the
mean diameters of the cylinders, if such diameters vary.
This form of absolute electrometer was employed in the researches of
Hurmuzescu*, who mounted the cylinders on a torsion balance, so that the
motion was horizontal and not strictly axial. Some advantages are attained
in this arrangement, especially perhaps that of being able to reverse the
force and so to double the rather inadequate value of the subject of measure
ment; but upon the whole it appears to me preferable to suspend the moving
cylinder C vertically in an ordinary balance, C and the upper fixed cylinder A
* Ann. d. Chim. x. p. 433, 1897. This author erroneously attributes Maxwell's reasoning
above, by which the unknown parts of the electrical distribution are eliminated, to Biohat and
Blondlot (1886).
1906] THE RATIO OF THE ELECTRICAL UNITS 337
being in connexion with earth by wires which may be very fine*. The
force is then evaluated in gravitation measure ; and it may of course in
effect be doubled by duplicating the cylinders on the other side of the
balance.
When we come to actual design the question at once obtrudes itself as to
how long the cylinders really need to be. In theory it is easy to treat them
as infinite, but in practice some concession must be made. In particular the
weight of G must not be increased unnecessarily.
The penetration of potential arising at the gap between B and A into
the annulus between C and A is easily investigated. For practical purposes
it suffices to treat the problem as in two dimensions. If r be the radius and z
(the axial coordinate) be measured from the end of A , the potential V in the
annulus may be taken to be
F=2ff e /'*' sin
where m is an integer and H an arbitrary constant variable with m. At the
surfaces of the cylinders where r = a or r = b, V vanishes. The term whose
influence extends furthest is that for which m = 1. Li raiting ourselves to
this, we take, since e w =23'2,
F =
,
b a
showing that when z=b a the value of V is already reduced to one
twenty third part of that at z = 0. When z = 4 (b a), that is at a distance
from the end equal to four times the thickness of the annulus, this term is
attenuated 290,000 times, and it is safe to conclude that the whole dis
turbance of potential may be neglected. A similar argument applies to the
annulus between B and C ; so that a total length of 8 or 9 times the thick
ness of the annulus 8 or 9 inches in the example spoken of above should
amply suffice.
There is less objection to increasing unnecessarily the length of the fixed
cylinders, but even here there must be some limit. It is easy to see that the
prolongation of the upper cylinder A at zero potential above C, also at zero
potential, is of little importance. But the case of the charged cylinder B
requires further consideration.
For the sake of defmiteness it is convenient to suppose B provided with
a metallic bottom as in the figure annexed, and the question is as to the
* Further screens in connexion with earth may be introduced to protect the moving parts of
the balance more effectually from the influence of the electricity upon B.
R. v. 22
338
ON THE EXPERIMENTAL DETERMINATION OF
[317
effect of the finite distance I of this bottom. If we regard the bottom as a
moveable piston at the same potential (taken as zero) as the cylinder B
which it closes, we may regard it as mechanically attached
to the suspended cylinder C, in spite of electrical attach
ment to B. In this case, i.e. when C and the bottom of B
move together as a rigid body, Maxwell's argument applies
exactly as if the cylinder B were infinite. The correction
of which we are in search is therefore equal to the
electrical force of attraction which tends to draw the
piston towards C.
The potential V in the interior of B is expressible by
means of Bessel's functions in terms of the values of V
over the plane which includes the bottom of C. Thus, if
r be the radius vector and z the vertical axial coordinate
measured downwards from the above plane, we may write
V= 2AJ (kr) sinh k (I  z),
where k has in succession a series of values such that
Fig. 3.
C
V=l
b being the radius of B. Each term in the above satisfies Laplace's general
equation and reduces to zero on the walls and on the bottom of the cylinder.
If a denote the density of electricity on the bottom (z = I), we have
and for the force of attraction upon the bottom as a whole
&JJW,
dS representing an element of area. Now
dV
and thus, since the products of the various terms must vanish when integrated
over the circle r 6,
Force
= i J*
[J 9 (fcr)] 9
by a known theorem. The values of kb are 2'404, 5'520, 8'654, &c.
When z = 0, V = ZAJ (kr) sinh kl ;
1906] THE RATIO OF THE ELECTRICAL UNITS 339
and the values of A can be found if we know that of V over the whole circle
r = b. As usual we have
f * VJ (kr) rdr = A sinh kl f * J 2 (kr) rdr
Jo Jo
so that Force = 2 .,,, r ,,, .
6 2 . sinh* &/ . J,,' 2 (6)
This would determine the force if we knew the value of V over the whole
circle r = b. We know that V = 1 from r = to r = a, and that from r = a to
r = 6 it falls from 1 to 0, but we do not know the precise law of this fall.
A pretty good estimate of the integral would be made by taking V = 1 up
to a radius (a + 6) and afterwards V= 0. In this case we get*
and Force 
ft* (
If the object be merely to find an upper limit to the force of attraction,
we may suppose the value V = 1 to extend up to r = b, and this in practical
cases will not really alter the result very much. Writing a = 6 in the above
formula, we get the simple expression
Force = 2 [sinh kl]~*.
This is the force with which the piston is attracted towards C when the
potentialdifference is unity, and it expresses the excess of the force by which
C is drawn in over that which would act if I = oo . In the case above con
sidered of b = 2a, the latter is equal to about .
As has been stated, the principal value of kl is 2  404 l/b. If we suppose
that the distance between C and the bottom of B is equal to the diameter of
the latter, J=2&, and sinh (4'808) may be identified with ^e 4808 , while the
other terms corresponding to higher values of k may be neglected. Hence
Correction to force = 2e~ 9>6 = (6500) 1 .
This compares with : so that if the correction be neglected altogether the
error would be less than 1 in 2200. This error, although finally halved, is too
great. It would be necessary either to increase the value of I a little, or to
calculate the correction and allow for it.
Perhaps the weakest point in the use of an absolute electrometer on
these lines is the rather high potential required to attain the necessary sensi
tiveness. There should be no difficulty over fa mg., especially if the necessary
* See for example, Theory of Sound, 204, equation (8).
222
340 THE RATIO OF THE ELECTRICAL UNITS [317
changes could be made without taking the moving parts of the balance off
their knife edges. But even then 2000 volts would scarcely suffice, and it is
likely that 3000 or 4000 would prove necessary*.
The objection to a high potential is not so much the difficulty of obtaining
it with steadiness as the risk of a brushlike discharge through the air, the
occurrence of which would probably be fatal to the success of the measure
ments. In order to diminish the risk, the edges of the cylinders A, B, C
should be rounded off, as can be done without theoretical objection.
I scarcely know whether the necessity of measuring a high potential in
electromagnetic measure (say in volts) is to be regarded as a disadvantage in
this method of determining v. It would seem that such measurements are
needed in any case and that they constitute a separate problem.
Upon the whole, while still disposed to give the preference to the con
denser, I am of opinion that the electrometer method is worthy of further
trial.
* Hurmuzescu employed about 2000 volts.
318.
ON THE INTERFERENCERINGS, DESCRIBED BY HAIDINGER,
OBSERVABLE BY MEANS OF PLATES WHOSE SURFACES
ARE ABSOLUTELY PARALLEL.
[Philosophical Magazine, Vol. xn. pp. 489493, 1906.]
THE importance which these rings have acquired in recent years, owing
to the researches of [Lummer], Michelson, and of Fabry and Perot, lends
interest to the circumstances of their discovery. It seems to be usually
supposed that Haidinger merely observed the rings, without a full apprecia
tion of the mode of formation. Thus Mascart* writes: "C'est par ce pro
c&Ie', que Haidinger les a observe'es le premier avec une lame de mica, mais
sans en dormer la veritable explication." A reference to the original papers
will, I think, show that Haidinger, in spite of one or two slips, understood the
character of the rings very well, and especially the distinction between them
and the rings usually named after Newton and dependent upon a variable
thickness in the thin plate.
In the first memoir (Pogg. Ann. LXXVII. p. 219, 1849) the bands formed
by reflexion are especially discussed. A spiritflame with salted wick, seen by
reflexion at considerable obliquity in a mica plate, is traversed by approxi
mately straight bands running perpendicularly to the plane of incidence.
Talbot had observed phenomena in many respects similar... . But the yellow
and black lines, observed by Talbot in thin blown glass, differ in character
from the lines from mica, though both are dependent upon the interference of
light. In the case of the glass the interference is due to the fact that the
thickness of the glass is variable, and the lines are localised at the plate.
The lines from the mica behave differently. However the plate may be
turned round in its own plane, the yellow and dark lines remain perpendicular
to the plane of incidence. The two surfaces of the mica are absolutely
* Ann. de Chim. t. xxm. p. 128 (1871).
342 ON INTERFERENCERINGS OBSERVABLE ON PLATES [318
parallel to one another, and accordingly the phenomenon is the same in all
azimuths. The lines appear sharper and more distinct, the nearer the mica
be held to the eye, in contradistinction to the lines from glass which then
become more and more indistinct and finally disappear.
The bands are due to interference of light reflected at the two surfaces.
The difference of path for the rays reflected at the front and back surfaces
amounts for the bright bands to a whole number of wavelengths plus a half
wavelength, for the dark bands to a whole number of wavelengths simply.
The dark parallel lines are seen with the greatest distinctness in the
reflected light. There is then a striking contrast between the reflected
bright light and the black due to its absence, the plate being backed by a dark
ground. If the plate is held in an oblique position between the eye and the
flame, the parallel lines are seen directly, but there is a much less striking
contrast with the bright parts.
At this time Haidinger had not succeeded in seeing the complete rings,
but in a later memoir (Pogg. Ann. xcvi. p. 453, 1854) he returns to the
subject and shows how the obstacle to the incident light caused by the head
of the observer may be overcome with the aid of a glass plate inclined at 45.
The incident light on its way to the mica is reflected at the glass plate, while
on its return it traverses the plate and so reaches the eye.
The observation of the transmitted rings is of the simplest possible
character. It is sufficient to look through the plate of mica at a sheet of
white paper illuminated either from in front or from behind by the homo
geneous light of the spiritflame. The rings are complementary to those seen
by reflexion. They are, however, much less intense, being due to the inter
ference of the powerful directly transmitted light with the much feebler light
twice reflected in the interior of the plate.
The distinction between " Beriihrungsringe " and " Plattenringe " is
again emphasised, the former depending upon a variable thickness, the latter
upon a variable obliquity. We may well agree with Haidinger when he
concludes : " Die Plattenringe am Glimmer bilden also eine Classe von Inter
ferenzErscheinungen fur sich, die einfachste, die es geben kann, wie ich diess
in der vorhergehenden Zeilen mit hinreichender Evidenz nachgewiesen zu
haben glaube*."
It is interesting to remark that Haidinger's rings, rather than Newton's,
are those directly explained by the usual calculation due to Young, Poisson,
It should perhaps be noted that Haidinger omits the factor ft (refractive index) in the
expression for the retardation on which the interference depends, viz. 2/u cos 6, where e denotes
the thickness of the plate and e the angle of refraction. Also that, probably by a slip of the pen,
he speaks of the retardation as increasing with the obliquity.
1906] WHOSE SURFACES ARE ABSOLUTELY PARALLEL 343
and Airy, where plane waves of light are supposed to be incident upon a
parallel plate. The application to a plate of variable thickness cannot be
more than approximately correct. That the indirect rather than the direct
application should have been (until lately) the more familiar may be attri
buted to the great difficulty of preparing artificial surfaces of the necessary
accuracy. The demand for equality of thickness is satisfied naturally in
plates of mica obtained by cleavage, and again when a layer of water rests
upon mercury*.
There is no difficulty in repeating Haidinger's observation. The trans
mitted rings are best seen by holding the mica close to the eye (focused for
infinity) and immediately in front of a piece of finely ground glass behind
which is placed a salted Bunsen flame f. If the mica be very thin, of the
kind sold by photographic dealers perhaps '05 mm. thick, the rings are on
too large a scale. But if the plate be inclined to the line of vision, the
circular arcs are well seen and, owing to the enhanced reflexions, exhibit more
contrast than is attainable at perpendicular incidence. When it is desired to
examine the complete rings, the plate should be much thicker. I have
experimented especially with two plates, "185 mm. and '213 mm. thick, and
have observed some novel effects, evidently dependent upon the double
refraction of the mica, hitherto it would seem not taken into account.
Very cursory observation on these plates, held squarely, showed that with
the thinner '185 mm. plate the inner rings were well seen, while with the
thicker one they were not. Familiarity with Fabry and Perot's apparatus at
once suggested that the complication might be due to a double system of
rings, corresponding to the two D lines, accidentally coincident in the first
case but interfering with one another in the second. It soon appeared, how
ever, that the duplicity of sodium light was not the cause. The substitution
of a helium vacuumtube for the salted Bunsen made no material difference.
And further, calculation showed that the two soda systems would be practically
in coincidence in both cases. Thus, if we take as the mean thickness '20 mm.
and a refractive index of T5, the relative retardation is 2 x 1/5 x 20 or '60 mm.
The wavelength for soda light is 5'9 x 10~ 4 mm., so that the order of the
rings under observation is about '60 r 5'9 x 10~ 4 , or very near 1000. Now
the wavelengths of the two soda lines differ by about onethousandth part,
and thus the two ringsystems are almost in coincidence. As the thickness
increases from '20 mm., the concordance would be lost, but complete discord
ance would not occur until a thickness of '30 mm. was reached. Practically
in both cases the ringsystems may be considered to be in coincidence. But
* Nature, XLVIII. p. 212 (1893); Scientific Papers, iv. p. 54.
t According to Prof. Wood's recommendation the salting is best effected with the aid of a
piece of asbestos, previously soaked in brine, wrapped round the tube of the Bunsen and forming
a prolongation of it.
344 ON INTERFERENCERINGS OBSERVABLE ON PLATES [318
although the duplicity of the soda line is not the cause, there is in fact a
second ringsystem, owing to the double refraction of the mica. That this is
the case is easily proved with the aid of a nicol capable of rotation about its
axis. When the ringsystem of the thicker plate is examined with the nicol,
there are four positions at right angles to one another at which the inner
rings become distinct. But in adjacent positions, i.e. positions distant 90, of
the nicol the ringsystems seen are different. If one has a bright centre, the
other has a dark centre. When no nicol is used, or when the nicol occupies
positions at 45 to those above mentioned, the ringsystems interfere, and
little or nothing is visible, at any rate near the centre. When the thinner
plate is employed, the ringsystem is really double, but does not appear to be
so, since the components are approximately in coincidence. In this case the
appearance is but little altered by the use of a nicol however held.
It is only near the centre of the system that the rings are obscured when
the thicker plate is used without a nicol. Further out, the rings become
distinct enough. A closer examination shows, however, that this statement
needs qualification. Along four directions, apparently at right angles,
radiating from the centre, there are regions of no definition. These regions
are narrow, so narrow that they might at first escape observation, and they
constitute, as it were, spokes of the ringsystem. It was natural to suppose
that these spokes represented places where the rings of each system bisected
the intervals between the rings of the other system a conjecture supported
by the fact that the spokes disappeared when a nicol was introduced in the
positions suitable for rendering the inner rings distinct. The effect was
to make distinct the outer rings all round the circumference. Further con
firmation was afforded by the introduction behind the mica of crosswires and
a collimatorlens, serving to indicate a fixed direction. This was pointed at a
spoke, so that without a nicol no bands were visible in the neighbourhood of
the crosswires. The nicol was then introduced in such a position as to give
maximum distinctness, and the crosswires adjusted to coincide with the
centre of a bright band. A rotation of the nicol through 90 showed that in
the bandsystem then visible the crosswires marked the centre of a dark in
place of a bright band.
The disappearance of the rings at the places where the brightest places in
one system bisect the intervals between the brightest places of the other
system depends of course upon the width of the bright rings being not much
less than half of the complete period. If, as in the FabryPerot apparatus,
the bright rings are much narrower, both systems should become visible.
I thought therefore that it would be of interest to silver lightly on both sides
a portion of the thicker plate, the more as, apart altogether from the spokes,
the whole effect would be improved owing to the enhanced reflexions. By
the chemical method, as ordinarily used for silvering glass surfaces, I did not
1906] WHOSE SURFACES ARE ABSOLUTELY PARALLEL 345
succeed ; but there was no difficulty in getting the required deposits by the
method of electrical discharge in vacuo using a silver cathode*. With the
silvered plate the nature of the whole phenomenon, including the character of
the spokes, was evident on simple inspection. At the spokes both ring
systems could now be seen, forming a compound system of half the original
period. If we neglect the circular character of the rings, the effect may be
imitated with straight bands, as shown [below]. From a piece of striped
stuff, in which the bright bands are of width equal to about a quarter of the
complete period, a photographic negative was taken. The prints from this
negative are made with two exposures, between which the paper is slightly
rotated. In this way two systems of bands are impressed, crossing one
another at a small angle.
The fact that the two concentric ringsystems cross one another proves of
course that they cannot both be strictly circular. Complete circularity in all
* Wright, Amer. Journ. Sci. xiv. p. 169 (1877).
346 ON INTERFERENCERINGS OBSERVABLE ON PLATES, ETC. [318
would require a wavesurface in the form of two concentric spheres, and
such is not the character of optical double refraction. An exception would
occur in the case of a uniaxal crystal cut perpendicularly to the axis, which
would then be an axis of symmetry for the whole phenomenon. Another
comparatively simple case would arise if the surfaces of the plate were
parallel to the axis of a uniaxal crystal. In general one at least of the ring
systems would be elliptical as in the observations ; and it would seem that the
systems need not even be concentric. A more minute theoretical examination
might be of some interest, especially in comparison with observations upon a
plate whose optical constants were known.
319.
ON OUR PERCEPTION OF SOUND DIRECTION*.
[Philosophical Magazine, Vol. xm. pp. 214232, 1907.]
IT is some thirty years ago since I executed a rather extensive series of
experiments in order to ascertain more precisely what are the capabilities of
the ears in estimating the direction of sounds^. It appeared that when the
alternative was between right and left, the discrimination could be made with
certainty and without moving the head, even although the sounds were pure
tones. Nor was any difficulty introduced by the requirement that the ears
should be stopped at the moment when the sounds commenced.
On the other hand, if the question was whether a sound were situated in
front or behind the observer, no pronouncement could be made in the case of
pure tones. The impossibility of distinguishing front and back carries with
it further confusions relating to cases where the sound may be obliquely
situated. But with sounds of other character and notably with the speaking
voice, front and back could often be distinguished. It is understood, of
course, that the head was kept still. A slight rotation, bringing a pure tone
(originally situated exactly in front or exactly behind) to the right or the left,
gives the information that was previously lacking.
The discrimination between right and left is usually supposed to be
explicable by the greater intensity of sensation experienced by the ear which
lies nearer to the sound. When the pitch is pretty high, there is no doubt
that this explanation is adequate. A whistle of pitch / IV , preferably blown
from a gasbag, is much better heard with the nearer than with the further
ear. " A hiss is also heard very badly with the averted ear. This observation
* This paper formed the substance of the Sidgwick lecture given at Cambridge on November 10,
1906, and (except the last two or three pages) was written before the delivery of the lecture.
I have learned since from Dr L. More that three years ago at Cincinnati he made experiments
which led him to similar conclusions. It is to be hoped that Dr More will publish an account of
his work, the more as it was conducted on lines different from mine.
t Nature, xiv. p. 32 (1876); Phil. Mag. m. p. 546 (1877); Phil. Mag. xin. p. 340 (1882).
Scientific Papers, i. pp. 277, 314 ; n. p. 98.
348
ON OUR PERCEPTION OF SOUND DIRECTION
[319
may be made by first listening with both ears to a steady hiss on the right or
left, and then closing one ear. It makes but little difference when the further
ear is closed, but a great difference when the nearer ear is closed. A similar
observation may be made upon the sound of running water." In a modified
form of the experiment the ear, say the right, nearest to the falling water is
stopped with the right hand. The comparatively feeble sound then heard
may be much increased if the left hand be so held at a little distance out as
to reflect the sound into the left ear. The effect remains conspicuous even
when the hand is held out at full arm's length. Of course a reflector larger
than the hand is still more effective.
The discrimination between the right and left situations of high sounds is
thus easily explained upon the intensity theory ; but this theory becomes less
and less adequate as the pitch falls. At a frequency of 256 (middle c = c f )
the difference of intensities at the two ears is far from conspicuous. At 128
it is barely perceptible. But although the difference of intensities is so small,
the discrimination of right and left is as easy as before.
There is nothing surprising in the observation that sounds of low pitch
are nearly as well heard with the further as with the nearer end. When the
wavelength amounts to several feet, it is not to be expected that the sound
(originating at a distance) could be limited to one side of the head. The
question is well illustrated by calculations relating to the incidence of plane
waves upon a rigid spherical obstacle, and the results may conveniently be
repeated here from Theory of Sound, 328. [See also p. 151.]
2irc
X
M
F+iG
*****
1
522 + 139 1
294

1
159 '484 t
260
430 217 t
232
1
668 + '238 f
503
1
1
440 '303 1
285
+ 322 365 t
237
1
797 + '234 1
690
2
1
250 + 506
318
 164 577 t
365
In this table 2?rc is the circumference of the sphere, and X is the wave
length of the sound. The symbol /* denotes the cosine of the angle at the
centre of the sphere between the direction of the sound (fi = 1) and the point
1907] ON OUR PERCEPTION OF SOUND DIRECTION 349
upon the sphere at which the intensity is to be reckoned. F+ iG denotes
the (complex) condensation, and F'*+G 2 the intensity. In the present
question of a sound situated say to the right of the observer, the intensity at
the right ear corresponds to p = + 1, and at the left ear to /* = 1. In the case
of the head the circumference (2?rc) may be taken at about 2 feet or a little
less, so that 27rc/\ = ^ corresponds about to middle c, or frequency 256. It
will be seen that the difference of intensities for /* = + 1 is only about 10 per
cent, of the whole intensity.
For still smaller values of 27rc/X, i.e. in the present application for still
graver notes, the difference of intensities may be adequately expressed by a
very simple formula. It appears that
while at the same time the total value of F z + G* approximates to '25.
A fall in pitch of an octave thus reduces the difference of intensities 16
times. At frequency 128 the difference would be decidedly less than one
per cent, of the whole ; and from this point onwards it is difficult to see how
the difference could play any important part.
So far as I am aware no explanation of the above difficulty, emphasised in
1876, has been arrived at. A few months since I decided to repeat, and if
possible to extend, the observations, commencing with frequency 128. Two
forks of this pitch were mounted in the open air at a considerable distance
apart, and were electrically maintained, one driving the other. In connexion
with each was a resonator which could be put out of action by interposing
(without contact) the blade of a knife or a piece of card. An observer with
eyes closed, placed between the two forks and so turned as to have them upon
his right and his left, could tell with certainty which resonator was in action.
The ears may be open all the time ; or, what is in some respects better, they
may be closed while the changes at the resonators are being made, and after
wards opened simultaneously. If one ear be opened, the sound will appear to
be on that side ; but when the second ear is also opened, the sound assumes
its correct position, whether or not this involves a reversal of the earlier
judgement. When the sounds were in front and behind, instead of to the
right and the left, several observers agreed that a discrimination between
front and back could not be made.
In another method of experimenting a single resonator and fork (which
need not be electrically maintained) suffice, but more than one assistant may
be required. The observer, either on his feet, or more conveniently seated
upon a rotating stool, is turned round until he loses his bearings. There is no
difficulty in this*, but some precautions are needed to prevent the bearing
* The process is aided by the illusion of a reverse rotation when the real rotation has
stopped.
350 ON OUR PERCEPTION OF SOUND DIRECTION [319
being afterwards recovered. The wind may act as a telltale. It is often
necessary to cover the eyes with the hands as well as to close the eyelids in
order sufficiently to exclude the light. Until all is ready for a judgement,
the ears are kept closed by pressure with the thumbs, and it is usually
advisable to keep the thumbs in motion and thus to cause miscellaneous
noises loud enough to drown any residue of the sound under observation.
Pure tones of pitch 128 and 256 yielded by this method results in agreement
with those already described.
The turntable facilitates observation upon the question as to the relative
loudness with which a sound is heard according as the source is on the same,
or the opposite, side as the ear in use. In my own case I thought I could
detect an advantage when the source was on the same side as the open ear,
while others could detect no difference. This relates to pitch 128. At 256
the advantage is quite marked.
In considering whether the discrimination between right and left at pitch
128 can really be attributable to the small intensitydifference, it occurred to
me that, if so, the judgement might perhaps be disturbed by the introduction
of an obstacle, such as a piece of board, near the head of the observer and on
the same side as the sound. But it was found that no mistakes could thus be
induced, although in each trial the observer did not know whether the board
were in position or not. Another circumstance, unfavourable to the intensity
theory, may also be mentioned. It was found that the observer on the turn
table could sometimes decide between the right and the left before un
stopping his ears.
The next step in the investigation appeared to be the examination of pure
tones of still graver pitch. A globe, such as are sold to demonstrate the
combustion of phosphorus in oxygen gas, was sounded with the aid of a
hydrogen flame*. Careful observation revealed little or no trace of overtones.
The frequency was about 96 vibrations per second, thus by the interval of the
Fourth graver than the 128 forks. At the temperature of the observations
this would correspond to a wavelength of about 12 feet. If we use this value
in the formula already given, we find for the proportional difference of in
tensities on the two sides of a sphere of 2 feet circumference only about
2 parts per thousand.
Observation in the open air showed that there was no difficulty whatever
in deciding whether this low sound was on the right or the left. Several
observers agreed that the discrimination was quite as easy as in the case of
forks of pitch 128. On the other hand, as was to be expected, the front and
back situations could not be discriminated.
At this stage a reconsideration of theoretical possibilities seemed called
for. There could be no doubt but that relative intensities at the two ears
* Phil. Mag. vii. p. 149 (1879); Scientific Papert, i. p. 407.
1907] ON OUR PERCEPTION OF SOUND DIRECTION 351
play an important part in the localization of sound. Thus if a fork of what
ever pitch be held close to one ear, it is heard much the louder by that ear,
and is at once referred instinctively to that side of the head. It is impossible
to doubt that this is a question of relative intensities. On the other hand, as
we have seen, there are cases where this explanation breaks down. When a
pure tone of low pitch is recognized as being on the right or the left, the only
alternative to the intensity theory is to suppose that the judgement is
founded upon the difference of phases at the two ears. But even if we admit,
as for many years* I have been rather reluctant to do, that this difference of
phase can be taken into account, we must, I think, limit our explanation
upon these lines to the cases of not very high pitch. For what is the differ
ence of phase at the two ears when a sound reaches the observer say from
the right ? It is easy to see that the retardation of distance at the left ear is
of the order of the semicircumference of the head, say one foot. At this rate
the retardation for middle c (c = 256) is nearly one quarter of a period ; for
c" (512) nearly half a period; for c'" (1024) nearly a whole period, and so on.
Now it is certain that a phaseretardation of half a period affords no material
for a decision that the source is on the right rather than on the left, seeing
that there is no difference between a retardation and an acceleration of half
a period. It is even more evident that a retardation of a whole period, or of
any number of whole periods, would be of no avail. In the region of some
what high pitch a judgement dependent upon phase would seem to be hardly
possible, especially when we reflect that the phasedifferences enter by degrees,
rising from zero when a sound is directly in front or behind to a maximum or
minimum in the extreme right and left positions.
As to whether there is any difficulty in localizing to the right or left a
tone of pitch 512, an early observation, conducted with the aid of two forks
and resonators of this pitch, gave an answer in the negative. The localization
was as easy and as distinct at pitch 512 as at pitch 256. But it is quite
possible that 512 is not sufficiently near the particular pitch for which the
retardation would have the value of precisely half a period.
The calculations for a spherical obstacle already quoted give some informa
tion upon this point. From the values of F+iG corresponding to p,= 1 for
the case of 2?rc/\ = 1, it appears that there is approximate opposition of
phase. A closer examination and comparison of the three cases shows that
exact opposition will occur at a somewhat higher value, say 2irv/X~l*l.
Calculating from this and taking 2?rc at 2 feet, I find at the temperature of
the observations a pitch about a minor third above 512, as that corresponding
to a phasedifference of half a period.
Naturally, in its application to the head, the calculation is not very
trustworthy, and I thought it important to make sure by actual experiment
* Conf. Theory of Sound, 385.
352 ON OUR PERCEPTION OF SOUND DIRECTION [319
that there is no pitch in this region for which the discrimination of right and
left is at all uncertain. Tuningforks not being available, I fell back upon
" singingflames," i.e. tubes, usually of metal and about 1 inch in diameter,
maintained in vibration by hydrogen flames. In order to eliminate overtones,
the tubes were provided near their centres with looselyfitting rectangular
blocks about two diameters long, held in position by the friction of attached
springs. In this way and with the precaution of not sounding them more
loudly than was necessary, the tones were, it is believed, sufficiently pure.
Trials were made in the open air on many occasions, the pitch ranging in all
from d' to g", and there was never the smallest suspicion of a difficulty in
discriminating right and left. In the region from c" to g" of special interest,
the pitch was varied by half semitones with the aid of sliding prolongations
of the tube. During most of the observations the listener was placed upon
the rotating stool and was in ignorance of the real position of the source.
This precaution is of course desirable; but after a good deal of practice I
found that I was able to trust the direct sensual impression. In the case of
right or left the impression is always distinct and always correct; but in
trying to discriminate front and back there is usually no distinct impression,
and when there is it often turns out to be wrong.
We may fairly conclude that in this region of pitch (above c" = 512) the
discrimination of right and left is not made upon the evidence of phase
differences, or at any rate not upon this evidence alone. And this con
clusion leads to no difficulty ; for, as has already been explained, the difference
of intensities at the two ears gives adequate foundation for a judgement. It
would seem that at high pitch, above c", the judgement is based upon in
tensities; but that at low pitch, at any rate below c (128), phasedifferences
must be appealed to.
It remained to confirm, if possible, the suggestion that not only are we
capable of appreciating the phasedifferences with which sounds of equal
intensity may reach the two ears, but that such appreciation is the founda
tion of judgements as to the direction of sounds in particular of the right
and left effects. The obvious method is to conduct to the two ears separately
two pure tones, nearly but not quite in unison. During the cycle, or beat,
the phasedifferences assume all possible values ; and the mere recognition of
the cycle is evidence of some appreciation of phasedifferences. Experiments
on these lines are not new. In 1877 Prof. S. P. Thompson* demonstrated
"the existence of an interference in the perception of sounds by leading
separately to the ears with indiarubber pipes the sounds of two tuningforks
struck in separate apartments, and tuned so as to ' beat ' with one another
the ' beats ' being very distinctly marked in the resulting sensation, although
the two sources had had no opportunity of mingling externally, or of acting
* Phil. Mag. Nov. 1878.
1907] ON OUR PERCEPTION OF SOUND DIRECTION 353
jointly, on any portion of the aircolumns along which the sound travelled.
The experiment succeeded even with vibrations of so little intensity as to be
singly inaudible." And in an observation of my own *, where tones supposed
to be moderately pure were led to the ears with use of telephones, a nearly
identical conclusion was reached. But although the cycle was recognized, in
neither case apparently was there any suggestion of a right and left effect.
In repeating the experiment recently I was desirous of avoiding the use
of telephones or tubes in contact with the ears, under which artificial condi
tions an instinctive judgment would perhaps be disturbed. It seemed that
it might suffice to lead the sounds through tubes whose open ends were
merely in close proximity one to each ear, an arrangement which has the
advantage of allowing the relative intensities to be controlled by a slight
lateral displacement of the head towards one or other source. Two forks of
frequency 128, independently electrically maintained, were placed in different
rooms. Associated with each was a suitably tuned resonator from the interior
of which a composition (gas) pipe led the sound through a hole in a thick wall
to the observer in a third room. With the aid of closed doors and various
other precautions of an obvious character, the sounds were fairly well isolated.
But each resonator emits a rather loud sound into its neighbourhood, a little
of which might eventually reach the other. To eliminate this cause of
disturbance more completely, a second resonator } of like pitch was employed
in association with each fork, so situated that the phases of vibration in the
two resonators were opposed. By a little adjustment it was possible to
provide that but little sound radiated externally from the combination,
though the internal vibrations might be as vigorous or more vigorous than
when one resonator was employed alone. These arrangements were so
successful that when one fork singly was in action, the sound was impercept
ible from the tube belonging to the other, even though the open end were
pressed firmly into the ear, by which the effect is enormously increased. The
open ends of the two pipes may thus be regarded as sources of sound of
constant intensity.
In the greater number of experiments the observer, leaning over a table
for the sake of steadiness, placed his head between the pipes, which were at
such a distance apart that one or two inches separated the open ends from the
adjacent ears. At the very first trial on July 31, the period of the cycle
being 5 seconds, Lady Rayleigh and I at once experienced a distinct right
and left effect, the sound appearing to transfer itself alternately from the one
side to the other. When the effect was at its best, the sound seemed to lie
entirely on the one side or on the other.
* Phil. Mag. n. p. 280 (1901) ; Scientific Papers, iv. p. 553.
t Three out of the four resonators consisted of "Winchester" bottles from which the necks
had been removed.
R v. 23
354 ON OUR PERCEPTION OF SOUND DIRECTION [319
The beat may be slowed down until it occupies 40 or even 70 seconds,
thus giving opportunity for more leisurely observation. The position of the
head should be so chosen that the right and left effects are equally distinct.
Under these circumstances it is found that the sound seems to be pre
dominantly on the right or on the left for almost the whole of the cycle, the
transitions occupying only small fractions of the whole time. The observa
tions may be made with the ears continuously open, or, as in some of the out
door experiments, the ears may be opened and closed simultaneously at short
intervals. It is perhaps better still, keeping the ears open, to close period
ically with the thumbs the open ends of the tubes from which the sounds
issue. When the tubes are closed, no sound is audible. It should be said that
although the best results require the position of the head to be carefully
chosen, right and left effects are perceived through a considerable range. It
is only necessary that the intensities be approximately equal.
These results are quite decisive, if we can assume that the sounds were
sufficiently isolated that nothing appreciable could pass from the open end
of a pipe to the wrong ear. It was easy to verify that when one pipe and the
opposite ear were closed, next to nothing could be heard ; but it may perhaps
be argued that the test is not delicate enough. The risk of error from this
cause is diminished by approximating the open ends to their respective ears.
Many experiments of this kind were made, but without influencing the
results. Finally, short lengths of rubber tubing were provided, by means of
which the ears could be connected almost airtight with the pipes. In this
case, to avoid being deafened, it was necessary to reduce the sounds by with
drawing the resonators from their respective forks. The right and left effect
remained fully marked. Another argument to show that the effects cannot
be explained by sounds passing round the head will be mentioned presently.
A question of great importance still remains to be considered. Before the
laboratory experiments can be accepted as explanatory of the discrimination
of right and left when a single sound is given in the open, it is necessary to
show that in the former the sensation of right (say) is associated with a phase
difference such that the vibration reaching the right ear leads. There was no
difficulty in obtaining a decision. While one observer listens as described for
right and left effects, a second observes the maxima and minima of the beat
as heard by one ear situated symmetrically with respect to the two sources.
In the case of sounds of higher pitch to be considered later some precaution
is required here ; but for the present sounds, corresponding to a wavelength
of nearly 9 feet, there is no difficulty. Under good conditions the minimum,
represented by a silence, is extremely well marked, and can often be signalled
to within half a second. This signal, corresponding to opposition of phases,
gives the required information to the first observer. If a signal for the
1907] ON OUR PERCEPTION OF SOUND DIRECTION 355
maximum, representing phaseagreement, is desired, it is best made by
halving the intervals between the silences.
The results can be stated without the slightest ambiguity. The transitions
between right and left effects correspond to agreement and opposition of
phase, not usually recognized by the first observer as maxima and minima of
sound. When the vibration on the right is the quicker, the sensation of
right follows agreement of phase, and (what is better observed) the sensation
of left follows opposition of phase. And similarly when the vibration on the
left is the quicker, the left sensation follows agreement, and the right follows
opposition of phase. The question which fork is vibrating the quicker is
determined in the usual way by observing the* effect upon the period of the
beat of the addition or removal of a small load of wax. If for example the
beat is slowed by loading the right fork, we may be sure that that fork was
originally the quicker. A large number of comparisons of this kind have
been made at various times, and in no case (at this pitch) has the rule been
violated. It is not a little remarkable that by merely listening to right and
left effects aided by signals giving the moment of phaseopposition, it is
possible conversely to pronounce which fork is the quicker, although the
difference of frequencies may not exceed '02 vibration per second.
It should not pass unnoticed that the laboratory experiments cover a
wider field than the observations in the open. In the latter case, if the
single sound of pitch 128 is in front or behind, there is agreement of phases
at the two ears. As the position becomes more and more oblique, the phase
difference increases ; but it can never exceed a moderate amount, about one
eighth of a period, which is attained when the position is precisely to the left
or right. Phasedifferences in the neighbourhood of half a period do not
occur. From the laboratory experiments it appears that the right and left
effects are not subject to this limitation ; but that, for example, a right effect
is experienced when the vibration reaching the right ear leads, whether the
amount of the lead be small or whether it approaches the half period.
The right and left or, as I shall sometimes say for brevity, the lateral,
sensations observable in this way are so conspicuous that I was curious to
inquire how I had contrived to miss them in the earlier experiments with
telephones already alluded to. The apparatus was the same as before. In
the neighbourhood of the electromagnet driving each fork (128) was placed a
small coil of insulated wire whose circuit was completed through a telephone.
The double wires connected to the telephones were passed through the
perforated wall. In order to weaken the higher overtones, thick sheets of
copper intervened between the electro magnets and the coils. But when the
telephones were held to the ears, the sounds were perceived to be of a more
mixed character than I had expected ; and I am forced to the conclusion that
I must formerly have overestimated their approximation to the character of
232
356 ON OUR PERCEPTION OF SOUND DIRECTION [319
pure tones. Although the cycle could be recognized, a distinct lateral effect
was not perceived, and the failure was evidently connected with the composite
character of the sounds. By loading the disks of the telephones with penny
pieces (attached at the centres with wax) the higher components could be
better eliminated. It was then possible to fix the attention upon the funda
mental tone and to recognize its transference from left to right during the
cycle. But the effect was by no means so conspicuous as with the tubes, and
might perhaps be missed by an unprepared observer*.
The subject now under consideration is illustrated by a curious observation
Accidentally made in the course of another inquiry. A large tuningfork of
frequency about 100, mounted upon a resonancebox, was under examination
with a Quincke tube. This consisted of a piece of lead pipe more than one
quarter of a wavelength long, one end of which was inserted into the
resonancebox. At a distance of oneeighth of a wavelength from the outer
end a lateral tube was attached which communicated with one ear by means
of an indiarubber prolongation. When the second ear was closed, it appeared
to make no difference to the sound whether or not the outer end of the
Quincke tube was closed with the thumb. But when the second ear was
open, marked changes in the sound accompanied the opening and closing of
the Quincke tube. On the view hitherto held, it would appear very para
doxical that a change not affecting the sound heard in either ear separately
should be able to manifest itself so conspicuously. It was easily recognized
that the alterations observed were of the nature of right and left effects, and
that they could be explained by the local reversal of phase which accompanied
the closing of the Quincke tube.
The conclusion, no longer to be resisted, that when a sound of low pitch
reaches the two ears with approximately equal intensities but with a phase
difference of onequarter of a period, we are able so easily to distinguish at
which ear the phase is in advance, must have far reaching consequences in
the theory of audition. It seems no longer possible to hold that the vibratory
character of sound terminates at the outer ends of the nerves along which the
communication with the brain is established. On the contrary, the processes
in the nerve must themselves be vibratory, not of course in the gross
mechanical sense, but with preservation of the period and retaining the
characteristic of phase a view advocated by Rutherford, in opposition to
Helmholtz, as long ago as 1886. And when we admit that phasedifferences
* Subsequently by a much heavier loading (53 gms.) the telephoneplates were tuned approxi
mately to pitch 128, as could be verified by tapping them with the finger. To find room for these
extra loads, the earpieces of the telephones had to be modified. The sounds now heard were
very approximately pure tones, and the lateral effects were as distinct as those observed when the
sounds were conveyed through tubes. It is easy to understand that considerable complication
must attend an accompaniment of octave and higher harmonics, which would transfer themselves
from right to left more rapidly than the fundamental tone.
1907J ON OUR PERCEPTION OF SOUND DIRECTION 357
at the two ears of tones in unison are easily recognized, we may be inclined
to go further and find less difficulty in supposing that phaserelations
between a tone and its harmonics, presented to the same ear, are also
recognizable.
The discrimination of right and left in the case of sounds of frequency 128
and lower, so difficult to understand on the intensity theory, is now satis
factorily attributed to the phasedifferences at the two ears. The next
observations relate to pure tones of pitch 256. Two large forks of this pitch
were used, such as are commonly to be found in collections of acoustical
apparatus. Tuned with wax so as to give beats of 3 or 4 seconds period,
they may be held (after excitation) by their stalks one to each ear, preferably
by an assistant. The sensation of transference from right to left was fully
marked, and when the conditions were good, especially in respect of equality
of intensities at the two ears, the whole of the sound seemed to come first
from one side and then from the other. The method of holding in the fingers
is satisfactory as regards the isolation of the sounds. Practically nothing of
either fork can be heard by the further ear. But there is a little difficulty in
maintaining quite constant the relative positions of forks and ears.
In another arrangement, which has certain advantages, the forks are
mounted, stalks upwards, on a sort of crown, in such a fashion that the free
ends of the forks are about opposite the ears. A sketch by Mr Enock is re
produced in the figure (p. 358). If the crown be sufficiently large, it can be
adapted to various sized heads with the aid of pads. At the back, attached to
the crown, is a forked tube of brass, symmetrically shaped, whose open ends
abut upon the faces of the forks. The short limb, forming the stalk, is pro
longed by indiarubber tubing and so connected with the ear of the assistant
observer. If the forks are vibrating equally, a welldefined silence marks the
moment of phaseopposition. To excite and maintain the vibrations, a violin
bow is employed in the usual manner.
Very good observations may be made in this way if the vibrations of the
forks are equally and sufficiently maintained. The assistant, listening
through the forked tube, is able to give a sharp signal at the moment of
phaseopposition. A transition in respect of right and left effect occurs at
this moment, and the sequence rule already stated defining whether the
transition is from right to left or from left to right is found to be obeyed.
There are some advantages, of course, in an experimental arrangement
allowing the sounds to be uniformly maintained. As in Helmholtz's vowel
investigations, the 256 forks can be driven by the 128 interrupter forks
already employed, and in each case the frequency of the driven fork is the
exact double of the frequency of the driving fork. The observations may be
made in two ways. Either the 256 forks may themselves be brought close to
the ears, leading wires being conveyed through the wall ; or, what is on the
358
ON OUR PERCEPTION OF SOUND DIRECTION
[319
whole preferable, the method employed for the 128 per second tones may be
followed. In this case each 256 fork is associated with two resonators,
vibrating in opposite phases, with one of which the pipe leading to the
observation room is connected. The isolation was good, each sound being
inaudible through the tube provided for the other.
Excellent results were obtained in this way. With good adjustment the
transitions between right and left were sharply marked and the sequence rule
already formulated was obeyed. It is again to be noted that right and left
effects are observable in the neighbourhood of phaseopposition, a situation
which does not arise when one sound in the open influences both ears.
Unless the open ends are pretty close to the ears, the sound has more
tendency to travel round the head to the wrong ear than was observed at
pitch 128. This may raise a question whether, after all, the right and left
effects may not be due to small differences of intensities at the two ears
varying periodically. The best answer to this objection is to consider what
would be the consequence of such invasions. Suppose that at one moment
the vibration on the right is in advance by one quarter period, so that a full
right sensation is being experienced. The retardation in travelling round
the head will at this pitch be about one quarter period, so that the sound
starting from the right in advance will on arrival at the left be in approximate
phaseagreement with the principal sound there. On the other hand, the
sound starting from the left, already a quarter period in arrear, will on
arrival at the right be in approximate phaseopposition with the principal
sound on the right. The effect of travel round the head is therefore to
1907] ON OUR PERCEPTION OF SOUND DIRECTION 359
augment the sound on the left and to dimmish the sound on the right. This
would evidently tend to cause a sensation of sound on the left, and cannot
therefore be the explanation of the observed sensation of a sound on the right.
The same considerations will apply, if in less degree, to sounds of pitch 128,
and to sounds somewhat higher in pitch than 256.
The next sounds to be experimented upon in order of pitch were from
forks giving e' of 320 vibrations per second. These could not be driven from
the 128 per second interrupters, and were merely held to the ears in the
fingers of an assistant. The right and left effect was very marked, but there
was a little difficulty at first in fixing the moment of phaseopposition. After
a few trials it became sufficiently clear that the rule was the same as at the
lower pitch, viz. that the quicker fork asserts itself after the maximum of the
beat, corresponding to phaseagreement.
From this point, as the pitch rises, the observations become more difficult,
partly no doubt on account of purely experimental complications, but also, I
believe, because the effects are themselves less well marked. From two forks
of pitch g', electrically driven from the 128 interrupters and provided with
resonators in connexion with pipes, fairly distinct right and left effects were
obtained, but at first there were discrepancies as to which effect followed
phaseopposition. These appear to have been due to faulty observation of
the phase of opposition. As might have been anticipated, the moment of
silence representing phaseopposition varies with the position in the room of
the ear of the assistant observer. To secure a satisfactory signal at the
higher pitches, this position requires to be carefully chosen. In the later
experiments a resonator was always employed, whose mouth was symmetrically
situated with respect to the open ends of the pipes, which are the proximate
sources of sound, connexion with the ear of the assistant observer being
through a suitable rubber tube. After this there was no ambiguity, the rule
of the lower pitch being uniformly followed. But when the open ends of the
pipes were not very close to the ears, perhaps 2 inches distant, the right and
left effects seemed to two observers (including myself) not only to be rather
obscured, but to be concentrated into the neighbourhood of phaseopposition.
A third observer, however, heard the right and left effects more strongly, and
with less apparent concentration towards phaseopposition.
On the theory that passage of sound round the head had something to do
with these complications, the open ends of the pipes were brought much
closer to the ears, but without fitting airtight, the resonators being re
adjusted so as to diminish the loudness. In this condition of things the two
observers experienced the right and left effect more normally and without
special concentration in the region of phaseopposition. But the observation
is certainly more difficult than at lower pitches, and I believe that the effects
are really less pronounced.
360 ON OUR PERCEPTION OF SOUND DIRECTION [319
Experiments similarly conducted with forks of pitch c" (512) gave results
of the same character. When the open ends of the pipes were quite close to
the ears, the right and left effect was pretty good and fairly distributed. In
this three observers concurred. But a slight withdrawal of the pipes intro
duced confusion, the extent of which, however, appeared to vary with the
observer. In all cases the right and left effects, when sufficiently marked to
be observed, obeyed the sequence rule.
At pitch e' (640) the results were not very different. The open ends of
the pipes being close to the ears, but not fitted airtight, only pretty good
right and left effects could be observed, and these appeared to be crowded
towards phaseopposition. The sequence rule was obeyed.
Finally, trials by the same method were made with forks of pitch g"
(6 x 128 = 768). No particular difficulty was encountered in satisfying the
necessary experimental conditions, but the results were of a nondescript
character. Even when the open ends of the pipes were close to the ears, I
could not satisfy myself that I experienced any right and left effect. Another
observer thought he heard a little. It seems clear that at any rate the limit
was being approached.
It will be understood that some of these observations were not made
without difficulty. Probably an experimenter new to the work would feel
misgivings with respect to some even of the easier decisions. But all the
more important results have the concurrence of at least three observers*.
I regard it as established that up to pitch g' phasedifferences are attended
with marked lateral effects. They are probably the principal basis on which
discriminations of right and left are founded at any rate below c (256).
As has already been suggested, it was reasonable to anticipate that phase
difference would cease to avail as an indicator at high pitch. Up to about e'
the conditions are favourable. At this pitch the phasedifference at the ears
affected by a distant sound increases from zero when the source is in front or
behind to a maximum of a quarter period (in one or other direction) when the
source is on the right or the left in the line of the ears. This is the phase
difference for which one would expect the lateral sensation to be most
intense, so that up to this pitch the lateral sensation would keep step with
the true lateralness of the source. At a point somewhat higher in pitch it
would seem that complications must enter. The maximum sensation (corre
sponding to a phasedifference of a quarter period) would occur while the
source was still in an oblique position, and the sensation of lateralness would
diminish while the true lateralness was still increasing. At a pitch in the
neighbourhood of e" (640) the maximum phasedifference would rise to half a
period, a phasedifference which could not give rise to lateral sensation at all.
* Lady Rayleigh, Mr Enock, and myself.
1907] ON OUR PERCEPTION OF SOUND DIRECTION 361
Thus, although there might be right and left sensations from sources obliquely
situated, these sensations would fail when most needed, that is when the
source is really in the line of the ears. In this case a perception of phase
differences would seem to do more harm than good. At a pitch a little
higher, ambiguities of a misleading and dangerous kind would necessarily
enter. For example, the same sensations might arise from a sound a little on
the left and from another fully on the right.
On the whole it appears that the sensation of lateralness due to phase
difference disappears in the region of pitch where there would be danger of its
becoming a misleading guide. It is not suggested that there is any precise
numerical coincidence. If it were a question of calculating a pitch precisely,
it might be necessary to look beyond the size of modern adult heads to those
of our ancestors, perhaps in a very distant past. It is fortunate that when
difference of phase fails, difference of intensity comes to our aid. Perhaps it
is not to be expected that we should recognize intuitively the very different
foundations upon which our judgment rests in the two cases.
A rather difficult question arises as to whether in the laboratory experi
ments it is possible to distinguish the phases of agreement and of opposition.
Not unnaturally perhaps, the apparent movements of the sound from right to
left and back are liable to be interpreted as parts of a general movement of
revolution, so that, for example, phaseagreement may correspond to the front
and phaseopposition to the back position. In a particular case the question
is as to the direction of the revolution, whether clockwise or counterclockwise.
With respect to this, my observers frequently disagreed, from which I am
disposed to conclude that in these experiments phaseagreement and opposi
tion are not definitely connected with front and back sensations.
At this point there seems to be some discrepancy with the observations of
Prof. S. P. Thompson, who found * that " when two simple tones in unison
reach the ears in opposite phases, the sensation of the sound is localized at the
back of the head." In Prof. Thompson's most striking experiment a micro
phone is connected in series with a battery and two similar Bell telephones,
one of the telephones being provided with a commutator by which the
direction of the current through it can be reversed. When the current flows
similarly through the telephones, a light tap near the microphone is heard in
the ears; but when the current is reversed in one of them a sensation is
experienced " only to be described as of some one tapping with a hammer on
the back of the skull from the inside." In some (rather inadequate) experi
ments I have not succeeded in repeating this observation.
The other branch of the subject, which I had hoped to treat in this paper,
is the discrimination between the front and back position when a sound is
* Phil. Mag. November 1878, p. 391.
362 ON OUR PERCEPTION OF SOUND DIRECTION [319
observed in the open ; but various obstacles have intervened to cause delay.
Among these is the fact that (at 64 years of age) my own hearing has
deteriorated. Thirty years ago it was only pure tones, or at any rate musical
notes free from accompanying noises, that gave difficulty. Now, as I find to
my surprise, I fail to discriminate, even in the case of human speech. It is to
be presumed that this failure is connected with obtuseness to sounds of high
pitch, such as occur especially in the sibilants. For some years I have been
aware that I could no longer hear as before many of the high notes from
birdcalls, such as I employ with sensitive flames for imitating optical
phenomena. If, as seems the only possible explanation, the discrimination of
front and back depends upon an alteration of quality due to the external ears,
it was to be expected that it would be concerned with the higher elements of
the sound. In this matter it would not be surprising if individual differences
manifested themselves, apart from deafness. A " paddlebox " formation of
the external ear, if not ornamental, may have practical advantages.
My assistant, Mr Enock, is able to make discriminations between front
and back, though I think not so well as I used to be able to do. Experiments
of this kind are easily tried on a lawn in the open, the observer closing his
eyes and ears, with if necessary a movement of the thumbs over the latter to
drown residual external sounds. At the moment of observation the ears are
of course opened. In observing sounds from sources not conveniently move
able, such as the ticking of a clock, the rotating stool is useful.
As had been expected, Mr Enock's judgment was liable to be upset by
the operation of little reflecting flaps situated just outside the ears. The
arrangement was that of Prof. Thompson's " pseudophone"*, whereby the
reflectors, whose planes were at an angle of 45 with the line of the ears,
could be rotated in a manner unknown to the observer about that line as
axis. In my use of it the two reflectors were always adjusted symmetrically.
Thus, if the reflectors were so turned as to send into the ears sounds from the
front, no mistakes were made, as if the action were cooperative with the
natural action of the external ears. On the other hand, if the collars carrying
the reflecting flaps were turned through 180 so as to reflect into the ears
sounds from behind, frequent mistakes ensued. I hope before long to be able
to confirm and extend these observations.
In conclusion, I will remark that the facts now established have a possible
practical application. In observing fogsignals at sea it is of course of great
importance to be able to estimate the bearing. If a sound is of sufficiently
long duration (5 or 6 seconds), it is best by turning the body or head to
bring it apparently to the right and to the left, and to settle down into the
position facing it, where no lateral effect remains. If, as for most fogsignals,
" Phil. Mag. November 1879.
1907] ON OUR PERCEPTION OF SOUND DIRECTION 363
the duration be decidedly less than this, it may be preferable to keep still ;
but we are then liable to serious errors, should the signal happen to come
from nearly in front or nearly behind. A judgment that the signal is to the
right or left may usually be trusted, but a judgment that it comes from in
front or behind is emphatically to be distrusted. If, for example, the sound
seems to come from a position 45 in front of full right, we must be prepared
for the possibility that it is really situated 45 behind full right. A com
bination of 3 or 4 observers facing different ways offers advantages. A
comparison of their judgments, attending only to what they think as to
right and left and disregarding impressions as to front and back, should lead
to a safe and fairly close estimate of direction.
[1911. See further on the subject of phaseperception Myers and Wilson, Proc. Roy. Soc.
Vol. 80 A, p. 260, 1908; Rayleigh, ibid. 83 A, p. 61, 1909.]
320.
ACOUSTICAL NOTES.
[Philosophical Magazine, Vol. xm. pp. 316333, 1907.]
Sensations of Right and Left from a revolving Magnet and Telephones. Multiple Har
monic Resonator. TuningForks with slight Mutual Influence. Mutual Reaction
of Singing Flames. Longitudinal Balance of TuningForks. A TuningFork Siren
and its Maintenance. Stroboscopic Speed Regulation. Phonic Wheel and Com
mutator.
Sensations of Right and Left from a revolving Magnet and Telephones.
AMONG the methods available for the production of a pure tone in a
telephone circuit is that where the electromotive force has its origin in the
revolution of a small magnet about an axis perpendicular to its length, the
magnet acting inductively upon a neighbouring coil which forms part of the
telephone circuit. It was by experiments made partly in this manner that
I formerly* determined the minimum of current necessary for audibility in
the telephone. In connexion with recent work upon the origin of the lateral
sensation in binaural audition f I have again employed this method, and I
now propose to give a brief account of the results, which were not available
in time for incorporation in the paper just cited.
The object of the experimental arrangements is the separate presentation
to the two ears of pure tones, practically in unison, in such a manner as to
allow the effect of a variation in the phaserelationship to be appreciated.
When the sounds proceed from tuningforks vibrating independently, the
phasedifference passes cyclically through all degrees, and if the beat be slow
enough, there is good opportunity for observation. But it is not possible
to stop anywhere, nor in some uses of the method to bring into juxtaposition
phaserelationships which differ finitely. I thought that it would be of
interest to observe under conditions which would allow any particular phase
relation to be maintained at pleasure, and to this the revolving magnet
method naturally lends itself.
* Phil. Mag. VoL xxxvm. p. 286 (1894) ; Scientific Papers, Vol. iv. p. 109.
t Phil. Mag. [6] Vol. xiu. p. 214 (1907). [See preceding paper.]
1907] ACOUSTICAL NOTES 365
The propulsion is by means of wind (under about two inches water
pressure) from a well regulated bellows. The blade forming the magnet
may be bent windmill fashion and receive the wind directly, but in the
present experiments it was combined with a diminutive turbine, the whole
revolving about a vertical axis. The speed was about 190 per second, giving
in the telephones a note of pitch g. Two inductorcoils* were used, the
circuit of each being completed through a telephone. The planes of the
coils were vertical, their centres being at the same level as the magnet.
One was fixed and the other was so mounted that it could revolve about an
axis coincident with that of the magnet and turbine. The angle between the
planes represents of course the phasedifference of the periodic electromotive
forces, subject it may be to an ambiguity of half a period, dependent on the
way the connexions are made. If the circuits are similar, as is believed, the
phasedifference of the currents and of the electromotive forces is the same.
The telephonediscs were loaded, but not so heavily as to bring them into
tune with the sounds employed. The circuit of one telephone included a
commutator by which the current through the instrument could be reversed,
corresponding to a phasechange of 180.
In commencing observations the first step is to adjust to equality the
sounds heard from the two telephones. This can be effected by varying the
distances between the magnet and the inductorcoils. The telephones are
then brought into simultaneous action at the two ears, and the effect is
observed. A rotation of the movable coil may then be made, or the current
in one telephone may be reversed by means of the commutator. The results
were for the most part in harmony with what had been expected from the
experiments with forks. But one anomaly must be noted, relating to the
neutral condition where no pronouncement can be made in favour of either
right or left. This should occur when the phases of vibration at the ears
are either the same or precisely opposed; and it had been expected that the
condition would be realized when the planes of the inductorcoils were strictly
parallel. There is no difficulty in determining the neutral position, where
neither right nor left has the advantage in either state of the commutator;
but I was surprised to find that according to my own judgment the neutral
position deviated very appreciably, perhaps 10 and on one occasion even more,
from that of parallelism. At first I supposed that the explanation of the
anomaly was to be sought in the behaviour of the telephone plates, whose
vibrations may not have the same relation in the two cases to the electric
currents actuating them. It is possible that this cause of disturbance may
have been operative to some extent ; but that it was not a complete account
of the matter became evident later when it was found that in Mr Knock's
* They were also employed in the 1894 experiments and are there spoken of as "wooden
coils " from the fact that the wire is wound upon wood.
366 ACOUSTICAL NOTES [320
judgment the neutral position did coincide sensibly with parallelism of the
coils. There is no doubt at all but that the judgments of the two observers
really differed; each repudiated the setting of the coil satisfactory to the
other.
In the judgment of the individual observer, the neutral position can be
determined with considerable precision by the observed absence of lateral
effect, in conformity with the results of the tuningfork experiments. In
using the commutator in order to ascertain that no change in respect of
lateral effect accompanies reversal, a complication arises from the fact that
during reversal one circuit is momentarily broken. The telephones may be
removed from the ears during commutation. If this be not done, care must
be taken that the judgment made relates to the permanent effect, or errors
may ensue due to the momentary action of the sound on one ear only.
When the neutral position of the coil is departed from, a lateral
sensation say to the right is experienced, and this increases until the
displacement reaches 90. A reversal at the commutator changes the right
into a left sensation, having the same effect as a rotation of the coil
through 180.
When the adjustment is such that the combined lateral sensation (to the
right) is a maximum, it is interesting to observe the effect of applying the
telephones to the ears consecutively. If the right telephone be the first
applied the sensation of course is of a sound to the right. When the left
telephone follows, the sound remains on the right and appears louder.
If on the other hand the left telephone be the first applied, the sound
appearing originally to be on the left transfers itself to the right as the
second telephone comes into action. Under the best conditions there seems
to be nothing remaining over on the left.
The results are thus confirmatory of those obtained from tuningforks.
Unquestionably we are able to take account of the phasedifference at the
two ears, and this in the case of low pitch is the foundation of the secure
judgment as to direction that we are able to form when a single sound is
heard from the right or from the left. With respect to the convenience
of the two methods of experimenting, much of course depends upon what
appliances are available. In most laboratories, I suppose, the tuningforks
would be preferred.
Multiple Harmonic Resonator.
The use of Helmholtz resonators to demonstrate the compound character
of a musical note is now familiar. The harmonic component tone which has
the pitch of the resonator is specially reinforced and so rendered conspicuous
even to untrained ears. By changing the resonator, the fundamental tone or
any of the harmonics may be intensified in succession.
1907] ACOUSTICAL NOTES 367
Such effects are rendered far more striking if the necessary changes of pitch
can be brought about in a single resonator, which then remains continuously
connected with the ear. We may do a little in this direction with a resonator
of the usual Kbnig pattern. Choosing one of somewhat high pitch and
listening to a harmonium note two or three octaves down, we find that
various harmonics swell out in turn as we pass the finger over the aperture,
thereby gradually lowering the pitch to which the resonator responds.
The idea is carried out more completely if the resonator is provided with
a number of separate apertures, any or all of which can be completely closed
with the fingers. According to the simple approximate theory* the natural
frequency (JV) of the resonator is given by
where a is the velocity of sound in air, 8 the volume included in the resonator,
and c the electrical conductivity between the interior and exterior calculated
upon the supposition that air is a conductor of unit specific conducting
power and that the walls behave as insulators. For a circular aperture in a
thin wall c is equal to the diameter of the aperture. If there are several
apertures, well separated from one another, c is equal to the sum of the
diameters, or as we may write it,
c = d t +d 2 + d s + ............................ (2)
Hence, if the first aperture acting alone give the fundamental tone, the
first and second together the octave, the first three together the twelfth and
so on, we have so far as relative magnitudes are concerned,
d l = l, d 1 + d 2 =4>, ^ + ^ + 4=9, &c.;
or d,:d,:d s ...... = 1 :3:5 ............................... (3)
The ratios (3) may give some idea of the proportions, but for many
reasons among them the neglect of the thickness of the walls they are
only roughly applicable. There is no reason for insisting on a circular form
of aperture ; indeed, when the aperture is large, an elongated form lends
itself better to closure by a finger. Extreme cases excluded, the effectiveness
of an aperture depends mainly upon its area.
When, as would be especially likely to happen in the case of the funda
mental tone, a simple aperture would be very small, it may be well to replace
it by a channel of finite length. If R be the radius of a tube of circular
section and L its length,
7T.R 2
Theory of Sound, 304, 305, 306.
368 ACOUSTICAL NOTES [320
from which it will be seen that the area of aperture may be much increased
as compared with what would be admissible if L = 0.
Two compound resonators on this principle have been constructed. The
first was made from the upper part of a glass bottle which had been cut off
square near the neck. Over the wide opening a rather stout zinc plate was
cemented through which the various apertures were bored. Through a cork,
fitted into the neck, passed a short piece of brass tubing, by means of which
and a suitable indiarubber prolongation connexion between the ear and the
interior of the resonator was established. In tuning the instrument it is
necessary to begin with the lowest tone and care must be taken to complete
the adjustment of each aperture, or channel, before the next is attempted.
Further details are hardly required. If the resonance is improved by shading
an aperture with the finger, it is a sign that the aperture is already too
large. If on the other hand the resonance improves up to complete with
drawal of the finger, the aperture may still be too small. Before finally
enlarging the aperture it may be well to ascertain that the resonance is
improved by a partial use of the one next in order.
The second resonator was constructed entirely of metal. It consists of
an elliptical box of sheet zinc, with top and bottom also of zinc and slightly
dished for the sake of enhanced rigidity (fig. 1). The capacity (S) is about
Fig. 1.
140 c.c. The apertures for the fundamental tone (F) and for the octave
Z(th) are formed of brass tubing soldered into position. The other passages
are simple perforations in the wall and in the top of the box. E represents
a short length of brass tubing over which is slipped an indiarubber attach
ment passing to the ear. To sound the fundamental tone of about 128
1907] ACOUSTICAL NOTES 369
vibrations per second (B of my harmonium), F, about 8 mm. in diameter
and 20 mm. long, is alone open. For the octave tone Z(th), 11 mm. in
diameter and 53 mm. long, is opened in addition. R 3 then gives the twelfth,
R. 2 the double octave, R l the higher third, L 3 the sixth component (octave
+ twelfth), L 2 the harmonic seventh, and finally L^ the triple octave. The
diameter of R s is about 5 mm. and that of L^ (the largest aperture) say
13 mm. The letters are intended to indicate the fingering. Thus L 1} L a _, L,
denote the first, second, and third fingers of the left hand; R l} R 2 , R 3 the
corresponding fingers of the right hand. The octave tube (th) is closed
with the thumb of the left hand.
The performance of this instrument is very satisfactory. The seventh
and eighth components are a little weak, but the others, and especially
the twelfth and higher third, are loudly heard. The experimenter should
bear in mind that when working indoors much depends upon the precise
position. The room is intersected with nodes and loops of approximately
stationary vibrations, whose position varies from one tone to another. If
a particular harmonic is ill heard, it may only be that the situation of the
resonator is unfavourable. A motion of a few inches will often make a great
difference. Usually the effects are best when the resonator is held pretty
close to the reed in action.
In general a harmonium note is the most convenient for experiment and
demonstration, but other instruments are of course available. A man's voice
singing the proper note (B as above) gives excellent results.
TuningForks with slight Mutual Influence.
Two forks giving 128 vibrations per second are independently maintained,
each making and breaking its own contacts at a mercury cup. If mutual
influence be altogether excluded, the "beat" may be made as slow as we
please. But although the electric circuits may be entirely distinct, if the
forks stand on the same table there may be enough mutual influence to
bring about absolute unison. The best method of observation is by Lissajous'
figures. The permanence of the ellipse is a sign that there is mutual control
and that absolute unison is established; otherwise the ellipse undergoes
more or less slowly the usual transformations.
A series of observations on this subject were made in 1901. Mutual
influence may arise from both forks being connected with the same battery.
If the electric circuits are in a series which includes two Grove cells, the
forks keep together indefinitely; but this arrangement is rather akin to the
familiar one in which a single interrupter fork drives another, the second
having no break of its own. Even when the fork circuits are in parallel
and are fed from two Grove cells*, there is, or may be, sufficient reaction to
* The internal resistance of the cells comes into play here.
E. v. 24
370
ACOUSTICAL NOTES
[320
maintain absolute unison. The feebler the reaction, the more nearly must
the natural frequencies approach to identity. When the reaction is just
insufficient for control, it is interesting to watch the cycle of the beat, as
revealed by Lissajous' figure. At one part of the cycle the changes are very
slow and at the opposite part relatively very quick.
In another set of experiments the electric circuits were quite distinct,
each fork being driven by a separate Grove cell. A sufficient mutual reaction
could be obtained through the air. To this end a large resonator was con
structed by cementing a wooden plate over the opening of a bellglass.
In the plate were two similar apertures, to which the free ends of the forks
were presented, the pitch of the resonator being equal to that of the forks
(128). In this way an adequate control was secured, but the margin was
Fig. 2.
A more powerful controlling reaction accompanies a connexion between
the two forks by means of slender cotton threads. The arrangement employed
is indicated in the figure. A and B are the free ends of the upper prongs
seen from above. To them is attached a Yshaped thread ABCD, the
tension of which can be adjusted at D. When the control is established, the
Lissajous' ellipse is stationary and usually open. An ellipse closed in upon
its major axis would indicate that the natural frequencies were identical,
independently of the control. By touching a fork judiciously with the rubber
tip of the exciting hammer, the phase may be disturbed without stopping
the electric maintenance. If one fork be touched, the ellipse closes in, while
a similar operation upon the other fork opens it out. In a short time the
ellipse settles back, showing that the original phaserelationship is recovered.
1907]
ACOUSTICAL NOTES
371
Mutual Reaction of Singing Flames.
In a former paper* I discussed the mutual influence of organpipes
nearly in unison, showing that the disturbances depend upon the approxi
mation of the open ends, and not sensibly upon the circumstance that they
may take their wind from the same source. When the reaction suffices, only
one note is sounded, and that is usually higher in pitch than the notes
of either pipe separately. It is proposed to record the results of some
observations of a similar character recently made upon socalled singing
flames, i.e. tubes caused to speak by means of hydrogen flames.
The tubes were of glass from the same length, each 30 cm. long and
16 mm. internal diameter. The hydrogen bottles were also similar arid were
provided with burners formed from 14 cm. lengths of glass drawn down at
the upper ends. Very small flames suffice. The tubes were held vertically
and so that their upper (and lower) ends were at the same level.
Fig. 3.
When the distance between the tubes is considerable, say 30 cm., and
draughts are avoided, fairly slow beats may be obtained by suitable tuning,
as by approach of the finger to one end of that tube which vibrates the
quicker. But when the distance is reduced to perhaps 8 or 10 cm., a
difficulty begins to be experienced in producing slow beats. Either they are
rather quick or else, when the tuning does not allow of that, they disappear
altogether, the vibrations as it were engaging. On the margin where beats
still occur, their character is peculiar. They appear unsymmetrical, the
swell being protracted and the fall hurried. The phenomenon is the same
as that observed optically in the case of forks (p. 370). When the tubes are
as close as possible they may conveniently be tied together with string,
even moderately slow beats are excluded. In this situation the sound is
much attenuated, indicating that the phases of vibration are opposite, at any
rate in the ideal case. The ideal case is, however, rather difficult to attain.
Phil. Mag. Vol. vn. p. 149 (1879) ; Scientific Papers, Vol. r. p. 409.
242
372 ACOUSTICAL NOTES [320
There should be complete cessation of the principal tone in a resonator ((7)
whose mouth is held near the upper ends (A, B) symmetrically in the
median plane (fig. 3). Frequently a better silence may be reached by
moving round a little, and even then it is not absolute. By use of the finger
to give a finishing touch to the tuning, the most silent position may be
driven to the median plane, but even so the residual tone may not be quite
extinguished. It is evident that the ideal condition is easily disturbed a
little by slight failures of symmetry, probably connected with the flames.
As so far described, the disappearance of the principal tone, sometimes
very nearly realized, leaves a considerable amount of octave outstanding.
A remedy may be applied by the insertion, at the middles of the tubes, of
rectangular blocks of wood, about two diameters long and forming a loose fit.
They are held in their places by springs. In this way the outstanding octave
may be very much reduced.
Longitudinal Balance of TuningForks.
The vibrations of a well  constructed tuningfork are approximately
isolated and are conveyed to the stalk in only a limited degree. When, as
in the ordinary use of small forks, the stalk is pressed against a sounding
board, the principal tone is attended by a considerable accompaniment of
octave, especially at first when the vibrations are vigorous. The substi
tution of a suitably tuned resonancebox for the soundingboard may easily
render the octave sound preponderant*. The experiments now to be recorded
were an attempt to ascertain how far it was possible to carry the isolation
of the principal tone. It should be remembered that however complete may
be the isolation as regards the stalk, there is necessarily a certain small
amount of direct communication from the vibrating prongs to the surround
ing air. For our present purpose this is to be disregarded.
At first sight it may appear that the desired state of things must be very
approximately attained in the usual construction where the prongs are
parallel. Something will depend upon the manner in which the transition
takes place between prongs and stalk. In what follows I have more par
ticularly in view a construction in which the prongs form a U of tolerably
uniform thickness, to which a cylindrical stalk is attached without much
exceas of metal at the junction. As a rough approximation we may suppose
that the inertia of the fork is concentrated at the ends of the prongs. Then
if the fork be free in space, these ends can move only backwards and
forwards along the line joining them. The question we have to ask is
Phil. Mag. Vol. ra. p. 456 (1877) ; Scientific Papen, VoL i. p. 318. Even when the box
is tuned to the fundamental note, the octave and twelfth are often easily audible. I have
observed this effect with three different 256forks when mounted upon a particular resonancebox.
There was no suggestion of looseness or chatter.
1907]
ACOUSTICAL NOTES
373
Fig. 4.
does the stalk remain at rest? A little consideration makes it fairly clear
that in the case of parallel prongs the answer is in the negative. As the
prongs approach one another the curvature of the bend is increased and the
stalk moves along its length outwards, i.e. away from the prongs*. Similarly
half a period later the opening of the prongs is accompanied by an approach
of the stalk. Under these conditions if the stalk be brought into contact
with a soundingboard, a motion of the first order is communicated and the
principal tone is heard.
It is evident that the effect to be expected when the
prongs are parallel may be compensated by a suitable per
manent bending of the prongs inwards, or what comes to (
the same by a suitable loading on the inner sides. The
motion of the stalk during the vibration is then composed
of two parts which have opposite signs the one already
considered depending on the variable curvature at the bend,
the other on the obliquity of the prongs to the line of motion
at the ends. It would appear then that by this adjustment
it should be possible to secure that the stalk remains at rest,
so far as motion of the first order is concerned. It is
assumed that everything is symmetrical, so that the stalk, if
it moVes at all, does so along its length and (in view of its
dimensions relatively to the wavelength of vibration in
steel) practically as a rigid body.
For the purposes of the experiments a large fork was
constructed by Mr Enock. The U was from a single length
of steel 60crn. long and of section T275 cm. square. The
prongs were parallel, 5'35 cm. apart (inside measurement),
and the stalk was attached by brazing (fig. 4).
In its unloaded condition it gave 128 vibrations per
second and could be screwed to a resonancebox appertaining
to a large fork by Konig of the same pitch. When excited
by bowing it emitted a very powerful sound and, largely in
consequence, came somewhat rapidly to rest. The isolation
of the vibrations was thus far from complete.
The principal loads, of 40 gms. each, were in the form of nuts and
travelled along screws passing through the prongs near their ends and
parallel to the direction of vibration. Suitable locknuts kept all tight. In
consequence of the loading, the pitch fell about a major third, and the tuning
of the resonancebox had to be readjusted by a piece of board obstructing
the mouth. It soon appeared that, as had been expected, when the loads
* Somewhat as if by a violent local bending at the middle of the U the prongs were brought
into contact throughout their whole length.
374 ACOUSTICAL NOTES [320
were outside the prongs the sound diminished as they were moved inwards
as far as possible. To obtain a minimum, the loads must be i)iside the
prongs; and a great falling off was readily achieved by adjustment of their
position.
At this stage considerable difficulty was experienced in appreciating the
quality of the residual sound, but it was suspected that most of it was octave,
in spite of the fact that the resonancebox was tuned to the fundamental
tone. The device appropriate to stop tones of a particular pitch from
gaining access to the ear is a Quincke tube. A straight length of com
position metal tubing, open at both ends, was provided with a lateral
connexion at a distance from the outer end amounting to ^X of the octave
tone. The whole length was nearly the double of this, and the other end was
inserted in the resonancebox. At the same time the ear was connected
with the lateral branch with the aid of an indiarubber prolongation. When
the outer end of the straight tube is closed with the thumb, that end
becomes a node of the stationary vibrations of octave pitch executed therein,
and as the junction with the lateral tube is distant \ from the end that
place is a loop, and consequently no (octave) vibration is propagated to the
ear. The application of the thumb accordingly has the effect of freeing
the possibly compound sound from its octave component, while it leaves the
fundamental tone in full vigour.
The application of this test proved at once that by far the greater part
of the residual sound heard when the loads were in approximate adjustment
was in fact octave. Immediately after bowing, when the vibrations of the
fork are vigorous, a loud sound is heard when the outer end of the Quincke
tube is open, but comparative silence ensues when the thumb is applied. As
the vibration dies down, closing the end has less effect.
In this way it appeared that in reality a great measure of success had
been already attained in isolating the fundamental tone, only obscured by
the accompaniment of octave in unexpected amount. A sensible revival of
the fundamental tone ensued when the loads were rotated from their best
adjustment through a quarter turn each, corresponding to a lateral shift
inwards or outwards of fa inch. Since the test is of such delicacy, we
may perhaps consider the isolation of the fundamental tone to be practically
complete.
The fact remains, and must not be slurred over, that it was not possible
by any adjustment of the loads to eliminate the fundamental tone entirely.
The residual sound did not come directly through the air from the prongs,
but was propagated through the stalk to the resonancebox. It is a little
difficult to trace the nature of this residue. Upon the supposition that the
vibrations of the various parts of the fork are all in one phase, and of
complete geometrical and mechanical symmetry in the construction of the
1907] ACOUSTICAL NOTES 375
fork, it would appear that some adjustment of the loads must eliminate the
fundamental tone. There was, indeed, evidence of actual lack of symmetry,
which could not in any case be mathematically perfect. When with the aid
of a handle the fork was held horizontally so that its stalk rested upon a
wooden edge supported in its turn upon the top of the resonancebox, sound
was heard from the box, which varied as the fork rotated round its stalk as
axis and in fact nearly vanished in two asymmetrical positions. It would
seem that the residual fundamental tone heard in the more normal use may
be connected with a lateral movement of the stalk, dependent upon some
failure of symmetry.
As a variation upon the above arrangement the prongs were now bent
inwards so that at the outer ends the distance from metal to metal was
reduced from 53*5 mm. to 38 mm., a bending intended nearly to represent
the effect of the loads. With the 40 gm. loads it was no longer possible to
reduce the fundamental tone to silence and, as soon appeared, for this reason
that the proper position for the loads was unattainable, being that occupied
by the metal of the prongs themselves. When smaller (20 gm.) loads were
substituted, interior positions could be found allowing the elimination of
the fundamental tone to about the same degree of perfection as before.
In listening with the Quincke tube to the dying sound with alternate appli
cation and removal of the thumb at the outer end, it was recognized that
the low tone was practically gone (thumb on) while the octave (thumb off)
was still fairly audible. About the same displacement of the loads as before
( s \j inch) was sufficient to cause a perceptible augmentation of the residual
fundamental tone.
As to whether these results can be turned to practical account in the
construction of forks, we must remember that if a fork is to be used in
conjunction with a soundingboard or resonancebox a too complete isolation
of the fundamental tone would defeat the intention. On the other hand if,
as in Helmholtz's vowel experiments, a fork is to be employed to excite an
air resonator placed near the ends of its prongs, a suitable turning inwards
of these prongs and consequent quiescence of the stalk would be of advantage.
In conclusion attention may be drawn to the circumstance that a sym
metrical bell with stalk attached would not need any particular adjustment
in order to ensure the isolation of the vibrations of the first order. If the
stalk tend to move outwards when contraction occurs along one particular
diameter of the circumference, the same tendency must repeat itself half
a period later when the contraction is transferred to the diameter at right
angles to the first. A similar remark would apply to a symmetrical com
pound fork, such as we may imagine to be produced by cutting away all the
material of the bell, except in the neighbourhood of two perpendicular
meridians.
376 ACOUSTICAL NOTES [320
A TuningFork Siren and its Maintenance.
When in 1901 I was experimenting upon the work absorbed in various
cases of the production of sound* I had at my disposal a Trinity House
"Manual" Foghorn. In this instrument the wind is generated by cylinders
and pistons, and a much higher pressure is available than is usual in
laboratory apparatus. Among the experiments then tried was the substi
tution of what I called a tuningfork siren for the natural reed and conical
horn of the instrument. Fitted to a windchest was a metal plate, carefully
faced internally and carrying a rectangular aperture about 10 mm. broad.
This aperture could be nearly closed by a plate 3 mm. wider, which vibrated
laterally in front of it. The vibrating plate was mounted upon the side of
one prong of a fork making 128 vibrations per second. When the fork was
at rest, the aperture was obstructed and the fit of the moving and fixed
plates was so good that the leakage of wind was not serious. But as the
fork vibrated the aperture was in part uncovered, and that twice during
each complete vibration of the fork, so that the pitch of the instrument,
considered as a siren, was 256. The fork was driven electrically from another
interrupterfork of the same pitch situated outside. Fitted to the aperture
externally was a resonating tube whose length could be adjusted to give the
maximum effect. One of the objects was to be able to vary the resonance
without disturbing the maintenance or the pitch of the siren. The pressure
employed was sometimes as low as 2'5 cm. of mercury, and the consumption
of wind about 3 litres per second, corresponding to '015 horsepower. At
this rate of working the pumps could be kept going by hand, or rather by
legs, for a moderate length of time.
One unexpected effect presented itself, which seems worthy of record.
As has been mentioned, the intention had been to keep the fork in motion
electrically. But it was found that, at any rate after being once started, it
remained in vigorous vibration under the action of the wind alone, although
the electric connexion was cut off. It will be observed that this case is
altogether different from that of a reed, where the tongue approaches and
recedes from the aperture normally. It is more analogous to the aeolian
harp, where, as I have formerly shown f, the vibration is executed in a plane
perpendicular to the direction of the wind. So far as I am aware, no
adequate mechanical explanation of this singular behaviour has been given.
* Phil. Mag. Vol. vi. p. 292 (1903). [Vol. v. p. 129.]
t Phil. Mag. Vol. vn. p. 149 (1879) ; Scientific Papen, Vol. i. p. 413.
Fig. 5.
1907] ACOUSTICAL NOTES 377
Stroboscopic Speed Regulation.
The stroboscopic method* has often been employed for testing and
regulating the speed of revolving shafts. I used it extensively in my deter
minations of absolute electrical unitsf, referring the speed of revolving coils
to the frequency of vibration of an electromagnetically maintained fork.
But I doubt whether even now the convenience of this method for general
purposes is appreciated. A few years ago the late Mr Gordon drew for
me upon card rows of alternate black and white "teeth" from 20 to 40.
Photographs from this upon flexible paper could be mounted upon a
revolving shaft so as to form reentrant circles of teeth for observation by
intermittent view. In my use of it an electrically maintained fork of large
dimensions and of home construction was employed. The fork was provided
with solid (platinum) contacts and made 64 vibrations per second, At the
ends of the prongs were blackened plates of thin metal perforated with slits,
so disposed as to be opposite to one another in the equilibrium position.
When the vibrations were excited by one or two cells, there were 128 views
per second through the slits.
Viewed past the fork, some of the circles of revolving teeth appear nearly
stationary. Usually two neighbouring circles can be picked out, which
appear to revolve slowly in opposite directions. From these data the neces
sary information is obtained in a way that need not be further explained.
It is thought that it may be of service to give a reproduction of Gordon's
drawing (see Plate, fig. 5). Photographic copies can easily be made upon
any desired scale adapted to the shafts round which it is intended to mount
them. Care should be taken to effect the junction properly, so that the
circles of teeth are continued through it without irregularity.
Phonic Wheel and Commutator.
By the use of the phonic wheel, invented independently by La Cour and
myself J, the speed of revolving shafts may be not merely compared with
a fork but automatically governed thereby. I have used this method for
driving a commutator of the kind required for passing a regular succession
of condenser charges through a galvanometer, as for example in determining
the ratio of the electrical units (velocity of light). The contacts required
are such that a piece A in connexion with the insulated pole of the condenser
shall make contacts alternately with a piece B representing the insulated
* Plateau (1836) ; Topler, Phil. Mag. Jan. 1867.
t See for example Proc. Roy. Soc. Vol. xxxii. p. 104 (1881) ; Scientific Papers, Vol. n. p. 8.
J Phil. Trans. 1883, p. 295. Scientific Papers, Vol. i. p. 355 ; Vol. n. p. 179.
378
ACOUSTICAL NOTES
[320
Fig. 6.
pole of the battery and another C connected to earth and to the other pole
of the battery and condenser. Of course the contacts must be good, and it is
essential that both be not made at the same time.
As in Thomson and Searle's work*, the commutator was of the usual type
adapted to a revolving motion, except that the cycle of contacts was repeated
so as to occur twice in each revolution. The developed form is shown in the
accompanying diagram (fig. 6) where the shaded parts represent
brass pieces, separated by ebonite insulation. Springs lightly
bearing against the exterior continuous portions of metal
correspond to B and C, while A corresponds to a brush bear
ing near the centre and making contact alternately with the
two metal pieces. Provision was made for varying the pres
sures at these contacts during the running and without
disturbing the insulation. The problem is to secure a uni
form rotation of this commutator, whose diameter was 28 mm.
The phonic wheel, mounted on the same shaft as the com
mutator, takes its time from a vibrating fork (44 per second f)
acting as interrupter of an electric current. The current
(about 4 amperes) is from three secondary cells and excites not
only the electromagnet by which the vibrations of the fork are maintained
but also the electromagnet of the phonic wheel. Four soft iron armatures
are mounted round the circumference of the drum and in their passage
complete approximately the magnetic circuit. The holes through which the
fork is viewed are also four in number.
The most advantageous action of the regulating current occurs when one
armature passes for each complete vibration of the fork. Under these
circumstances the prong, or rather a projecting wire attached for the purpose,
is seen stationary and single. There are then Il(=x44) revolutions of
the wheel per second and 22 charges and discharges of the condenser. But
the wheel may also be run at double or triple this speed, and then the
projecting wire is in general seen doubled or tripled. The regulating current
from the fork is of itself capable of maintaining the rotation at single or
double speed when once the necessary engagement has been secured. For
this purpose the speed must be raised to the required point by means of j^
string passed round the shaft and worked with the fingers, and even then 'it
may be only after many trials that engagement ensues. For the triple
* Phil. Trant. 1890, p. 607.
t For the design of steel vibrators and for rough determinations of frequency, especially
when below the limit of hearing, the theoretical formula is often convenient. We may take
frequency = 84600t/P,
where Ms the total length of a prong and t the thickness in the plane of vibration, both being
reckoned in centimetres. (Theory of Sound, 177.) At any rate the octave is never uncertain.
1907] ACOUSTICAL NOTES 379
speed the power of the fork current is insufficient, and recourse must be had
to independent driving by an electric motor or waterpower engine. If the
driving power be in excess, the fork currents are equally capable of holding
the wheel back. The whole behaviour is evident on observation of the fork
through* the revolving apertures, and so long as the engagement lasts the
wheel can never gain nor lose a complete cycle relatively to the fork.
I have used the commutator thus driven to observe the charges of a
condenser in their passage through a galvanometer. The only inconvenience
was the necessity of a considerable separation between the galvanometer and
the rest of the apparatus to obviate magnetic disturbance. The galvano
meter deflexion was steady and apparently independent of the force exercised
at the springs of the commutator. I believe that the arrangement might
be used with advantage in such work as determining the ratio of the
electrical units.
321.
ON THE PASSAGE OF SOUND THROUGH NARROW SLITS.
[Philosophical Magazine, Vol. xiv. pp. 153161, 1907.]
THEORY leads fco the curious conclusion that plane waves of sound incident
upon a parallel infinitely thin reflecting screen in which is perforated a
narrow slit, are transmitted in a degree depending but little upon the width
of the slit. If the plane of the screen be at a; = 0, and if the waves incident
on the negative side be denoted by
<j> = cos(ntkx), .............................. (1)
then the waves diverging from the slit upon the positive side have the
expression
COB (nt fcr jir)
7+log(iA;6)
in which 26 is the width of the slit, r the perpendicular distance of any point
from it, 7 = Euler's constant ('5772), and as usual k = 2?r/X, the wavelength*.
These equations apply also to the case of light incident upon a perfectly
reflecting screen, provided that the electric vector is perpendicular to the
length of the slit.
I thought that it would be of interest to examine the question experi
mentally, but the difficulties in the way have turned out to be more
considerable than had been expected. In dealing with soundvibrations
we have a large choice of wavelengths, down to 1 inch, or less if a sensitive
flame be employed as percipient. But, even so, there are formidable obstacles
in the way of realizing the theoretical conditions. In practice the screen
must be limited ; and then if the source be situated far behind, sound readily
passes the boundary, and with the aid of reflexions reaches the ear more
effectively by this course than through the slit; while if the source be near,
the waves incident upon the slit are not sufficiently plane. Even with the
shortest waves an adequate approximation to the infinitely wide and at the
same time infinitely thin screen of theory seems unrealizable. An attempt
was next made to obviate the difficulty by boxing up the source of sound so
* Phil. Mag. Vol. XLIII. p. 259 (1897) ; Scientific Papers, Vol. iv. p. 291.
1907] ON THE PASSAGE OF SOUND THROUGH NARROW SLITS 381
as to ensure that nothing could reach the ear otherwise than through the
slit. Even this demand is not met without much care, and when it is met
there is no security that the amplitude and phase of the vibration at
different parts of the length of the slit shall be the same. As to this length
itself, it is evident that it ought to amount to a considerable multiple of the
length of a wave.
An effective test on these lines of the escape of sound through a narrow
slit seeming hopeless, attention was turned to the opposite extreme where
the length of the slit is regarded as a small, in place of a large, multiple of
the wavelength. The expression replacing (2) is now*
(3)
where M denotes the electrical capacity of a plate having the size and shape
of the aperture, and situated at a distance from all other electrified bodies.
So far as I am aware, M has not been calculated for a rectangular aperture ;
but for an ellipse of semimajor axis a and eccentricity e
F being the symbol of the complete elliptic function of the first kind.
When e = 0, F(e) ^TT; so that for a circular aperture M = 2a/7r.
If the ellipse be very elongated,
log >r log ...................... (5)
if b be the semiaxis minor, so that in this case
The introduction of this value into (3) shows the same comparative
independence of the magnitude of the width of the small aperture as was
manifested in (2). It is understood that the longer dimension 2a of the
ellipse, as well as the shorter, is to be a small fraction of A.
In the earlier experiments the latter condition was but imperfectly
fulfilled. The source of sound was a birdcalif giving a wavelength of
1 inch or 30 mrn. The wavelength is ascertained in the usual way by
placing a highpressure sensitive flame in the stationary system compounded
of the direct waves and of those reflected perpendicularly from a movable
reflector. The displacement of the reflector required to pass from a maximum
to a maximum or from a minimum to a minimum of excitation, measures the
half wavelength. When the sensitiveness of the flame is suitably adjusted,
* See equations (14), (15) of memoir cited, or Theory of Sound, 292.
t Theory of Sound, 371, 2nd edition.
382 ON THE PASSAGE OF SOUND THROUGH NARROW SLITS [321
the observation of the minimum, characterized by recovery of the flame,
admits of great precision. The birdcall was mounted in a glazed earthen
ware drainpipe. This was closed at the hinder end with a wooden disk
perforated by two tubes, one serving as an inlet and the other as an outlet
for the wind, and in front by a second disk upon which the adjustable slit was
mounted. As has already been hinted, the isolation of the sound requires
much precaution. Although the jaws of the slit fitted well, the sound could
not be prevented from escaping without the aid of grease. The wind was
from a bag of heavy rubber cloth suitably weighted and controlled by obser
vation of a manometer, and the outlet tube was continued to a point outside
the window by a rubber prolongation. It was necessary to cement up every
joint until the whole was airtight, otherwise sound could be heard through
a rubber tube connected with the ear and presented at the outer end to the
place under test. I should say that my own ears are not now effective for
observations at this pitch, but Mr Knock's are very sensitive to it.
The slit was of tinplate with jaws carefully filed to a knifeedge. The
length was half an inch. The width could be reduced to about 2 thousandths
of an inch without too great an uncertainty of measurement. With these
arrangements it had been expected that the desired observations could be
made without difficulty. An intensity observed whether by ear or flame at
a certain short distance in front and with a certain width of slit might be
recovered at a greater distance with a wider slit. On the basis of the usual
law of attenuation of sound with distance a measure would be attained of
the effect of widening the slit.
But it soon appeared that nothing of value could be obtained on these
lines. In listening with the ear through a rubber tube whose open end
(usually provided with a conical termination of glass) was moved to and from
the slit, the intensity on receding was found not to fall in a continuous
manner, but to be subject to alternate risings and fallings, almost as if it were
due to a system of stationary waves. The effect was not equally apparent
with the flame, but it is difficult to make good observations with a flame
unless it can be maintained in a fixed position.
The earlier experiments were made in the laboratory where the floor,
ceiling, and walls might act as reflectors, but I was surprised at the vigour of
the alternations in view of the proximity of the place of observation to the
source. A transfer of the apparatus to the outside proved indeed to be no
remedy. With the slit facing upwards, observations along the vertical line
passing through it indicated alternations nearly as marked as before.
Although it is impossible to avoid obstacles altogether the observer himself
and the windbag would be the principal ones the result seemed unfavour
able to the reflexion theory.
1907] ON THE PASSAGE OF SOUND THROUGH NARROW SLITS 383
At this stage I thought that the limitation of the disk upon which the
slit was mounted its diameter was about 9 inches might be a source of
complication, and the apparatus was modified so as to permit the disk to
form part of the general floor of the laboratory. For this purpose a smaller
containing jar was necessary and the opportunity was taken to replace the
birdcall by a small whistle or open organpipe whose pitch lay more within
the capacity of my own ears. Preliminary experiments with the flame and
movable reflector showed that (as had been expected from the pitch f v and
dimensions of the pipe) the complete wavelength was 2 inches, giving
\\ inch periods between nodes or between loops.
Experiments with the apparatus thus mounted exonerated the disk, for
alternations of a marked kind were still recognized commencing at a few
inches' distance from the slit, which might now be regarded as situated in
an infinite impermeable plane. The character of these alternations was
peculiar. Sometimes they occurred in a period of 1^ inch, as if due to a
reflector in front which might perhaps be the ceiling of the room. But as a
rule the more conspicuous feature was a period of about 2 inches, and this
sometimes manifested itself in a condition of great purity.
Many times the thought occurred that I had misestimated the octave,
and that the fundamental wavelength was really double what had been
supposed, but this suggestion could not be maintained. After all, if reflexions
are admitted, the period of the alternations is not limited to the half wave
length. Even when stationary waves are formed truly in one dimension,
i.e. in parallel planes, it is possible by crossing them obliquely to encounter
periods which may exceed the half wavelength in any proportion. The
whole wavelength would in fact be a reasonable average value. In particular,
the whole wavelength would be the period due to the interference of the
direct sound with one arriving from a distant reflector in a perpendicular
direction.
An origin of this kind would afford an explanation of a several times
suspected difference of behaviour between the ear and the sensitive flame.
As regards the former, it makes but little difference in which direction a
sound arrives at the end of the hearing tube ; but it is otherwise with the
latter, whose excitation is due, not to variable pressure, but to variable
motion. If, as usual, the flame burns vertically, it is insensitive to sounds
arriving in a vertical direction, and even in the horizontal plane the sensitive
ness is of a semicircular character, vanishing in two opposite azimuths*.
It may thus easily happen in the above experiment that the flame may be
insensitive to reflected sounds, which are nevertheless capable of influencing
the ear.
* Nature, Vol. xxxvin. p. 208 (1888) ; Scientific Papers, Vol. in. p. 24.
384 ON THE PASSAGE OF SOUND THROUGH NARROW SLITS [321
The difficulty of accepting the explanation by reflected waves is in great
degree that of understanding how they can be powerful enough in compari
son with the direct sound at such short distances from the source. In
forming a judgment we must bear in mind that it is amplitudes that are
compounded and not intensities, so that as in the theory of Newton's rings
seen by transmission, a sound which would be inaudible by itself may be
competent to cause a very perceptible variation in the loudness of another.
Fig. 1.
In order to obtain further evidence a modification was introduced in the
terminal of the hearing tube. Thus if this be a Tpiece (fig. 1) of such
dimensions that the head of the T measures half a wavelength, discrimi
nation will be made between different directions somewhat as in the case of the
flame. If the head be parallel to a wavefront, the two openings cooperate ;
but on the other hand if the head be perpendicular to the wavefront, the
phases at the ends are opposed and nothing is propagated to the ear. This
is the reciprocal of the effect noticed on .a former occasion* that open organ
pipes emit little sound in the direction of their length. The Tpiece may be
used to eliminate a reflected sound travelling at right angles to the one
which it is desired to isolate.
Fig. 2.
For my purpose a more symmetrical arrangement was preferable which
should treat similarly sounds arriving in all directions perpendicular to the
length of the stalk. In the sketch (fig. 2) AA and BB are circular disks of
tinplate of which A is complete while B is perforated to receive the stalk C.
The former is held in position by three distancepieces extending across, and
Phil. Mag. Vol. vi. p. 304 (1903). [Vol. v. p. 139.]
1907] ON THE PASSAGE OF SOUND THROUGH NARROW SLITS 385
the diameter A A measures 1 inch (47 mm.). The distance between the
plates is  inch. When the hearingtube is provided with this appliance
as a terminal, sounds are heard with full intensity when they arrive in a
direction parallel to the stalk inasmuch as phaseagreement then obtains all
round the circumference, but if the pitch be that of the whistle a sound
arriving in any direction perpendicular to the stalk fails to penetrate G on
account of interference. The theory is given in the appendix to this paper.
I may note in passing that a similar apparatus has been constructed upon
a larger scale with disks of thick millboard 10 inches in diameter and about
half an inch apart. This is adapted to pitch c" of 1024 vibrations per second,
and it works well. The note may be given upon a harmonium at a few
feet distance. When the stalk points towards the reed the sound is very
loud, but it falls off in oblique positions and becomes faint when the reed
lies in the plane of the disks. In the open air the apparatus may be used to
find the direction in which a sound arrives.
With these aids I had hoped to be able to eliminate reflexions sufficiently
to realize the continuous diminution of intensity which should attend
recession from a single source whether situated in the open or on the
bounding wall of a semiinfinite space; but these hopes have been dis
appointed. In the laboratory an adjustment of the disks to parallelism
with the floor (in which the slit was situated) should eliminate reflexions
from the walls. There remains the ceiling; but so far as this was flat it
should give rise to alternations with a period of the half wavelength. As
a fact the whole wavelength was often observed, but the ceiling was some
what curved and there were of course other obstacles in the room. Not
much better success was attained out of doors, the apparatus being placed
upon a lawn and the slit facing upwards. Here again the horizontal position
of the terminal disks should have eliminated reflexions from obvious obstacles,
but alternations in the period of the full wavelength were usually apparent.
The question would sometimes suggest itself whether visible obstacles were
necessary at all to cause reflexions. In fogsignalling, echos, sometimes up
to 30 seconds' duration, have been observed when the sea was smooth and
there was no visible cause of reflexion. But these are usually attributed
to a streaky condition of the air, a cause which could scarcely have operated
in evening experiments over a lawn. That reflected waves, arriving more
or less horizontally, were still concerned is suggested by the observation
that the anomalous alternations were intensified by holding the terminal
disks nearly vertical so as to attenuate the direct sound. Possibly there was
diffuse reflexion from the grass. Whatever the cause of disturbance may
have been, it rendered hopeless any attempt to compensate width of slit by
alteration of distance, as had been the original intention. Indeed it would
hardly have been worth while to describe at so much length the difficulties
B. v. 25
"""! ON THE PASSAGE OF SOUND THROUGH NARROW SLITS [321
encountered, were it not that they may probably embarrass other observers
unprepared for them, and that the terminal diskapparatus has an inde
pendent interest.
The best that I have been able to do is by altering the length of the slit
so as to compensate variations of width. For this purpose a sliding plate
was provided cutting off equal lengths from the two ends. Observations
have been made both in the laboratory and outside upon the lawn. In both
cases the slit faced upwards, and the sound (/ r ) was observed by ear through
a hearingtube of rubber, provided at the further end with the diskapparatus
already described and held in a clip with stalk vertical. In the first
arrangement of the slit the width might be *002 inch and the length one
half an inch. The sound reaching the ear would be observed and as far
as possible retained in the memory. An assistant would then alter the
width, say to '010 inch, and the length say to onequarter inch, and the
observer would endeavour to decide which of the two sounds was the louder.
It was found as the mean result of two observers that compensation ensued
when the length was reduced from '5 inch to '28 inch. A similar change of
length compensated an alteration of width from *004 to *020 inch.
It will be seen that at any rate the sound is much less sensitive to an
alteration of width than to one of length. If we apply the formula (6)
applicable to an ellipse, identifying the length of the slit with 2a and its
width with 26, we get using logarithms to base 10
so that the formula in question gives results not very wide of the mark.
Other observations also were in fair accordance.
APPENDIX.
Mean potential over the circumference of a circle whose plane is parallel
to the direction of propagation of plane waves.
P is any point on the circle. OP = r, OM = x. Potential at P
= e**
Hence for the mean we have
J,(kr) vanishes when kr = 2?rr/\ = 2'404, or 2r = '77\.
1907]
ON THE PASSAGE OF SOUND THROUGH NARROW SLITS
387
If the plane of the circle be inclined at an angle a to the direction of
propagation, J (kr) is replaced by J (krcos a). If a = TT, so that the plane
is parallel to the waves, we find unity for the mean value, as was to be
expected.
When the potential to be averaged varies in two dimensions only, even
though the waves may not be plane, we may proceed by the method of
Theory of Sound, 339. Using polar coordinates and omitting as before the
timefactor, we have
i/r = A J (kr) + . . . + (A n cos n0 + B n sin n0) J n (kr) + . . . ,
n being integral.
Thus mean i/r over circle of radius r = A J (kr). At the centre of the
circle, ^r r=0 = A ; so that mean i/r over circle r = i/r r=0 x J (kr\ and vanishes
as before when J (kr) = 0.
Fig. 3.
A similar method applies in three dimensions. Thus (Theory of Sound,
330) we have in general
81J T
=  2ikS Q
+ terms in spherical harmonics of orders 1, 2, &c.,
vanishing when integrated over a sphere and also vanishing when ? = 0.
Hence
sin AT
Mean ^ over sphere r = i/r r=0 x r .
Accordingly mean ty vanishes if kr = mr, or r = \ n\. If AT = (n + ) TT,
over r = =0
For an example, reference may be made to the case of plane waves treated
in 334.
252
322.
ON THE DYNAMICAL THEORY OF GRATINGS.
[Proceedings of the Royal Society, A, Vol. LXXIX. pp. 399416, 1907.]
IN the usual theory of gratings, upon the lines laid down by Fresnel, the
various parts of the primary wavefront after undergoing influences, whether
affecting the phase or the amplitude, are conceived to pursue their course as
if they still formed the fronts of waves of large area. This supposition,
justifiable as an approximation when the grating interval is large, tends to
fail altogether when the interval is reduced so as to be comparable with the
wavelength. A simple example will best explain the nature of the failure.
Consider a grating of perfectly reflecting material whose alternate parts are
flat and parallel and equally wide, but so disposed as to form a groove of
depth equal to a quarter wavelength, and upon this let light be incident
perpendicularly. Upon Fresnel's principles the central regularly reflected
Fig. l.
image must vanish, being constituted by the combination of equal and
opposite vibrations. If the grating interval be large enough, this conclusion
is approximately correct and could be verified by experiment. But now
suppose that the grating interval is reduced until it is less than the wave
length of the light. The conclusion is now entirely wide of the mark.
Under the circumstances supposed there are no lateral spectra and the whole
of the incident energy is necessarily thrown into the regular reflection,
which is accordingly total instead of evanescent. A closer consideration
shows that the recesses in this case act as resonators in a manner not covered
by Fresnel's investigations, and illustrates the need of a theory more strictly
dynamical.
The present investigation, of which the interest is mainly optical, may be
regarded as an extension of that given in Theory of Sound *, where plane
* Second edition, 272 a, 1896.
1907]
ON THE DYNAMICAL THEORY OF GRATJNGS
389
waves were supposed to be incident perpendicularly upon a regularly
corrugated surface, whose form was limited by a certain condition of
symmetry. Moreover, attention was there principally fixed upon the case
where the wavelength of the corrugations was long in comparison with that
of the waves themselves, so that in the optical application there would be
a large number of spectra. It is proposed now to dispense with these
restrictions. On the other hand, it will be supposed that the depth of the
corrugations is small in comparison with the length (\) of the waves.
The equation of the reflecting surface may be taken to be z = f, where is
a periodic function of x, whose mean value is zero, and which is independent
of y. By Fourier's theorem we may write
= d cospx + c 2 cos 2px + s. 2 sin 2px + ... + c n cos npx + s n sin npx + ...
= id (&* + e**) + i (c n  is n } e in v* + (c n + is n ) e~ in ^ +..., (1)
the wavelength (e) of the corrugation being 2rr/p. Formerly the 8 terms
Fig. 2.
were omitted and attention was concentrated upon the case where d was
alone sensible. The omission of the s terms makes the grating symmetrical,
so that at perpendicular incidence the spectra on the two sides are similar.
It is known that this condition is often, and indeed advantageously, departed
from in practice.
The vibrations incident at obliquity 6, POZ, fig. 2, are represented by
^ _ e ik(Vt+zoos8+xa\nO) (2)
where & = 27r/\, and V is the velocity of propagation in the upper medium.
Here i/r satisfies in all cases the same general differential equation, but its
390 ON THE DYNAMICAL THEORY OF GRATINGS [322
significance must depend upon the character of the waves. In the acoustical
application, to which for the present we may confine our attention, ^r is the
velocitypotential. In optics it is convenient to change the precise inter
pretation according to circumstances, as we shall see later.
The waves regularly reflected along OQ are represented by
in which A is a (possibly complex) coefficient to be determined. In all the
expressions with which we have to deal the time occurs only in the factor
e* yt , running through. For brevity this factor may be omitted.
In like manner the waves regularly refracted along OR into the lower
medium have the expression
faB f*iiraM++*u+> t ........................... (4)
<J> being the angle of refraction ; and, by the law of refraction,
k, :k = V: V^smO : sin ......................... (5)
In addition to the incident and regularly reflected and refracted waves,
we have to consider those corresponding to the various spectra. For the
reflected spectra of the nth order we have
ilr = AnC* ( ~ z "^ e + xs]ne ) f A'nC* <*<s fl '''+ a; '"n e '''> .(5')
where, by the elementary theory of these spectra,
e sin 6 n e sin 6 = n\, or sin 6 n sin = + n\/e = np/k. . . .(6)
We shall choose the upper sign for B n and the lower for O' n . In virtue
of (6) the complete expression for ^ in the upper medium takes the form
I A n e in ^ e*""* + A' n
where n has in succession the values 1, 2, 3, etc.
Similarly, in the lower medium the spectra of the nth order are repre
sented by
fa = B n e ik * (ZCM *+*"in<M + B'nB*' fcoo^Vhwin*'.^ ............... (g)
where sin <f> n  sin <f> = up/h ......................... (9)
Accordingly, for the complete expression of fa, we have with use of (5),
". (10)
We must now introduce boundary conditions to be satisfied at the
transition between the two media when z = . It may be convenient to
commence with a very simple case determined by the condition that ^ = 0.
The whole of the incident energy is then thrown back, and is distributed
between the regularly reflected waves and the various reflected spectra.
1907] ON THE DYNAMICAL THEORY OF GRATINGS 391
We proceed by approximation depending on the sraallness of . Expanding
the exponentials on the right side of (7), we get
+ A n e in x (1  i&f cos 6 n +...) + A' n r***(l  ifcfcos ^ n +...) = 0. (11)
In this equation the value of f is to be substituted from (1), and then
in accordance with Fourier's theorem the coefficients of the various
exponential terms, such as e inpx , e~ inpx , are to be separately equated to
zero. As the first approximation, we get from the constant term (inde
pendent of a?)
1+A = 0, ................................ (12)
and from the terms m.e inpx , e~ inpx ,
A n = ik cos (CM is n ), A' n = ik cos (CM + is n ) ....... (13)
Thus, as was to be expected, A n , A' n are of the first order in , and if we
stop at the second order inclusive, (11) may be written
1 + A + 2ifcf cos 6 + A n e invx (1  t'&fcos n ) + A' n e in *' x (1  ik% cos #') = 0.
...... (14)
For the second approximation to A we get
1 + A  P 2 cos 6 2 (c M 2 + ) (cos e n + cos 0' n ) = ....... (15)
By means of (13) and (15) we may verify the principle that the energies
of the incident, and of all the reflected vibrations taken together, are equal.
The energy corresponding to unit of wavefront of the incident waves may
be supposed to be unity, and for the other waves mod 2 A , mod 2 ^, mod 2 J.'i,
etc. But what we have to consider are not equal areas of wavefront, but
areas corresponding to the same extent of reflecting surface, i.e., areas of
wavefront proportional to cos0, cos0 u cos ff l , etc. Hence,
cos 6 . modMo + 2 cos Q n . mod 2 A n \"S, cos ff u . mod 2 A' n = cos 6, . . .(16)
with which the special approximate values already given are in harmony.
In the formation of (16) only real values of cos O n , cos ff n are to be
included. If p>k, no real values exist, i.e., there are no lateral spectra.
The regular reflection is then total, and this without limitation upon the
magnitude of the c's. The question is further considered in Theory of
Sound, 272 a.
In pursuing a second approximation for the coefficients of the lateral
spectra, we will suppose for the sake of brevity that the s terms in (1) are
omitted. From the term involving e inpx in (14), we get with use of (13),
A n = ik cos 0Cn + & 2 cos 6 cos ff n . c n c m
+ ^k 2 COS {(Ci COS # w i + CZMI COS ^'ni) c ni
+ (C, COS n 2 + Cjna COS f n _ s ) C n _ 2 + . . .
+ (C, COS n+1 + C2M+, COS 0' n+ i) <Wi
+ (Co COS ll+ ., + C 2 ,, +2 COS 0' n+a ) C n+3 + ...] ............ (17)
392 ON THE DYNAMICAL THEORY OF GRATINGS [322
in which the first (descending) series is to terminate when the suffix in
cos n _ r is equal to unity.
The value of A' H may be derived from (17) by interchange of & and 6
in cos0 n _ r , costf'nr, cos n+r , cos0' n +r> cos ^ remaining unchanged. As
a particular case of (17), we have, for the spectra of the first order,
A ! = ik&i cos 6 + AT'CJ c a cos 6 cos 0\
+ A cos {c a (c, cos 0i + c, cos ^.,)
+ c 3 (c a cos 8 + c 4 cos ff 3 ) + ...}, (18)
.4 ', = lArCj cos 9 + & 2 CiC 2 cos cos l
+ A,* 2 COS JGa (Cj COS ^, + C 3 COS 0o)
+ c 3 (c 2 cos tf's + c 4 cos 3 ) + ...}, (19)
the descending series in (17) disappearing altogether.
If the incidence is normal, cos = 1, cos0' n = cos0 M , and thus A n , A' n
become identical and assume specially simple forms. Referring to (7), we
see that in this case
ijr = e* 2 + A o e** + 2^ 1 e' fccosfl ' cos px + . . . + 2^1 n e *z* cos npx +.,., (20)
in which, to the second order,
A =  1 + & 2 2c n 2 cos0 n (21)
A n =  ikc n + 2 cos 9 n . c n c m
+ W {(Ci + CJB.,) C n _, COS n _, + (C 3 + Co,,.:) C n _ 2 COS M _ S I ...
+ (C, + Cjn+i) C, l+1 COS M+1 f (c 2 + Can+a) C n+2 COS n+2 + . . . } (22)
If we suppose that in (1) only Cj and c 2 are sensible, we have
c 2 2 cos0 2 , (23)
! + cos 2 ), (24)
(25)
(26)
while A t , A s , etc. vanish to the second order of small quantities inclusive.
There is no especial difficulty in carrying the approximations further. As
an example, we may suppose that c, is alone sensible in (1), so that we may
write
=ccos^, (27)
and also that the incidence is perpendicular. For brevity we will denote
A;cos0 n or fccostf 7 ,, by /A,,. The boundary condition (i/r = 0) becomes by (7)
in this case,
e*<  er*t + (A + 1) er*< + 24, e~^ cospx +
+ 2A n er**S cos npx + = 0, (28)
in which
4i {J t (kc) cospxJ 3 (kc) cos 3/*r + / 5 (A;c) cos 5px ...},.. .(29)
J (kc)  2J, (kc) cos 2px + . . .
...}, (30)
1907] ON THE DYNAMICAL THEO11T OF GRATINGS 31)3
with similar expressions for e" 1 '^, e~ lti ^, etc. By Fourier's theorem the terms
independent of #, in cos px, cos 2px, etc., must vanish separately. The first gives
...(31)
The term in cos px gives
2tV, (Arc)  * (A, + 1) /, (kc) + A, {J (^c)  J, 0*ic)j
 t4 a {J, (i^c)  J 3 foe)}  4, { J, (^c)  J t (^c)} 4 . . . = 0. . . .(32)
The term in cos *2px gives
(A. + 1) J,(kc)  iA, (J,(w) J,(w)}
+ At[J fac) + J 4 (toc)} + ...... = ...................... (33)
The term in cos 3px gives
 J, O^c)}  iA 2 {J, M + J 5 (P, c){
...... = ......................... (34)
We see from these that A + 1 is of the second order in kc, that A! is of
the first order, A 2 of the second order, A 3 of the third order, and so on.
Expanding the Bessel's functions, we find, to the second order inclusive, as in
(23), (24), (25), (26),
A = l+kfrc*, A^ikc, } ,
A^lkfr*, ^ = 0, f
A 4 , etc., vanishing. To the third order inclusive (34) now gives
A 3 = & ikc s (tf 3^ + 6^^) ...................... (36)
From (33) to the same order we have still for A t)
(37)
and from (32)
A 1 = ikc + ^ikc 3 (k^4>k f jL 1 + 2 fJ L 1 to^) ............. ( 38 )
These are complete to the third order of kc inclusive. To this order
A 4 , A 6 , etc., vanish.
So far as the third order of kc inclusive, A remains as in (35); but it is
worth while here to retain the terms of the fourth order. We find
from (31)
A = l+ k^c 2 + &c 4 (>,  4&/v + 2/V  Ztfto + /*i/v) (39)
It is to be noted that k, /AJ, // are not independent. By (6), with 6 = 0,
f i n ^k 2 cos' 2 n = k 2 nY, ........................ ( 40 )
so that tf = k 2  p, fj,<? =k 4>p",
and 3Ar J 4/i, 3 + /i 2 2 = ............................ (41)
394 ON THE DYNAMICAL THEORY OF GRATINGS [322
By use of (41) it may be verified to the fourth order that when fi lt p., are
real, so that the spectra of the second order are actually formed,
(42)
expressing the conservation of vibratory energy.
When ^ is real, but not /*,, we may write ji. 2 = iV 3 , where v* is positive.
In this case
A, = 
and in virtue of (41) to the fourth order,
a ...................... (43)
Again, if /i, , /t 2 are both imaginary, equal, say, to IJA,, iv y , we have
from (39) with separation of real and imaginary parts,
A = 1 + ^A^c 4  i(kv l (^+ terms in c 4 ),
so that, to the fourth order,
mod4 =l, .............................. (44)
expressing that the regular reflection is now total.
In the acoustical interpretation for a gaseous medium >Jr represents the
velocitypotential, and the boundary condition (ir = 0) is that of constant
pressure. In the electrical and optical interpretation the waves are incident
from air, or other dielectric medium, upon a perfectly conducting and,
therefore, perfectly reflecting corrugated substance. Here ty represents the
electromotive intensity Q parallel to y, that is parallel to the lines of the
grating, the boundary condition being the evanescence of Q.
We now pass on to the boundary condition next in order of simplicity,
which ordains that d^/dn = 0, where dn is drawn normally at the surface of
separation. Since the surfaces z = 0, ^r = constant, are to be perpendicular,
the condition expressed in rectangular coordinates is
?0, ........................... (45)
dz dx dx
i/r being given by (7) and f by (1 ).
For the purposes of the first approximation, we require in d^jdx only the
part independent of the c's and 8\ since d/dx is already of the first order.
Thus at the surface
= ik sin 6 e'to.ino ^ +
1907] ON THE DYNAMICAL THEORY OF GRATINGS 395
Also, correct to the first order,
j = fa e ik*me j cog 0{lA + (l + A ) ik cos 0}  ......
 cos O n A n e in ?*  cos ffnA' n e~ inpx \.
Thus (45) gives
cos (1  A ) + cos 2 0(I+A ) ik$  cos O n A n e in *> x
 cos 6' n A' n e in P x  ......  sin 6 (1 +A ) ^ = .......... (46)
From the term independent of a; we see that, as was to be expected,
A=l .................................. (47)
Also A n cos O n = (c n  w) {& cos 2 O^np&nO}, ............... (48)
A' n cos ffn = t (c + w fl ) {& cos 2 + /> sin 0} ................ (49)
When n = 1 in (48), (49), we may put s, = 0. These equations constitute
the complete solution to a first approximation.
For the second approximation we must retain the terms of the first order
in d^jdx. Thus from (5), (7)
e .tn = & [ s i n (i + 4 o + (i _
+ sin OnAntf"** + sin 0' n A' n e~ in v x ]
= t'A; {2 sin ^ + sin O n A n e in * x + sin 0' n A' n er in P*},. . .(50)
since to the first order inclusive A = 1. Also
= ^ cos x _ ^ o + 2ifcf cos ^}
 tl cos OnA n &w* (1  ^^cos n )  t'A; cos ^' n A'n e~ inpx (1  t'A;^ cos 0' H ).. . .(51 )
Thus by (45) the boundary condition is
cos 6(1 A ) + 2ik cos 2  2 sin
cos O n  ik cos 2 0f + sin ^ n ^
* cos ^' n ~ ^ cos 2 ^'r + sin & n = ....... (52)
In the small terms we may substitute for A n , A' n their approximate values
from (48), (49).
In (52) the coefficients of the various terms in e in *> x , cr in ** must vanish
separately. In pursuing the approximation we will write for brevity
f = &***+_,***+... + .***"+? r*v, ............ (53)
where ?, = Ci = ^c,,
and ='*'  = C +
396 ON THE DYNAMICAL THEORY OF GRATINGS [322
The term independent of a; gives A to the second approximation. Thus
cos 6 (1  A ) + iA n (k cos 3 H + up sin H ) ^
+ iA' n (k cos 2 ff n  np sin ff n ) & = 0. . . .(55)
In (55), as follows from (6),
k cos 8 n + np sin 6 n = k cos 5 6 np sin 0,
and k cos 2 0' H np sin ff n = k cos 2 6 + np sin 0.
Hence with introduction of the values of A H , A' H from (48), (49),
(56)
as might also be inferred from (48), (49) alone, with the aid of the energy
equation
cos = cos mod 2 A + . . . + cos 0, t mod: 2 A + cos 0' n mod*A' H . . . .(57)
From the term in e inpx in (52) we get
cos n A n = 2i (k cos 2  np sin 0) t
+ iA' n (k cos 2 0' u  2np sin 0' rt ) 2/l + ......
+ iA n _ r (k cos 2 n . r  rp sin n . r ) r
+ iA n+r (k cos 2 n+r + rp sin n + r ) _ r
+ iA' n _ r {k cos 2 0',,_ r  (2n r)p sin 0' 7l _ r } ? 2/l _ r
+ iA' n+r {k cos 2 ^ n+r  (2n + r) p sin (9 / n+r  r 2 , l+ r .......... (58)
In (58) r is to assume the values 1, 2, 3, etc., the descending series
tenninating when n r = 1.
The corresponding equation for A' n may be derived from (58) by changing
the sign of n, with the understanding that
A. m = A' m , A'_ m = A m  ^_ w = ^, 0', n = m . .(59)
If the incidence be perpendicular, so that 0' m = m > and if m = m>
which requires that *, n = 0, the values of A' n and A lt become identical.
If n = 1, the descending series in (58) make no contribution. We have
cos 6 l A l = 2i (k cos 1  p sin 0) & + iA\ (k cos 2 0\  2p sin ^,) &
+ \A t (k cos 2 3 + p sin 3 ) ^ + iA 3 (k cos 2 2 + 2/> sin 0J f_ 2 + . . .
I iA' 9 (k cos ^,  3p sin ^,) f, + i4' 3 (Jfc cos 8 ^,  4p sin ^,) ^ + (60)
We will now introduce the simplifying suppositions that = 0, s m = 0,
making A' lt = A H , and also that only c, and c a are sensible, so that
1907] ON THE DYNAMICAL THEORY OF GRATINGS 397
$ = 4 = ...... = 0. We will also, as before, denote k cos 6 n or k cos & u by /x n .
Accordingly (60) gives, with use of (6), (48), (49),
In like manner, we get from (58)
<>
(64)
after which A s , A 6 , etc., vanish to this order of approximation. In any
of these equations we may replace /*,, 2 by its value from (6), that is
2 _ n *p2 t
The boundary condition of this case, i.e., d^/dn = 0, is realised acoustically
when aerial waves are incident upon an immovable corrugated surface. In
the interpretation for electrical and luminous waves, ty represents the
magnetic induction (b) parallel to y, so that the electric vector is perpen
dicular to the lines of the grating, the boundary condition at the surface of a
perfect reflector being db/dn = 0.
We have thus obtained the solutions for the two principal cases of the
incidence of polarised light upon a perfect corrugated reflector. In comparing
the results for the first order of approximation as given in (13) for the first
case and in (48), (49) for the second, we are at once struck with the fact that
in the second case, though not in the first, the intensity of a spectrum may
become infinite through the evanescence of cos#, 4 or cos0' 7l , which occurs
when the spectrum is just disappearing from the field of view. But the effect
is not limited to the particular spectrum which is on the point of dis
appearing. Thus in (61) 4, , giving the spectrum of the first order, becomes
infinite as the spectrum of the second order disappears (/ti 2 =0). Regarded
from a mathematical point of view, the method of approximation breaks down.
The problem has no definite solution, so long as we maintain the suppositions
of perfect reflection, of an infinite train of simple waves, and of a grating
infinitely extended in the direction perpendicular to its ruling. But under
the conditions of experiment, we may at least infer the probability of abnor
malities in the brightness of any spectrum at the moment when one of higher
order is just disappearing, abnormalities limited, however, to the case where
the electric displacement is perpendicular to the ruling*. It may be remarked
* See a " Note on the Remarkable Case of Diffraction Spectra described by Professor Wood,"
recently communicated to the Pliilostnpliical Magazine, Vol. xiv. p. 60, 1907. [Art. 323.]
398 ON THE DYNAMICAL THEORY OF GRATINGS [322
that when the incident light is unpolarised, the spectrum about to disappear
is polarised in a plane parallel to the ruling.
In both the cases of boundary conditions hitherto treated, the circum
stances are especially simple in that the aggregate reflection is perfect, the
whole of the incident energy being returned into the upper medium. We now
pass on to more complicated conditions, which we may interpret as those
of two gaseous media of densities a and <r,. Equality of pressures at the
interface requires that
r^ = r,fk, ................................. (65)
and we have also to satisfy the continuity of normal velocity expressed by
d^/dn = d^,fdn, .............................. (66)
or, as in (45),
rf(**)_gftA)g. a (67)
dz das dx
^ and ^ being given by (7), (10). We must content ourselves with a
solution to the first approximation, at least for general incidence.
From (65),
^ {1 + A. + (1  A ) ik cos 6 + A n e?* + A' n <r**>}
= B (1 + ik, f cos <J>) + B n e in ^ + & n trtae*. . . .(68)
Distinguishing the various components in as in (53), we find
B 0> .............................. (69)
, ......... (70)
A' n B' n =iS 1l \k 1 cos<t>B (lA )kcos8\ ....... (71)
"i ( \ }
In forming the second boundary condition (67) we require in
only the part independent of f. Thus
s 0}
 ik cos eA n #*v*  ik, cos t n A' n e 4n **,
i^cos <f>B (1 + ik, cos </,)
+ ik, cos <j> n B n ? n v* + ik, cos <j> f n R n e~ in P x .
1907] ON THE DYNAMICAL THEORY OF GRATINGS 399
Thus (67) takes the form
ik cos 6 (1 A ) t&j cos <f>B
 k z cos 2 6 (1 + A n ) + A, 2 cos 2 <J>B
 e in v* {ik cos B n A n + ik t cos <f> n B n ]
[ik cos 6' H A' n + ik, cos <''}
A B )dyda: (72)
From the part independent of x we get
/?cos^(l^ )^cos<#>5 = 0, r (73)
and from the parts in e inpx , e~ inpx
k cos B n A n + k, cos <j> n B n
= i& [k* cos 2 (1 4 A )  k? cos 2 <f>B  npk sin (1 + A  B )}, . . .(74)
and a similar equation involving A' n , B' n .
From (69), (73) we find
O"! COt <f>
C0t0
Again, eliminating B n between (70), (74), we get, with the use of (5),
A n {k cos O n + &! cos <f> n . <T/<TI} = f~  1 cos 2 6  * sin
D denoting the denominators in (75).
The equations (75) for the waves regularly reflected and refracted are those
given (after Green) in Theory of Sound, 270. They are sufficiently general
to cover the case where the two gaseous media have different constants of
compressibility (m, mj as well as of density (<r, o^). The velocities of wave
propagation are connected with these quantities by the relation, see (5),
k t *: & 2 = sin 2 0:sin 2 <#>=F 2 : F, a = m/a : mj^ (77)
In ideal gases the compressibilities are the same, so that
oj : o = sin 2 : sin 2 < (78)
In this case (75) gives
_ sin 20 sin 2</> _ tan (6 <j>) ,KQ.
= sin 28 + sin 2< = tan (6 + <f>) ' "
Fresnel's expression for the reflection of light polarised in a plane perpen
dicular to that of incidence. In accordance with Brewster's law the reflection
vanishes at the angle of incidence whose tangent is F/F,.
400 ON THE DYNAMICAL THEORY OF GRATINGS [322
In like manner the introduction of (78) into (76) gives, after reduction,
A n (fccos H + , cos <t> . ff/<r,}
= 2tVfc" M cot tan (0  <j>) {cos (6 + <f>) cos (d  </>)
cos ^ (cos < cos n ) np/k . sin 0} ............. (80)
If the wavelength of the corrugations be very long, p = 0, cos </> becomes
identical with cos </>, and thus A n vanishes when cos (6 + <j>) = 0, that is at
the same (Brewsterian) angle of incidence for which A (I = Q, as was to be
expected. In general A n =0, when
cos (6 + <) cos (6  <) = cos <f> (cos <f>  cos <f> n ) + np/k . sin 0. . . .(81 )
If we suppose that np/k is somewhat small, we may obtain a second
approximation to the value of cos (d + <). Thus, setting in the small terms
e + <j> = $TT, we get
cos (0 + <) = ^ sec 6 {cos <f> cos <j> n + np/k}.
Here cos < n = cos <f> np/kj . tan < = cos <f> np/k . cot 2 0,
so that COS( , + <W = J_. ...................... (82)
This determines the angle of incidence at which to a second approximation
(in np/k) the reflected vibration vanishes in the ?ith spectrum.
Since, according to (82) with n positive, 6 + <f> < IT, and n > 0, it seemed
not impossible that (82) might be equivalent to cos (0 n + <)= 0, forming a
kind of extension of Brewster's law. It appears, however, from (6) that
< 83 >
so that the suggested law is not observed, although the departure from it
would be somewhat small in the case of moderately refractive media.
For the other spectrum of the wth order we have only to change the sign
of n in (82), (83).
When np/k is not small, we must revert to the original equation (81).
Even this, it must be remembered, depends upon a first approximation,
including only the first powers of the fs.
Another special case of interest occurs when <r, <r, so that in the
acoustical application the difference between the two media is one of com
pressibility only. The introduction of this condition into (75) gives
. _ tan <f>  tan __ sin (<ft  0)
tan <f> + tan sin (0 + 0) '
the other Fresnel's expression.
1907] ON THE DYNAMICAL THEORY OF GRATINGS 401
Again, from (76),
whence A.. .=. ...(85)
sin
In this case the vibration in the nth spectrum does not vanish at any
angle of incidence.
We have now to consider the application of our solutions to electro
magnetic vibrations, such as constitute light, the polarisation being in one or
other principal plane. In the usual electrical notation,
K, K l being the specific inductive capacities, and /u., /^ the magnetic
permeabilities ; while in the acoustical problem,
The boundary conditions are also of the same general form. For instance,
the acoustical conditions
may be written
and in the upper medium where er is constant it makes no difference whether
we deal with i/r or <n/r. Thus if in the case of light we identify ^ with /3, the
component of magnetic force parallel to y, the conditions to be satisfied at the
surface are the continuity of @ and of K^d^jdn*.
Comparing with the acoustical conditions, we see that K replaces a, and
consequently (by the value of F 2 ) p, replaces l/m. Hence, in the general
solution (75), (76), it is only necessary to write K in place of <r. For optical
purposes we may usually treat /* as constant. This corresponds to the special
supposition (78), so that (79), (80) apply to light for which the magnetic
force is parallel to the lines of the grating, or the electric force perpendicular
to the lines, i.e., in the plane of incidence.
From (76) we may fall back upon (48) by making K l = <x> t /^ = 0, in such
a way that V lt and therefore 9, remains finite.
The other optical application depends upon identifying ^ with Q, the
electromotive intensity parallel to y, i.e., parallel to the lines of the grating.
The conditions at the surface are now the continuity of Q and of /jr l dQ/dn.
Equations (75), (76) become applicable if we replace <r by //,. If /* be
invariable, this is the special case of (84), (85); so that these equations are
* See Phil. Mag. Vol. xn. p. 81 (1881) ; Scientific Papers, Vol. i. p. 520.
R. v. 26
402
ON THE DYNAMICAL THEORY OF GRATINGS
[322
applicable to light when the electric vibration is parallel to the lines of the
grating, or perpendicular to the plane of incidence. The associated Fresnel's
expression (79) or (84) suffices in each case to remind us of the optical
circumstances.
In order to pass back from (76) to (13), we are to suppose ^ = 00,
/A, (or <r,) = 0, so that </> remains finite. Thus D = cot </cot 0, and the
only terms to be retained in (76) are those which include the factor o/^.
The polarisation of the spectra reflected from glass gratings was noticed
by Fraunhofer: "Sehr merkwiirdig ist es, dass unter einem gewissen
Einfallswinkel ein Theil eines durch Reflexion entstandenen Spectrums aus
vollstdndig polarisirtem Lichte besteht. Dieser Einfallswinkel ist fur die
verschiedenen Spectra sehr verschieden, und selbst noch sehr merklich fur die
verschiedenen Farben ein und desselben Spectrums. Mit dem Glasgitter
e = 00001223 ist polarisirt: E(+I\ d.i., der grune Theil dieses ersten
Spectrums, wenn a = 49 ist ; E (+ II), oder der grime Theil in dem zweiten
auf derselben Seite der Axe liegenden Spectrum, wenn o = 40 betragt;
endlich E ( 7), oder der griine Theil des ersten auf der entgegengesetzten
Seite der Axe liegenden Spectrums, wenn a = 69. Wenn E (+ 7) voll
standig polarisirt ist, sind es die iibrigen Farben dieses Spectrums noch
unvollstandig *."
In Fraunhofer's notation a is the angle of incidence, here denoted by 6,
and X (E) = 0'00001945 in the same measure (the Paris inch) as that
employed for e, so that p/k = \/e = 0'159. If we suppose that the
refractive index of the glass was 1'5, we get
Order
sin*
*,
,+*
JB( + II)
49
40
69
0755
0643
0934
0503
0429
0623
30 11'
25 25'
107 33'
79 11'
65 25'
107 33'
On the other hand, from (82) we get for E(+I) + <=77 44', for
E(+II) 59 48', and E(l] 104 45', a fair agreement between the two
values of + <f>, except in the case of E(+II).
It appears, however, that the neglect of p 3 upon which (82) is founded
is too rough a procedure. By trial and error I calculate from (81) for
E(+I) 0=48 52'; for #( + //) = 42 17'; for E(I) = 65 46'.
These agree perhaps as closely as could be expected with the observed
values, considering that they are deduced from a theory which neglects the
square of the depth of the ruling. The ordinary polarising angle for this
index (15) is 56 19'.
Gilbert's Ann. d. Phytik, Vol. LXMV. p. 337 (1823) ; Collected Writing*, Munich, 1888, p. 134.
1907] ON THE DYNAMICAL THEORY OF GRATINGS 403
It would be of interest to extend Fraunhofer's observations ; but the work
should be in the hands of one who is in a position to rule gratings himself.
On old and deteriorated glass surfaces polarisation phenomena are liable to
irregularities.
In the hope of throwing light upon the remarkable observation of
Professor Wood*, that a frilled collodion surface shows an enhanced reflec
tion, I have pursued the calculation of the regularly reflected light to the
second order in f, the depth of the groove, limiting myself, however, to the
case of perpendicular incidence and to the supposition that i and its equal
_i are alone sensible. Although the results are not what I had hoped, it
may be worth while to record the principal steps.
Retaining only the terms independent of x, we get from the first
condition (65),
<r/<T l .{(I+A )(lJ<?tf)2ikZ 1 cos0 l A l }=B (} k 1 *tf) + 2ik&cvsfaB 1 , ...(86)
and from the second condition (67),
k(l A,) (1  k*tf) + 2ik* 1 A 1 cos 2 0,  k, B (I  kftf)  2dfe 1 a 1 fl 1 cos 2 fa
= 2t> 2 r i (^i5 1 ) ............................ (87)
Eliminating J5 (l  kftf), and remembering that
& 2 cos 2 6 l + p* = A?, kf cos 2 fa + p 2 = ki>,
we get k(l A )  cr/oj . k, (1 + A ) + 2itf ^
+ a/a, . 2ikk, & A, cos 0, + 2^ 8 & cos &  2^ 2 5, = 0, . . .(88)
in which we are to substitute the values of A 1} B l from (70), (74). From
this point it is, perhaps, more convenient to take the principal suppositions
separately.
Let, as in (78), ^ : a = sin 2 : sin 2 < = kf : fc\
k k 2k 2
we have ^0 = , B ' =
and accordingly, from (70), (74),
k 2 A,  k^B, = 2iA?^ (^  k), k cos 6 l A, + k, cos ^B, = 0;
so that A l {k cos fa + ^ cos 0J = 2ik& (h  k) cos fa.
Hence, from (88),
lA
_
^ h \ ^(kcOSfa +^*
Again, when <r l = a,
kk 2k
and from (70), (74),
2ik{ 1 (kk 1 )
* Physical Optics, p. 145.
262
404 ON THE DYNAMICAL THEORY OF GRATINGS [322
The introduction of these into (88) gives
f frfr 
'
l+A ~k k p k cosB. + k, cos </>,
The question is whether the numerical value of A is increased or
diminished by the term in '. In (89) it is easy to recognise that in the
standard case of &, greater than k (air to glass in optics) the term in f, 2
is positive, 0, and & being supposed real. The effect of the second term
is thus to bring the righthand member nearer to unity than it would
otherwise be, and thus to diminish the reflection. Again, in (91), the second
term is negative, even when cos l = 0, as we may see by introducing the
appropriate value of cos^, viz., V(l k^/k^). The effect is therefore to
subtract something from kjk, which is greater than unity, and thus again
to diminish the reflection.
If in (89), (91) we neglect the terms in (k*  & 2 ) 2 2 , which will be
specially small when the two media do not differ much, the formulae become
independent of the angles l and </>j. In both cases the effect is the same as
if the refractive index, supposed greater than unity, were diminished in the
ratio 12 (k^  k 2 ) ^ 2 : 1. It appears then that the present investigation
gives no hint of the enhanced reflection observed in certain cases by
Professor Wood.
[1911. For some extensions of this theory see Voigt, Proceedings of the
Gottingen Society, Jan. 1911.]
323.
NOTE ON THE REMARKABLE CASE OF DIFFRACTION
SPECTRA DESCRIBED BY PROF. WOOD.
[Philosophical Magazine, Vol. xiv. pp. 6065, 1907.]
IN the Philosophical Magazine for Sept. 1902 Prof. Wood describes the
extraordinary behaviour of a certain grating ruled upon speculum metal,
which exhibits what may almost be called discontinuities in the distribution
of the brightness of its spectra. Thus at a certain angle of incidence this
grating will show one of the Dlines of sodium, and not the other. In fig. 1,
p. 398, Prof. Wood gives ten diagrams fixing the positions (in terms of wave
length) of bright and dark bands in the spectrum at various angles of
incidence ranging from 4 12' on the same side of the normal as the spectrum
to 5 45' on the other side. In general there may be said to be two bands
which approach one another as the angle of incidence diminishes, coincide
when the incidence is normal, and open out again as the angle increases
upon the other side. In the tenth diagram there is a third band whose
behaviour is different and still more peculiar. In the movement of the two
bands the rate of progress along the normal spectrum is the same for each.
The above represents the cycle when the grating is in air. "If a piece of
planeparallel glass is cemented to the front of the grating with cedaroil
the cycle is quite different. In this case we have a pair of unsymmetrical
shaded bands which move in the same direction as the angle of incidence is
changed."
An important observation relates to polarization. " It was found that
the singular anomalies were exhibited only when the direction of vibration
(electric vector) was at right angles to the ruling. On turning the nicol
through a right angle all trace of bright and dark bands disappeared.
The bands are naturally much more conspicuous when polarized light is
employed."
The production of effects changing so suddenly with the wavelength
would appear to require the cooperation of a large number of gratinglines.
406 ON THE REMARKABLE CASE OF [323
But, as the result of an experiment in which all but about 200 lines were
blocked out, Prof Wood was compelled to refer the matter to the form of
the groove. To this cause one would naturally look for an explanation of
the difference between this grating and others ruled with the same interval,
but it does not appear how the discontinuity itself can have its origin in
the form of the groove.
The first step towards an explanation would be the establishment of a
relation between the wavelengths of the bands and the corresponding angles
of incidence; and at the time of reading the original paper I was inclined
to think that the determining circumstance might perhaps be found in the
passing off of a spectrum of higher order. Thus in the spectrum under
observation of the first order, an abnormality might be expected at a
particular wavelength if in the third order light of this wavelength were
just passing out of the field of view, i.e. were emerging tangentially to the
grating surface. The verification or otherwise of this conjecture requires a
knowledge of the grating interval (e). This is not given in the published
paper; but on hearing from Prof. Wood that there were 14,438 lines to the
inch, I made at once the necessary calculation.
If 6 be the angle of incidence for which light of wavelength \ is just
passing off in the nth spectrum,
e(l sin0) = nX. (1)
In the first diagram the angle of incidence is 4 12' and the wavelengths
of the bands are given as 609 and 517, or in centimetres 6'09 x 1Q 8 and
517 x 10~ 8 . Also e = 2540/14438 cm., and sin 6 = '0732. Using these data
in (1), we find for the larger wavelength n = 310, or n = 2'68, according as
the upper or the lower sign is taken. Again, for the smaller wavelength
we find with the upper sign n = 3'65, and with the lower n = 315. To
reconcile these numbers with the suggested relation it is necessary to suppose
that 609 is passing off in the third spectrum on the same side as that on
which the light is incident, and 517 in the third spectrum upon the other
side. But the agreement of 310 and 315 with the integer 3'00 seemed
hardly good enough, and so the matter was put aside until recently, when
my attention was recalled to it in reading an article by Prof. Ames* on
Rowland's rulingmachines, from which it appeared that gratings have been
ruled with three different spaces, viz. 14,438, 15,020, and 20,000 lines to the
inch. If we permit ourselves to suppose that the number of lines in the
special grating is really 15,020 to the inch in place of 14,438, the alteration
would be in the right direction, 310 becoming 2'98 and 315 becoming 3'03,
so that the mean would be about correct.
In view of this improved agreement it seems worth while to consider how
John Hopkins University Circular. Notes from the Physical Laboratory, Ap. 1906.
1907]
DIFFRACTION SPECTRA DESCRIBED BY PROF. WOOD
407
far the position of the bands recorded in the other diagrams would accord
with the formula
\ = e (lsin0), (2)
taking e to correspond with ruling at the rate of 15,020 to the inch. In one
respect there is a conspicuous agreement with Prof. Wood's observations.
For if \j, \2 are the two values of X in (2), we have at once
X! + X, = fe, (3)
so that the two bands move equally in opposite directions as 6 changes.
The results calculated from (2) for comparison with diagrams (2) ..(10)
(fig. 1) are given below.
Calculation
Observation
e
Xi
X 2
Xi
X 2
No.
2 37'
590
538
589
537
2
015'
566
561
566
559
3
5'
564
563
562
561
4
0'
564
564
561
561
5
5'
564
563
6
115'
576
551
575
549
7
153'
582
545
581
542
8
2 38'
590
538
589
538
9
5 45'
620
507
619
506
10
The numbers headed " observation " are measured from Prof. Wood's
diagrams; but owing to the width and unsymmetrical form of some of the
bands they are liable to considerable uncertainty. It would appear that
(with the exception of the third band in diagram (10)) all the positions are
pretty well represented by (2).
As regards the observations when the face of the grating was cemented
to glass with cedaroil, we have in place of (1)
where V is the wavelength and 6' the angle of incidence in the oil. Now
if fi be the refractive index of the oil,
V = X//J,, sin 6' = sin 6 . /p,
so that e(pam0) = n\, .............................. (4)
if as usual 6 and \ are measured in air.
In the diagrams of Prof. Wood's fig. (2) there are four angles of incidence.
The bands are markedly unsymmetrical and the numbers entered in the
following table are those corresponding to the sharp edge. The values for
n are calculated from (4) on the supposition that /*=1'5, the lower sign
4(>s
ON THE REMARKABLE CASE OF
being chosen if the angles on the first side are regarded as positive. The
wavelengths observed correspond pretty well with the passing off of
the fourth and fifth spectra on the opposite side to that upon which the
light is incident. There seems to be nothing corresponding to the passing
e
X
n
12 8'
541
403
7 8'
590, 469
394, 496
yw
610, 489
397, 495
2W
655, 529
399, 493
off of spectra on the same side. Upon the whole there appears to be
confirmation of the idea that the abnormalities are connected with the
passing off of higher spectra, especially if the suggested value of e can be
admitted.
The argument which led me to think that something peculiar was to be
looked for when spectra are passing off may be illustrated from the case of
plane waves of sound, incident upon a parallel infinitely thin screen in which
are cut apertures small in comparison with X. The problem for a single
aperture was considered in Phil. Mag. Vol. XLIII. p. 259, 1897 *, from which
it appears that corresponding to an incident wave of amplitude unity the
wave diverging from the aperture on the further side has the expression
where k = 2Tr/X, r is the distance from the aperture of the point where the
velocitypotential i/r is reckoned, and M represents the electrical capacity of
a conducting disk having the size and shape of the aperture, and situated
at a distance from all other electrical bodies. In the case of a circular
aperture of radius a,
Jlf=2a/7r .................................. (6)
The expression (5) applies in general only when the aperture is so small that
the distance between any two points of it is but a small fraction of X, It
may, however, be extended to a series of equal small apertures disposed at
equal intervals along a straight line, provided that the distance between
consecutive members of the series is a multiple of A, The condition is then
satisfied that any two points, whether on the same or on different apertures,
are separated by a distance which is very nearly a precise multiple of X.
The expression for the velocitypotential may be written
(7)
Or Scientific Paper*, Vol. nr. p. 283.
1907] DIFFRACTION SPECTRA DESCRIBED BY PROF. WOOD 409
where r l} r 2 , &c., are the distances of any point on the further side of the
screen from the various apertures, and M' is the electrical capacity of each
aperture, now no longer isolated, but subject to the influence of the others
similarly charged.
It is not difficult to see that if the series of apertures is infinitely
extended, M' approaches zero. For, if e be the distance between immediate
neighbours, and we consider the condition of the system when charged to
potential unity, we see that the potential at any member due to the charges
on the other members has the value
Accordingly M' = 0, indicating that the efficiency of each aperture in allowing
waves to pass to the further side of the screen is destroyed by the cooperative
reaction of the series of neighbours. The condition of things now under
contemplation is that in which one of the lateral spectra formed by the
series of holes (considered as a grating) is in the act of passing off, and it
is evident that the peculiar interference is due to this circumstance*. The
argument applies even more strongly, if less simply, to an actual grating
formed by a series of narrow parallel and equidistant slits cut in an infinite
screen.
The case of a reflecting grating differs in some important respects from
that above considered. An investigation applicable to light is now nearly
completedf. It confirms the general conclusion that peculiarities are to be
looked for at such angles of incidence that spectra of higher order are just
passing off, but (it is especially to be noted) only when the polarization is
such that the electric vector is perpendicular to the grating.
P.S. June 5. In anwer to further inquiry Prof. Wood tells me that he
thinks the ruling may perhaps be 15,020 to the inch, but (the grating being
for the time out of his hands) he is not able to speak with certainty.
[1911. In a letter of Nov. 2, 1911 Prof. Wood writes " I think that there is no doubt about
the grating constant. It must have been 15,020, as the 14,000 machine was not in use at the
time it was made. I am practically sure of this." And in a subsequent letter referring to more
recent observations " The thing is evidently a little more complicated than your first theory, but
I think you are on the right track. "]
* If e be not a precise multiple of X, the series 1+ ^ +  + ad infin. would be replaced by
e  t*e + ^ g  2t* + i g  Sike + ^
which is equivalent to
t [1911. See preceding paper.]
324.
ON THE LIGHT DISPERSED FROM FINE LINES RULED UPON
REFLECTING SURFACES OR TRANSMITTED BY VERY
NARROW SLITS.
[Philosophical Magazine, Vol. xiv. pp. 350 359, 1907.]
THE problem of the incidence of plane waves upon a cylindrical obstacle,
whose radius is small in comparison with the length of the waves and whose
axis is parallel to their plane, is considered in Theory of Sound, 343, also
Scientific Papers, Vol. IV. p. 314; but it is now desired to carry the approxi
mation further and also to make some applications. On the other hand, we
shall confine ourselves to the cases of perfect reflexion where the boundary
conditions are simplest.
The primary waves, travelling in the negative direction, are represented
by <f> = e ikl > at+x) , where a is the velocity of propagation and k= 27T/X, X being
the wavelength. Dropping the timefactor for brevity, we shall write
< f ) = e ikx ss e .*r cos* = j^fa.) + 2i/ I (ifer) cos + . . . + 2i n J n (kr) cos nd + ...,.. .(1)
J n being the Bessel's function usually so denoted, so that
In (1) x and r are measured from the centre of the section of the cylinder,
whose length is supposed parallel to the axis of z.
The secondary waves diverging from the obstacle are represented by
^ = BJ) (kr) + B,Dj (kr) cos + 5 2 D 2 (kr) cos 20 + . . . , (3)
where
1907] ON LIGHT DISPERSED FROM FINE LINES 411
7 being Euler's constant (5772...), and the other D's are related to D
according to
D n (*) = (2*)" (^) n D (z) ...................... (5)
The first expression in (3) is available when z is large and the second when
z is small. It should be remarked that the notation is not quite the same
as in the papers referred to.
The leading term in D Q when z is large is
and in finding the leading term in D n (z) by (5) it suffices to differentiate
only the factor e~ iz . Thus when z is great
y OV / * * * \ /
Accordingly when in (3) ty is required at distances from the cylinder very
great in comparison with \ we may take
(8)
We have now to consider the boundary conditions to be satisfied at the
surface of the cylinder r = c, and we will take first the condition that
</>+t = <> ................................. (9)
at this surface. We have at once from (1) and (3) in virtue of Fourier's
theorem
B = J (kc) r D (kc), B, =  2^ (kc) s A (kc),
and generally
B n = WJ n (kc) + D n (kc) ......................... (10)
In like manner if the condition to be satisfied at the surface of the cylinder is
*> .............................. <">
we get, using C , C 1 &c. in place of B , B l &c.,
C =J '(kc) + D '(kc),
C n = 2*J n '(kc) + D u '(kc), ..................... (12)
the dashes denoting differentiation.
The next step is to introduce approximations depending upon the small
ness of kc. In addition to (4) we have
AW 
412 ON LIGHT DISPERSED FROM FINE LINES [324
and so on. Also
A'(*)= ........................................... (17)
Using these, we find
............ (18)
B, i*V .............................................. (20)
Referring to (8) we see that when kc is small the predominant term is
the symmetrical one dependent on B . Retaining only this term, we have
as the expression for the secondary waves
as in the papers referred to. Relatively to this, the term in cos 6 is of
order A^c 8 , and that in cos 20 of order A^c 4 .
Passing on to the second boundary condition (11), we have
(22)
(23)
( 24 >
When these values are introduced into (8), we see that the terms in C and
C, are of equal importance. Limiting ourselves to these, we have
(25)
the symbolical expression which gives the effect of the incidence of aerial
waves upon a rigid and fixed obstacle (Theory of Sound, 343). Fully to
interpret it we must restore the timefactor and finally reject the imaginary
part, thus obtaining
(atrW, ............ (26)
corresponding to the primary waves
</> = cos?^(a< + a;) ............................ (27)
1907] OB TRANSMITTED BY NARROW SLITS 413
In the application to electric or luminous vibrations the present solution
is available for the case of primary waves
c* = e**, (28)
incident upon a perfectly conducting, i.e. reflecting, cylinder, c* denoting the
magnetic induction [parallel to z\, for which the condition to be satisfied at
the surface is dc*/dr=Q. Accordingly the secondary waves are given by
(25) with c* written fori/r. This is the case of incident light polarized in a
plane parallel to the length of the cylinder.
For incident light polarized in the plane perpendicular to the length of
the cylinder the primary waves have the expression
R = e ikx , (29)
where R denotes the electromotive intensity parallel to z. In this case the
secondary waves are given by (21) with R in place of i/r. It appears that
if the incident waves in the two cases are of equal intensity, the secondary
P'
POD=P'OD=a; QOD = <j>.
waves are of different orders of magnitude, R preponderating. Thus if
unpolarized light be incident, the scattered light is polarized in the plane
perpendicular to the length of the cylinder, and the polarization is complete
if the cylinder be small enough.
It is now proposed to make application of these solutions to meet the
problem in two dimensions of the incidence of plane waves upon a perfectly
reflecting plane surface from which rises an excrescence, also composed of
perfectly reflecting material, and having the form of a semicylinder whose
axis lies in the plane. We shall show that it is legitimate to substitute the
complete cylinder, provided that we suppose incident upon it two sets of
plane waves adjusted to one another in a special manner.
In the figure ABEGD represents the actual reflector, upon which are
incident waves advancing along PO, a direction making an angle a with the
414 ON LIGHT DISPERSED FROM FINE LINES [324
surface OD. The secondary disturbance is required at a great distance (r)
along OQ inclined to OD at an angle <f>. But for the present we suppose
the cylinder to be complete and the plane parts of the reflector AB, CD to
be abolished ; and in addition to the waves advancing along PO we consider
others of the same quality advancing along a line P'O equally inclined to
the surface upon the other side. The angle 6 of previous formulae is repre
sented now by POQ, P'OQ whose values are <  a and < + a. Thus we may
write
cos 6 = cos (</> a), cos 6 ' = cos (<f> + a),
so that
cos 6 + cos 6' = 2 cos a cos <f>, cos cos 0' = 2 sin a sin </>.
The two sets of waves advancing along PO, P'O will be supposed to be
of equal amplitude ; but we shall require to consider two distinct suppositions
as to their phases. In dealing with R the supposition is that the phases
are precisely opposed. In this case we obtain from (8) as the complete
expression of the secondary waves
R =  sr*"*" I 2 * 5 ' sin a sin ~ 25 ' sin 2a sin 20 +...}, . . .(30)
\Z / IKT/
the term in J5 disappearing, while the values of B l , B z are given by (19), (20).
Each of the two separate solutions here combined, primary and secondary
terms included, satisfies the condition R = Q at the surface of the cylinder
and so of course does the aggregate. It is easy to see that the aggregate
further satisfies the condition R = Q along AB, CD where 6, 0' are equal,
the contributions from the two solutions being equal and opposite. Hence
(30) gives the secondary waves due to the incidence of primary waves along
PO upon the reflecting surface ABECD ; and the expression for the primary
waves themselves is
x being parallel to OD and y parallel to OG, so that
In like manner the c* solution may be built up. In this case we have
to give the same phase to the two component primaries. Corresponding to
the incident
C* = e *(*COa+y8ina) ^. gtifzcosay sina^ ............... /32)
we have for the secondary disturbance
c* =  (2^) *~* r ( 2C? + * ic * cos c <>s </>  2G, cos 2 cos 2< + ...}. . . .(33)
Each solution, consisting of primary and associated secondary, satisfies
over the surface of the cylinder dc*/dn = 0, dn being an element of the
normal. And over the plane part AB, CD the two solutions contribute
1907] OR TRANSMITTED BY NARROW SLITS 415
equal and opposite components to dc*/dn. Hence all the conditions are
satisfied for the incidence of waves along PO upon the compound reflecting
surface ABECD.
The problem is now solved for the two principal cases of polarized
incident light. If the incident light be unpolarized, the condition as regards
polarization of the scattered light turns upon the value of
_ iB l sin sin <j> B 2 sin 2a sin 20 f ...
~~ C + iC l cos a cos ^ C a cos 2a cos 20 +...'"
in which the values of B l9 B^,,..,C , C l} ... are to be substituted from (18),
(19), (20), and from (22), (23), (24).
If we stop at the first approximation, neglecting B 3 , C 2 , &c., we have
_ 2 sina sin
~l + 2cosacos0'
From (34) or (35) we see that the value of II is symmetrical as between
a and <j>, an example of the general law of reciprocity (Theory of Sound,
108 &c.).
If = 0, or if = 0, II vanishes without appeal to approximations. This
means that c* preponderates, or that the scattered light is polarized in a
plane parallel to the length of the cylinder. The conclusion follows ap
proximately although a be not very small, provided <f> be also small.
According to (35) II becomes infinite when
1 + 2 cos a cos = 0, ........................... (36)
for example when a = 45, = 135.
If we take a = 40, = 130 so as to avoid the directly reflected rays,
we have II = 67, so that there is nearly complete polarization in the plane
perpendicular to the length of the cylinder.
If we suppose = 180 a, so that observation is made nearly in the
direction of the regularly reflected rays, (35) becomes
The scattered light is unpolarized when II = 1. If we make this sup
position in (37) we find a =30. This angle separates the two kinds of
polarization. Thus when a is small, II = 2 sin 2 a ; when a = 30, II = 1 ; when
a = 45, n = oo ; when a = 90, II =  2.
By use of (34) the approximation may be carried further. As an example
we may take the case of perpendicular incidence and observation, so that
o = 90, = 90. Thus by (34)
(38)
416 ON LIGHT DISPERSED FROM FINE LINES [324
It may be well to recall that in the results which we have obtained the
angles o, <f> are measured from the surface and not, as is usual in Optics,
from the normal. Again, if it be desired to attach significance to the sign
of II, we must remember that in one case we were dealing with c* and in
the other with R.
The above given theoretical investigation was undertaken in order to
see how far an explanation could be arrived at of some remarkable obser
vations by Fizeau*, relating to the light dispersed at various angles from
fine lines or scratches traced upon silver and other reflecting surfaces. In
every case the incident and dispersed ray is supposed to be perpendicular
to the lines, so that the problem is in two dimensions. The most striking
effects are observed when the incident and dispersed rays are both highly
oblique and upon the same side of the normal to the surface. The dispersed
light is then strongly, sometimes almost completely, polarized, and the plane
of polarization is parallel to the direction of the lines, i.e. perpendicular to
the plane of incidence. A silver surface, polished by rubbing with ordinary
rouge in one direction, shows these effects well, and even a piece of tinplate,
treated similarly with cottonwool, suffices. The plate is to be held obliquely
and the incident rays should come from a window or skylight behind the
observer. It is of importance to avoid stray light and especially any that
could reach the eye by specular reflexion. Observation with a nicol shows
at once that the light is strongly polarized and in the opposite way to that
regularly reflected from a glass plate similarly held.
Under the microscope a single line may be well observed, especially
when strongly lighted by sunlight. Fizeau found that when the incidence
is oblique and the observation normal (a = small, < = 90), or equally when
the incidence is normal and the observation oblique (a = 90, <f> = small), the
above specified polarization obtains, provided the line be very fine ; otherwise
the polarization may be reversed. When the incidence is oblique and the
light nearly retraces its course (a and <f> both small and of the same sign),
the polarization is more complete and also less dependent upon extreme
fineness. When a and < are both in the neighbourhood of 90, the polar
ization becomes insensible. If the incidence is oblique and the angle of
observation in the neighbourhood of the regularly reflected light, traces of
reversed polarization are to be detected.
My own observations are in essential agreement with Fizeau. At first
accidental scratches upon silver surfaces which had been worked in one
direction were employed. Afterwards I had the opportunity of observing
specially fine lines ruled with a diamond by a dividingengine, for which
I am indebted to Lord Blythswood. In the latter case the plate was of
speculummetal.
* Annalet de Chimie, Vol. win. p. 385 (1861) ; Mascart's Traiti d'Optique, 645.
1907] OR TRANSMITTED BY NARROW SLITS 417
It will be seen that the theory agrees with observation well in some
respects, but fails in others. When a and cf> are both less than 90 and of
the same sign, the polarization expressed by (35) sufficiently represents the
facts. But there is little in the observations to confirm the strongly
reversed polarization which should occur when the denominator in (35)
becomes small. One defect of correspondence in the conditions of theory
and experiment is obvious. The former relates to semicylindrical excre
scences, while the observations are made upon light dispersed from scratches
which are mainly depressions. In order to examine the question thus
arising, a glass plate provided with suitable scratches was coated chemically
with silver upon which copper was afterwards deposited by electrolysis.
When the coating thus obtained was stripped from the glass, a highly
reflecting surface was obtained in which the original scratches are repre
sented by precisely fitting protuberances. But even with this I was unable
to find the strongly reversed polarization to be expected according to (35)
when (36) is nearly satisfied.
If we trace back the denominator in (35), we find that it is derived from
the factor (^ f cos 6) in (26), and that its evanescence depends upon the
antagonistic effects of the terms which are symmetrical and proportional
to cos 6. The precise form of this factor is doubtless connected with the
assumption of a circular crosssection, but the discrepancy from observation
seems almost too complete to be attributed to such departures from the
theoretical shape. As other possible sources of discrepancy we may note
the assumption of reflecting power which is absolutely complete, and again
that the dimensions of the line are small in comparison with the wave
length. It may be that lines sufficiently fine to justify (35) in its integrity
would not reflect enough light to be visible. At the same time the evane
scence of II with a or </> does not demand such a high degree of fineness.
In the memoir already cited Fizeau treats also the polarization impressed
upon light which traverses fine slits. Thus (p. 401): "Une lame d'argent
tresmince, depos6 chimiquement sur le verre, a e"t4 raye'e en ligne droite, avec
de I'e'meri tresfin; c'e'tait un fragment de la lame designee pr^ce'demment
par la lettre (A), et dont I'e'paisseur a te trouve"e de 1/3400 de millimetre.
Un grand nombre de stries avaient traverse la couche d'argent de maniere
a donner naissance a autant de fentes d'une tenuitd extreme. Ces lignes
lumineuses etant observees, a 1'aide d'un analyseur, au microscope eclair^
par la lumiere solaire, ont pr&ente' les phenomenes de polarisation deja
de'crits, c'estadire qu'un grand nombre d'entre elles e"taient polarised dans
un plan perpendiculaire a leur longueur.
" Mais en observant avec plus d'attention les moins lumineuses de toutes
ces lignes, c'estadire celles qui devaient etre les plus fines, on en a trouve'
un certain nombre qui pr^sentaient un phe"nomene de sens oppose", c'estadire
R. v. 27
418 ON LIGHT DISPERSED FROM FINE LINES [324
qu'elles 6taient polarises dans un plan parallele a leur longueur, les unes
totalement, les autres partiellement ; cet effet e'tant accompagne de pheno
menes de coloration semblables k ceux qui ont 6t6 signales dans les lignes
qui donnent la polarisation perpendiculaire."
The passage of electric or luminous waves through a fine slit in a thin
perfectly conducting screen was considered by me in a memoir published
ten years ago*. If the electric vector is parallel to the length of the slit,
the amplitude of the transmitted vibration is proportional to the square of
the width of the slit; but if the electric vector is perpendicular to the
length of the slit, the transmitted vibration involves the width only as a
logarithm see equation (46) much as in equation (21) of the present
paper. If the incident vibration be unpolarized and the slit be very fine,
the latter component preponderates in the transmitted waves, viz. the
direction of polarization is parallel to the length of the slit, in accordance
with Fizeau's observations upon light transmitted by apertures of minimum
width.
* Phil. Mag. Vol. xun. p. 259 (1897); Scientific Papers, Vol. iv. p. 283. See also Phil. Mag.
July 1907. [Scientific Papers, Vol. v. p. 405.]
325.
ON THE RELATION OF THE SENSITIVENESS OF THE EAR
TO PITCH, INVESTIGATED BY A NEW METHOD.
[Philosophical Magazine, Vol. xiv. pp. 596604, 1907.]
IN a former research* I examined the sensitiveness of the ear to sounds
of different pitch with results which were thus summarized :
c', frequency = 256, s = 6'0 x 10,
g', = 384, s = 46 x 10*,
c", = 512, s = 46 x 10,
no reliable distinction appearing between the two last numbers. "Even
the distinction between 6'0 and 4'6 should be accepted with reserve; so
that the comparison must not be taken to prove much more than that the
condensation necessary for audibility varies but slowly in the singly dashed
octave." Here s denotes the condensation (or rarefaction) which in one
respect is a maximum and in another a minimum. It is the maximum
condensation which occurs during the course of the vibration, but the
vibration (and s with it) is the minimum capable of impressing the ear
in a progressive wave. The method employed depended upon a knowledge
of the rate at which energy was emitted from a resonator under excitation
by a freely vibrating tuningfork. The amplitude of vibration of the prongs
of the fork was under continuous observation with the aid of a microscope.
From this could be inferred the energy in the fork at any time and the
rate at which it was lost. The loss was greatest when the resonator was
in action, and the excess was taken to represent the energy converted into
sound. From this again the condensation in the progressive waves at a
given distance could be calculated. It was remarked that the numbers thus
obtained were "somewhat of the nature of upper limits, for they depend
upon the assumption that all the dissipation due to the resonator represents
production of sound. This may not be strictly the case even with the
moderate amplitudes here in question, but the uncertainty is far less than
in the case of resonators or organpipes caused to speak by wind."
* Phil, Mag. Vol. XXXVIH. p. 365 (1894); Scientific Papers, Vol. IT. p. 126.
272
420
ON THE RELATION OF THE
[325
In a carefiil reexamination of this question, M. Wien*, working with
the telephone, finds not only a still higher degree of sensitiveness but a
much more rapid variation with pitch. In the following extract from his
Table xiv., N represents the frequency and A the proportional excess of
pressure, equal to 7*, where 7 is the ratio of specific heats of air (1'41).
The higher degree of sensitiveness may be partly explained by the greater
precautions taken to ensure silence and by the sounds under observation
being rendered intermittent; or, on the other hand, my estimates of sen
sitiveness may have been too low in consequence of the already named
assumption that all the excess of damping due to the resonator represented
production of sound. With respect to the dependence on pitch, Wien
remarks that my own observations! on the minimum current in the tele
phone necessary for audibility at various frequencies, support his view of the
question. Certainly this is their tendency. At the time when these obser
vations were made the whole modus operandi of the telephone was still
involved in doubt, and my object in these observations was rather to elucidate
N
A
N
A
50
l6xlO~ 7
1600
14x10"
100
11x108
3200
14x10"
200
1OxlOo
6400
23x10"
400
12 xlO 10
12800
80x10"
800
23x10"
the action of this instrument. Even now there are points which remain
obscure, for example the easy audibility of sounds when the iron disk is
replaced by one of copper or aluminium. It is to be presumed that the
movements of the disk then depend upon electric currents induced in it.
If so, they would follow different laws from those governing the behaviour
of a simply magnetized disk ; and in the case of iron complications would
ensue from the cooperation of both causes.
Again, though this is partly a matter of definition, I am of opinion that
the sensitiveness of the ear is best investigated with the ear free. When
a telephone, pressed closely up, is employed, the situation is materially
altered. For example, the natural resonances of the earpassage must be
seriously disturbed.
The above objections do not apply to some of Wien's determinations,
where the ear was placed at a distance from the telephone and the vibrations
* Pfliiger's Arch. Vol. xcvii. p. 1 (1903).
t Phil. Mag. Vol. xxxvm. p. 285 (1894); Scientijic Paperi, Vol. iv. p. 109.
1907] SENSITIVENESS OF THE EAR TO PITCH 421
of the plate were directly measured; and his conclusions must necessarily
carry great weight. But I could not forget that my own experiments in
1894 had been carefully made, and I was desirous, if possible, of checking
the results by some new method different from those previously employed
either by Wien or myself. The difficulties of the problem are considerable ;
but it occurred to me that, so far as the important question of the dependence
of sensitiveness upon pitch is concerned, they might be turned by calling
to our aid the general principle of dynamical similarity*. Thus if vibrations
are communicated to the air from the prongs of a tuningfork, we are unable
to calculate the theoretical connexion between the invisible aerial vibrations
and the visible amplitude of the prongs. But if there are two precisely
similar forks of different dimensions, and each communicating vibrations, a
comparison may be effected. In the first place, if the material is the same,
the times of vibration of the forks (regarded as uninfluenced by the air)
are as the linear dimensions. And further, what is more important for our
purpose, the condensations at corresponding points in the surrounding air
will be the same, provided the amplitudes of vibrations at the prongs be
themselves in the proportion of the linear dimensions. Corresponding
points are, of course, such as are similarly situated with respect to the
vibrating forks, the distances from corresponding points of the forks being
in proportion to the linear dimensions of the latter. Since times and
distances are altered in the same proportion, velocities are unchanged. In
conformity with this, the velocitypotentials in the two systems are as the
linear dimensions.
It appears then that by means of the principle of similarity we can
obtain aerial condensations which may be recognized to be equal in spite
of a change of pitch. As has been said, equality occurs when the amplitudes
of the solid vibrators are as the linear dimensions. In virtue of the principle
of superposition as applicable to the small vibrations of either system, we
are not limited to the case of equal condensations. The ratio of condensa
tions can be inferred from the ratio of amplitudes by introduction of the
factor expressing the linear magnification.
My first intention had been to use forks for the actual experiments. But
apart from the difficulty of obtaining them of the necessary geometrical
similarity over a sufficient range, it appeared that the communication of
vibration to the air was inadequate. At a suitable distance there was
danger that the sounds might prove inaudible. In connexion with this
the difficulty of supporting the forks has to be considered. It is essential
that no sound capable of influencing the results shall reach the ear by way
of the supports, to which the principle of similarity can hardly be extended ;
* The application of this principle to acoustical problems is discussed in Theory of Sound,
2nd ed. 381.
422 ON THE RELATION OF THE [325
and the danger of disturbance from this source increases if the direct com
munication of vibration to the air is too much enfeebled. On the other
hand the arrangements must, if possible, be such as will render adequate
the optically observed amplitude up to the point at which the ear is begin
ning to fail. We have in fact to steer as best we may between difficulties
on opposite sides.
The requirements of the case seem to be best met by using thin open
metal cans vibrating after the manner of bells. They were constructed by
Mr Knock from tinplate or ferrotype plate and were maintained in vibration
electrically. The wall is a simple cylinder and there is a flat bottom of
similar material. In the case of the largest can, giving 85 complete vibrations
per second, the height was 8 inches and the diameter 4'5 inches. During
the vibration the circular bottom bends, and thus the can must not be held
fast round the lower circumference. Support was usually given at the centre
only, by means of a short length of tube attached with solder. It is possible,
however, as will be explained more fully later, to give support at four points
of the lower circumference, or rather along two diameters of the base
perpendicular to one another. These diameters are at 45 to the line of
the electromagnet by which the vibrations are maintained. In order to
avoid communicating vibration, the metal handle of the can was attached
to a large bung, resting upon a leaden slab, supported in its turn from the
floor by a tall retortstand. Before or after observations the bung could be
lifted with the fingers and security taken that all the sound heard came
direct from the vibrating can.
The bar electromagnet, by which the iron substance of the can is at
tracted, lies just below a diameter of the upper rim and is supported from
the centre of the base. Since the electromagnet acts as an obstacle to the
aerial vibrations in the region where they are strongest, care must be taken
that in passing from one can to another geometrical similarity extends to
the external form of the magnet and its accessories. The current was
supplied by an interrupterfork and usually both cans under comparison
were driven (alternately) from the same fork, so that no question could arise
with respect to the accuracy of the musical intervals. In constructing the
cans the thickness of the plate employed was taken proportional to the
other linear dimensions, but this alone would not suffice to secure an
accurate tuning. The final adjustment for the greatest possible response
to the intermittent electric current was effected with wax, required only
in small quantities.
Trouble was sometimes experienced from the intrusion of undesired tones.
A shunt across the mercury break of the interrupterfork, as employed by
Helmholtz, was useful, and an improvement would often follow a readjust
ment of the position of the electromagnet. No observations were taken
1907]
SENSITIVENESS OF THE EAR TO PITCH
423
until false sounds had been rendered entirely subordinate, if not inaudible.
In every case the ear was placed in the plane of the rim opposite the loops
L (fig. 1), where the radial motion is 'greatest, two of which face the poles
of the electromagnet. All four positions were utilized, and either the
intensity of vibration or the common distance was varied until the average
intensity was judged suitable. The intensity aimed at was such that the
sound was just easily audible, but sometimes a little more was allowed.
The amplitude of vibration at a loop L was then measured by means of a
microscope provided with a micrometerscale and focused upon starch grains
carried by the rirn. In passing to the can under comparison the distance
of observation is no longer variable at pleasure but must be taken in pro
portion to the linear dimension. Thus if the second can be on half the
scale of the first and sound the octave above, the distance must be halved.
If, when listened to in the four positions, the sound is judged too strong or
Fig. l.
too faint, the vibration must be modified (by varying interposed resistance)
until the former audibility is reproduced. The amplitude of vibration at
a loop is then measured with a second microscope similar to the first. The
distances of the ear, measured from the rim, varied from 8 inches to 24 inches.
The sounds under comparison were usually estimated independently by
Mr Enock and myself. A slight tendency on my part to estimate the graver
sounds as the louder was suspected, but the difference was of no importance.
Other observers also have taken part occasionally, and there was sufficient
repetition on different days to eliminate chance errors. It will suffice to
record the mean results.
In the comparison of cans of dimensions in the ratio of 2 : 1 and making
128 and 256 vibrations per second, it was found that for equal audibility at
distances in the ratio of 2 : 1 the radial amplitude of the larger can required
to be 4'0 times that of the smaller. Equal aerial condensations at the points
424 ON THE RELATION OF THE [325
of observation require amplitudes in the ratio of 2:1, from which we infer
that for equal audibilities the condensation needed at pitch 128 is the double
of that needed at pitch 256. In like manner observations with another pair
of cans showed that the condensation needed at 256 was 1*6 of that needed
at 512 vibrations per second. It did not appear feasible by this method to
go to higher pitch, but the range could be extended at the other end. For
this purpose the largest can already spoken of was constructed, whose
dimensions relatively to the 128 can were as 3 : 2. In this case the interval
was a Fifth, and the comparison showed that the condensation necessary for
audibility at 85 per second was almost precisely the double of that needed
at pitch 128. So far no interval had been attempted exceeding the octave ;
but subsequently confirmation was obtained by a direct comparison between
the cans vibrating 256 and 85 per second. With large intervals the difficulties
are increased, as the amplitude of the smaller can is too minute for satis
factory measurement under the microscope.
Since the numbers have merely a relative value, we may call the con
densation necessary for audibility at pitch 512 unity. The results are then
summarized in the accompanying statement. It was rather to my surprise
N.
512
10
256
16
128
32
85
64
that I found my former conclusion as to the small variation of sensitiveness
in the octave 256 512 substantially confirmed. Below 256 and especially
below 128, it is evident that the sensitiveness of the ear falls off more
rapidly; but even here the differences appear much less than those calcu
lated by Wien from his own observations. I am much at a loss to explain
the discrepancy. Although doubtless criticisms may be made, I should have
supposed that both methods were good enough to yield fairly approximate
results.
To give a general idea of the trend, a plot of the values of s is given in
fig. 2, the logarithm of the periodic time being taken as abscissa. It would
appear that the minimum s, corresponding to maximum sensitiveness of the
ear, would not be reached under 1024 vibrations per second, and perhaps
not until an octave higher, in accordance with Wien's conclusions.
I take this opportunity of recording a few observations on the mode of
vibration of these cans, although the results have no immediate connexion
with the main subject of this note. The theory of the vibration of thin
cylindrical shells vibrating without extension of the middle surface* indicates
two distinct types, of which one is excluded by the action of the plane disk
forming the bottom of a can. The remaining type is defined by the equations
8r = az sin 8<j>, a&<f>= zcoss<f>, Bz = s~ l a sin s<f>, (1)
* Proceeding! Royal Society, Vol. XLV. p. 105 (1888) ; Scientific Papers, Vol. in. p. 217.
1907]
SENSITIVENESS OF THE EAR TO PITCH
425
in which z is measured upwards from the bottom, and the angle <j> is
measured round the circumference, the radius being a. Br, aS<f>, and Sz are
the radial, circumferential, and axial displacements of the point whose equi
librium position is denned by a, <>, z. When z = 0, i.e. at the junction of the
cylindrical and plane parts, 8r and a&<f> vanish, but Sz remains finite except
when sins< = 0. Thus if Bz be constrained to remain zero all round the
circumference of the bottom, no vibration of this kind is possible ; but the
bottom may be supported at the places defined by sins</> = 0, which are
situated under the nodes N (fig. 1) of the radial motion at the upper rim. .
Fig. 2.
1024
512
256
128
85
If, as in the above experiments, we limit ourselves to the principal
vibration for which s = 2, and if the height be I, we have for the maximum
amplitudes
These relations were verified by observation under the microscope. The
ratio of the maximum radial to the maximum circumferential motion at
the upper rim was found to be almost exactly 2:1. The accurate observation
of Bz, as found at the bottom, was more difficult on account of its relative
smallness. In the actual case (85 per second) I = 8 in., 2a = 4 in., so that
8/2a = 14. This is the theoretical ratio of the maximum radial motion at
the rim to the maximum axial motion. By observation the mean number
was 15, as close to theory as could be expected.
326.
EFFECT OF A PRISM ON NEWTON'S RINGS.
[Philosophical Magazine, Vol. xv. pp. 345351, 1908.]
WHEN Newton's rings are regarded through a prism (or grating) several
interesting features present themselves, and are described in the " Opticks."
Not only are rings or arcs seen at unusual thicknesses, but a much enhanced
number of them are visible, owing to approximate achromatism at least on
one side of the centre. The first part of the phenomenon was understood by
Newton, and the explanation easily follows from the consideration of the case
of a true wedge, viz. a plate bounded by plane and flat surfaces slightly
inclined to one another. Without the prism, the systems of bands, each
straight parallel and equidistant, corresponding to the various wavelengths
(X) coincide at the black bar of zero order, formed where the thickness is
zero at the line of intersection of the planes. Regarded through a prism
of small angle whose refracting edge is parallel to the bands, the various
systems no longer coincide at zero order, but by drawing back the prism, it
will always be possible so to adjust the effective dispersive power as to bring
the nth bars to coincidence for any two assigned colours, and therefore
approximately for the entire spectrum.
"In this example the formation of visible rings at unusual thicknesses
is easily understood ; but it gives no explanation of the increased numbers
observed by Newton. The width of the bands for any colour is proportional
to X, as well after the displacement by the prism as before. The manner of
overlapping of two systems whose nth bars have been brought to coincidence
is unaltered ; so that the succession of colours in white light, and the number
of perceptible* bands, is much as usual.
" In order that there may be an achromatic system of bands, it is necessary
that the width of the bands near the centre be the same for the various
colours. As we have seen, this condition cannot be satisfied when the plate
Strictly speaking the number of visible bands is doubled, inasmuch as they are now formed
on both tide* of the achromatic band.
1908] EFFECT OF A PRISM ON NEWTON'S RINGS 427
is a true wedge ; for then the width for each colour is proportional to X. If,
however, the surfaces bounding the plate be curved, the width for each colour
varies at different parts of the plate, and it is possible that the blue bands
from one part, when seen through the prism, may fit the red bands from
another part of the plate. Of course, when no prism is used, the sequence
of colours is the same whether the boundaries of the plate be straight or
curved."
In the paper* from which the above extracts are taken, the question was
further discussed, and it appeared that the bands formed by cylindrical or
spherical surfaces could be made achromatic, so far as small variations of X
are concerned, but only under the condition that there be a finite separation
of the surfaces at the place of nearest approach. If a den6te the smallest
distance, the region of the nih band may form an achromatic system if
a = i?iX ..................................... (1)
At the time pressure of other work prevented my examining the question
experimentally. Recently I have returned to it and I propose now to record
some observations and also to put the theory into a slightly different form
more convenient for comparison with observation.
For the present purpose it suffices to treat the surfaces as cylindrical, so
that the thickness is a function of but one coordinate as, measured along the
surfaces in the direction of the refraction. The investigation applies also to
spherical surfaces if we limit ourselves to points lying upon that diameter
of the circular rings which is parallel to the refraction f. If we choose the
point of nearest approach as the origin of x, the thickness may be taken to be
t = a + bx*, ................................. (2)
where b depends upon the curvatures. The black of the nth order for wave
length X occurs when
(3)
so that jfe .. **  . ............................ (5)
The nth band, formed actually at x, is seen displaced under the action of the
prism. The amount of the linear displacement () is proportional to the
distance D at which the prism is held, so that we may take approximately
d\
.(6)
* "On Achromatic Interference Bands," Phil. Mag. Vol. xxvn. pp. 77, 189 (1889); Scientific
Papers, Vol. in. p. 313.
t In the paper referred to the general theory of curved achromatic bands is considered at
length.
428 EFFECT OF A PRISM ON NEWTON'S RINGS [326
representing the dispersive power of the prism, or grating. The condition
that the nth band may be achromatic (for small variations of X) is accordingly
(7)
a quadratic in n. The roots of the quadratic are real, if
yS 2 D 2 6>a/X a .................................. (9)
If a be zero, the condition (9) is satisfied for all values of D, so that at
whatever distance the prism be held there is always an achromatic band.
And if a be finite, the condition can still always be satisfied if the prism be
drawn back far enough.
From (8) if ,, n a be the roots,
I+IA. ...do)
ni n, 2a
Again, if a = 0, that is if the plates be in contact, ^ = 0, and
n a = 8X/8D"6 ............................... (11)
The order of the achromatic band increases with the dispersive power of the
prism and with the distance at which it is held. The corresponding value
of x from (4) is
x = 2\{3D .................................. (12)
If a be finite, there is no achromatic band so long as D is less than the value
given in (9). When D acquires this value, the roots of the quadratic are
equal, and
or n 1 = w s = 4a/X ............................... (13)
This is the condition formerly found for an achromatic system of bands.
If D be appreciably greater than this, two values of n satisfy the condition,
viz. there are two separated achromatic bands, though no achromatic system.
From (8)
n lWf = 16a6/8D ............................ (14)
Thus if D be great, one of the roots, say n^, becomes great, while the other,
see (10), approximates to 2a/X, that is to half the value appropriate to the
achromatic system (13).
There is no particular difficulty in following these phenomena experi
mentally, though perhaps they are not quite so sharply defined as might be
expected from the theoretical discussion, probably for a reason which will
be alluded to presently. It is desirable to work with rather large and but
1908] EFFECT OF A PRISM ON NEWTON'S RINGS 429
very slightly curved surfaces. In my experiments the lower plate was an
optical " flat " by Dr Common, about six inches in diameter and blackened
behind. The upper plate was wedgeshaped with surfaces which had been
intended to be flat but were in fact markedly convex. In order to see the
bands well, it is necessary that the luminous background, whether from
daylight or lamplight, be uniform through a certain angle, and yet this
angle must not be too large. Otherwise it is impossible to eliminate the
light reflected from the upper surface of the upper plate, which to a great
extent spoils the effects. In my case it sufficed to use gaslight diffused
through a groundglass plate whose angular area was not so great but that
the false light could be thrown to one side in virtue of the angle between the
upper and lower surfaces of the wedge*. It will be understood that these
precautions are needed only in order to see the effects at their best. The
most ordinary observation and appliances suffice to exhibit the main features.
Another question which I was desirous of taking the opportunity to
examine was one often propounded to me by my lamented friend Lord
Kelvin, viz. the nature of the obstruction usually encountered in trying to
bring two surfaces nearly enough together to exhibit the rings of low order.
In favour of the view that the obstacle is merely dust and fibres, I remember
instancing the ease with which a photographic print, enamelled by being
allowed to dry in contact with a suitably prepared glass plate, could be
brought back into optical contact after partial separation therefrom. My
recent observations with the glass plates point entirely in the same direction.
However carefully the surfaces are cleaned by washing and wiping finally
with a dry hand, the rings of low order can usually be attained only at certain
parts of the surface f. If we attempt to shift them to another place chosen
at random, they usually pass into rings of higher order or disappear altogether.
On the other hand, when rings of low order have once been seen at a par
ticular place, it is usually possible to lift the upper glass carefully and to
replace it without losing the rings at the place in question. I have repeatedly
lifted the glass when the centre of the system was showing the white of the
first order or even the darkening (I do not say black) corresponding to a still
closer approximation, and found the colour recovered under no greater force
than the weight of the glass. Some time is required, doubtless in order that
the air may escape, for the complete recovery of the original closeness ; but
in the absence of foreign matter it appears that there is no other obstacle to
an approximation of say ^\.
In making the observations it is convenient to introduce a not too small
magnifying lens of perhaps 8 inches focus and to throw an image of the
* Compare "Interference Bands and their Applications," Scientific Papers, Vol. iv. p. 54.
t The plates are here supposed to be brought together without sliding. By a careful sliding
together of two surfaces, the foreign matter may be extruded, as in Hilger's echelon gratings,
where optical contact is attained over considerable areas.
430 EFFECT OF A PRISM ON NEWTON'S RINGS [326
source of light upon the pupil of the eye. With the glasses in contact it is
easy to trace the rise in the order of the achromatic band as the eye and
prism are drawn back. As regards the latter a directvision instrument of
moderate power (three prisms in all) is the most suitable. An interval
between the glasses may be introduced by stages. When the approximation
is such as to show colours of the 3rd or 4th orders at the centre, it becomes
apparent that the best achromatic effects are attained when the prism is
at a certain distance, and that when this distance is exceeded the more
achromatic places are separated by a region where the bands are fringed
with colour. This feature becomes more distinct as the interval is still
further increased, so that without the prism only faint rings or none at all
can be perceived. For the greater intervals the interposition of a piece of
mica at one edge is convenient. In judging of the degree of achromatism,
I found that narrow coloured borders could be recognized as such much more
easily by one of my eyes than by the other, and the difference did not seem
to depend on any matter of focusing.
In observing bands of rather high order, the question obtruded itself as
to whether the achromatism was anywhere complete. It will have been
remarked that the theoretical discussion, as hitherto given, relates only to
a small range of wavelength and that no account is taken of what in the
telescope is called secondary colour. So long as this limitation is observed,
the character of the dispersive instrument does not come into play. It
appeared, however, not at all unlikely that even with gaslight the range of
wavelength included might be too great to allow of this treatment being
adequate ; and with daylight, of course, the case would be aggravated. It is
thus of interest to examine what law of dispersion is best adapted to secure
compensation and in particular to compare the operation of a prism and a
grating.
As to the law of dispersion to be aimed at, we have from (4), if X = \ + B\,
If be the displacement due to the instrument, f should be a similar function
of SX. In this matter the constant terms (independent of 8X) are of no
account, and the terms in 8x may be adjusted to one another, as already
explained, by suitably choosing the distance D. In pursuing the approxima
tion, what we are concerned with is the ratio of the term in (8X) a to that in
SX. And in (15) this ratio is
i ^
............................. (16)
1908] EFFECT OF A PRISM ON NEWTON'S RINGS 431
thus in the particular cases
Corresponding expressions are required for the dispersive instruments. In
any particular case they could of course be determined ; but no very simple
rules are available in general. If the intrinsic dispersion be small the
necessary effect being arrived at by increasing D, we may make the com
parison more easily. Thus in the case of the grating the variable part of
is proportional to S\ simply, so that the ratio of the second and third terms,
corresponding to (16), is zero. And in the case of the prism if we assume
Cauchy's law of dispersion, viz. /A = A + B\~*, we get in correspondence
with (16)
So far as these expressions apply, it appears that the dispersion required is
between that of a grating and of a prism, and that especially when a = the
grating gives the better approximation. It would be possible to combine
a grating and a prism in such a way as to secure an intermediate law, the
dispersions cooperating although the deviations (in the case of a simple
prism) would be in opposite directions.
I have made observations with a grating, using for the purpose a photo
graphic reproduction upon bitumen*. This contains lines at the rate of
6000 to the inch and gives very brilliant spectra of the first order. I thought
that I could observe the superior achromatism of the most nearly achromatic
bands as compared with those given by the prism, but the conditions were
not very favourable. The dispersive power was so high that the grating had
to be held very close, and the multiplicity of spectra was an embarrassment.
If it were possible to prepare a grating with not more than 3000 lines to the
inch, and yet of such a character that most of the light was thrown into one
of the spectra of the first order, it might be worth while to resume the
experiment and, as suggested, to try for a more complete achromatism by
combining with the grating a suitable prism.
* Nature, Vol. LIV. p. 332 (1896) ; Scientific Papers, Vol. iv. p. 226.
327.
FURTHER MEASUREMENTS OF WAVELENGTHS, AND
MISCELLANEOUS NOTES ON FABRY AND PEROT'S
APPARATUS.
[Philosophical Magazine, Vol. xv. pp. 548558, 1908.]
IN a former paper* I described a modified form of apparatus and gave
the results of some measurements of wavelengths, partly in confirmation of
numbers already put forward by Fabry and Perot and partly novel, relating
to helium. I propose now to record briefly some further measures by the
same method, together with certain observations and calculations relating
thereto of general optical interest.
The apparatus was arranged as before, the only change being in the
interferencegauge itself. The distancepieces, by which the glasses are kept
apart, were now of invar, with the object of diminishing the dependence upon
temperature. The use of invar for this purpose was suggested by Fabry and
Perot, but I do not know whether it has actually been employed before. The
alloy was in the form of nearly spherical balls, 5 mm. in diameter, provided
with projecting tongues by which they were firmly fitted to the iron frame.
The springs, holding the glasses up to the distancepieces, were of the usual
pattern. The whole mounting was constructed by Mr Enock, and it answered
its purpose satisfactorily. There is no doubt, I think, as to the advantage
accruing from the use of invar.
The measurements were conducted as explained in the earlier paper.
The first set related to zinc which was compared with cadmium. Both
metals were used in vacuumtubes, of the pattern already described, with
electrodes merely cemented in. It was rather to my surprise that I found
ordinary soft glass available in the case of zinc, but no difficulty was
experienced. The former observations with the "trembler" suggested a
wavelength for zinc red about onemillionth part greater than that (6362'345)
given by Fabry and Perot. This correction has been confirmed, and I would
Phil. Mag. [6] VoL u. p. 685 (May 1906). [This Collection, Vol. v. p. 313.]
1908]
FURTHER MEASUREMENTS OF WAVELENGTHS
433
propose 6362350, as referred to Michelson's value of the cadmium red, viz.
6438'4722. No difficulty was experienced in identifying the order of the
rings by the method formerly described and dependent upon observations
with the gauge alone.
The results of the measurements upon helium were not in quite such
close accord with the earlier ones as had been expected. Both sets are given
below for comparison.
Wavelengths of Helium.
II.
III.
7065192
7065200
6678147
6678150
5875618
5875625
5015682
5015680
4921927
4921930
(4713173)
4713144
(4471480)
4471482
The two last entries under II., enclosed in parentheses, were obtained
with the 1 mm. apparatus, and could not be expected to be very accurate.
Preference may be given to III. throughout.
These measurements of wavelengths were not further pursued, partly
because it was understood that other observers were in the field and partly
because my own vision, though not bad, is less good than it was. In particular
at the blue end of the spectrum I found difficulty. It is evident that work
of this sort should be undertaken only under the best conditions.
One of the less agreeable features of the method is the complication
which arises from the optical distance between the surfaces being slightly
variable with the colour. In the earlier observations with a 5 mm. apparatus
I was surprised to find the change amounting to 2 parts per million between
cadmium red and cadmium green. In the light of subsequent experience
I am disposed to think that the silver surfaces must have been slightly
tarnished. At any rate in the later measurements I found the difference
much less, indeed scarcely measurable. It will be understood that no final
uncertainty in the ratio of wavelengths arises from this cause. Whatever
the change may prove to be, it can be allowed for.
Thirty Millimetre Apparatus.
In this instrument the object was to construct a gauge with a much
greater distance than usual between the plates, but otherwise on the same
general plan as that of Fabry and Perot. The distancepiece A A, fig. 1,
consisted of a 30 millimetre length of glass tubing, each end being provided
434
FURTHER MEASUREMENTS OF WAVELENGTHS
[327
with three protuberances, equally spaced round the circumference, at which
the actual contacts took place. The removal of the intervening material and
the shaping of the protuberances were effected with a file moistened with
turpentine.
Fig. l.
I
1908] FURTHER MEASUREMENTS OF WAVELENGTHS 435
Against this distancepiece the glass plates B B are held by the arrange
ment shown in fig. 1. The lower plate B rests upon a brass ring G to which
the brass castings D are rigidly attached. The upper ring E is connected
with the castings only through the steel springs F. Both rings are provided
with protuberances in line with those on the glass cylinder, and the pressure
is regulated by the screws G. The whole was constructed by Mr Enock.
Some little care is required in putting the parts together to avoid scratching
the halfsilvered faces ; but when once the apparatus is set up its manipula
tion is as easy as that of the ordinary type.
In all interferencegauges it is desirable that the distancepieces be
adjusted as accurately as possible. For although a considerable deficiency
in this respect may be compensated by regulating the pressures (see below),
the adjustment thus arrived at is less durable, at least in my experience.
Even when the distancepieces are themselves well adjusted, it is advisable
to employ only moderate pressures.
Observations with the 30 mm. gauge have been made upon helium,
thallium, cadmium, and mercury. In the first case the (yellow) rings are
faint, the retardation being not far from the limit. Indeed when at first it
was attempted to adjust the plates with helium, the rings could not be
found. With thallium also the rings were rather faint, but with mercury
and cadmium there was no difficulty.
Magnifying Power.
At a distance of 30 mm. the rings are rather small, and one is tempted to
increase the magnifying power of the observing telescope. As to this there
should be no difficulty if the aperture could be correspondingly increased.
But although the plates themselves may be large enough, an excessive strain
may thus be thrown upon the accuracy of the figuring and upon the adjust
ment to parallelism. If, on the other hand, the aperture be not increased,
the illumination of the image fails and the extra magnifying may do more
harm than good.
A means of escape from this dilemma is to effect the additional magni
fication in one direction only, which in the present case answers all purposes.
When straight interferencebands, or spectrum lines, are under observation,
there is no objection to astigmatism, and we may merely replace the ordinary
eyepiece of the telescope by a cylindrical lens or by a combination of spherical
and cylindrical lenses. This arrangement can be employed in the present
instance, but the result is not satisfactory. A complete focusing, leading to
a pointtopoint correspondence between image and object, may however be
attained by suitably sloping the objectlens of the telescope. In this way
excellent observations upon interferencerings are possible under a magnifying
282
436 FURTHER MEASUREMENTS OF WAVELENGTHS [327
power which otherwise would be inadmissible, as entailing too great a loss of
light. The subject will be more fully treated in a special paper*.
Adjustment for Parallelism.
If the surfaces are flat, and welladjusted, Haidinger's rings depend
entirely upon obliquity. A slight departure from parallelism shows itself
by an expansion or contraction of the rings as the eye is moved about so
as to bring different parts of the surfaces into play. In making this ob
servation the eye must be adjusted to infinity, if necessary with the aid of
spectacleglasses, and it may be held close to the plates ; but a telescope is
not needed or even desirable. If the departure from parallelism be con
siderable, no rings at all are visible ; but there is an intermediate state of
things where circular arcs may be seen by an eye drawn back somewhat and
focused upon the plates.
The character of these bands is intermediate between those of Newton's
and Haidinger's rings, the retardation depending both upon the varying
direction in which the light passes the plates and reaches the eye and also
upon the varying local thickness. If we take, as origin of rectangular
coordinates in the plane of the plates, the place corresponding to normal
passage of the light, the retardation due to obliquity is as (a 8 + y 2 ). The
retardation due to local thickness is represented by a linear function of x
and y, so that the variable part of it may be considered to be proportional
to x. Hence the equation of the bands is
ax a? y 2 = constant,
a being positive if x is considered positive in the direction of increasing
thickness. Accordingly the bands are in the form of concentric circles and
the coordinates of the centre are
When curved arcs are seen by an eye looking at the plates perpendicu
larly, the greatest thickness lies upon the concave side of the arcs. The
perpendicular direction of vision may be tested by observing the reflexion
of the eye itself in the silvered surface.
Behaviour of VacuumTubes.
The form of vacuumtube described in the first paper, and depending
on sealingwax for airtightness, continues to give satisfaction. As already
mentioned, though made of soft glass, they are available for zinc, and the
cadmium tubes have lasted well with occasional reexhaustion. It is ad
visable to submit them to this operation when the red. light begins to fall
* [See Proc. Roy. Soc. Vol. LXXXI. p. 26 (1908) ; This Collection, Vol. v. Art. 328, Part n.]
1908] FURTHER MEASUREMENTS OF WAVELENGTHS 437
off. After one or two reexhaustions the condition seems to be more
durable.
With thallium my experience has been rather remarkable. The green
light is very brilliant and offers a further advantage as being comparatively
free from admixture with other colours*. But Fabry and Perot found
thallium tubes to be very shortlived, sometimes lasting only a few minutes.
I have used but one thallium tube, of the same construction as the others,
and charged with a little thallium chloride. This tube has been used
without special care on many occasions I cannot say how many, but
probably seven or eight times, and it does not appear to have deteriorated
at all. It looks as though the chloride had decomposed and metal had
deposited upon the aluminium electrodes. But what the circumstances can
be that render my experience so much more favourable I am at a loss to
conjecture.
The same form of tube answers well for mercury, but with this metal
there is usually no difficulty.
Control of the figure of the glasses by bending.
Very good plates can now be procured from the best makers, but on
careful testing they usually show some deficiency, mostly of the nature of
a slight general curvature. Thus when in Fabry and Perot's apparatus the
adjustment for parallelism is made as perfect as possible, the rings may be
observed to dilate a little as the eye moves outwards in any direction from
the centre towards the circumference of the plates. This indicates a general
convexity.
It occurred to me that an error of this kind might be approximately
corrected by the application of bending forces to one of the plates it does
not matter which. The easiest way to carry out the idea is to modify the
apparatus in such a way that the points of application of external pressure
are not exactly opposite the contacts with the distancepieces, but are
displaced somewhat inwards or outwards in the radial direction (fig. 2).
If the plates are too convex, the points of pressure must be F  2
displaced outwards. In this form I have tried the experiment 
with a certain degree of success, but the displacements that
I could command (1 mm. only) were too small in relation to
the thickness of the plate. If it were intended to give this
plan a proper trial, which I think it would be worth in order
to render a larger aperture than usual available, the plates, or
at least one of them, should be prepared of extra diameter, so
that the bending forces could act with a longer leverage and at
* The green line is known to be itself double.
438 FURTHER MEASUREMENTS OF WAVELENGTHS [327
a greater distance from the parts to be employed optically. Such a con
struction need not involve a much enhanced cost, inasmuch as the outer
parts would not need to be optically accurate.
It may be worth while to consider the question here raised more
generally. The problem is so to deform one surface, by forces and couples
applied at the boundary, as to compensate the joint errors of the two
surfaces and render the distance between them constant. If we take rect
angular coordinates x, y in the plane of the surface with origin at the
centre, the deformation obtainable in this way is expressed by terms in the
value of f (the other coordinate of points on the surface) proportional to
x, y, x*, xy, y 8 , x 3 , a?y, xy 3 , y 3 . For such terms are arbitrary in the solution
of the general equation of equilibrium of a plate, viz.
Of these terms those in x and y correspond of course merely to the adjust
ment for parallelism, and those of the second degree to curvature at the
centre. The conclusion is that we may always, by suitable forces applied to
the edge, render the distance between the plates constant, so far as terms of
third order inclusive.
Another inference from the same argument is that, in any optical
apparatus, approximately plane waves of light may be freed from curvature
and from unsymmetrical aberration (expressed by terms of the third order)
by means of reflexion at a plate to the boundary of which suitable forces
are applied. And the surface of the plate need not itself be more than
approximately flat.
Figuring by Hydrofluoric Add.
It would be poor economy to employ any but the best surfaces in
measuring work needing high accuracy ; but there are occasions when all
that is needed can be attained by more ordinary means. Common plate
glass is rarely good enough*; but I have found that it can be refigured
with hydrofluoric acid so as to serve fairly well, and the process is one of
some interest. From what has been said already it will be understood that
it is not necessary to make both surfaces plane, but merely to fit them
together, which can be effected by operations conducted upon one only.
Pieces of selected plateglass, about inch thick and of a size suited to
the interferencegauges, were roughly shaped by chipping. The best surfaces
were superposed and the character of the fit examined by sodalight. One
If the surfaces are so shaped that the interferencebands presented on superposition are
hyperbolic, much may be gained by limiting the aperture to a narrow slit corresponding to one of
the asymptotes, especially if the magnification used is in one direction only.
1908] FURTHER MEASUREMENTS OF WAVELENGTHS 439
glass being rotated upon the other, the most favourable relative azimuth was
chosen; and by means of suitable marks upon the edges the plates were
always brought back to the chosen position.
The principles upon which the testing is conducted have been fully
explained in a former paper*. In the present case the surfaces are so close
to one another that no special precautions are required. With a little
management the contact is so arranged that a moderate number of bands
are visible. If the fit were perfect, or rather if the surfaces were capable
of being brought into contact throughout, these bands would be straight,
parallel and equidistant. Any departure from this condition is an error
which it is proposed to correct. The sign of the error can be determined
without moving the glasses by observing the effect of diminishing the
obliquity of reflexion, which increases the retardation. Thus if a band is
curved, and the change in question causes the band to move with convexity
forwards, it is a sign that material needs to be removed from the parts of
the glass occupied by the ends of the band. Such an operation will tend to
straighten the band. If, however, the movement take place with concavity
forward, then material needs to be removed from the middle parts. In every
case the rule is that by removal of glass the bands, or any parts of them, can
be caused to move in the same direction as that in which they move when
the obliquity of reflexion is diminished.
In carrying out the correction, the plate on which it is intended to
operate is placed below, and it is convenient if it be held in some form of
steady mounting so that the upper plate can be removed and replaced in
the required position without trouble. The acid, two or three times diluted,
is applied with a camel's hair brush and after being worked about for a few
seconds is removed suddenly with a soft cloth. Endeavour should be made
to keep the margin of the wetted region moving in order to obviate the
formation of hard lines. Success depends of course upon judgment and
practice, and the only general advice that can be given is to make a great
many bites at the cherry, and to keep a record of what is done each time
by marking suitably on one of a system of circles drawn upon paper and
representing the surface operated on. After each application of acid the
plates are reexamined by sodalight and the effect estimated. The difficulty
is that in most cases the bands are not reproduced in the same form. In one
presentation the error may reveal itself as a curvature of the bands and in
another as an inequality in the spacing of bands fairly straight. Often by
a little humouring the original form may be approximately recovered, and in
any case the general rule indicates what needs to be done.
* " Interference Bands and their Applications," Nature, Vol. XLVIII. p. 212 (1893) ; Scientific
Papers, Vol. iv. p. 54.
440 FURTHER MEASUREMENTS OF WAVELENGTHS [327
By this method I have prepared two pairs of plates which perform very
fairly well, but of course only when placed in the proper relative position.
The operations, though prolonged, are not tedious, and I doubt not that with
perseverance better results than mine might be achieved. The surface of
the glass under treatment suffers a little from the development of previously
invisible scratches in the manner formerly explained, but the defect hardly
shows itself in actual use. I have not ventured to apply the method to
surfaces already very good such as those supplied by the best makers for use
in Fabry and Perot's apparatus ; but I should be tempted to do so if I came
across a pair suffering from slight general concavity. The application of acid
would then be at the outer parts. In the best glasses that I possess the
error is one of convexity.
Effect of Pressure in Fabry and Perot's Apparatus.
The observation that the rings were more sensitive than had been
expected to the pressure by which the plates are kept up to the distance
pieces, led to a calculation on Hertz's theory of the relation between the
change of interval and the pressure. If two spheres of radii r x and r 2 and of
material for which the elastic constants in Lamp's notation are X,, /*,, X 2 , //>,
are pressed together with a force P, the relation between P and the distance
(a) through which the centres approach one another, as the result of the
deformation in the neighbourhood of the contact, is
''
In the case of materials which satisfy Poisson's condition, \ = p, and we
may take as sufficiently approximate
,3 3
so that
In the application that we have to make, one of the spheres is of steel
(invar) and of radius ^ = '25 cm., while the other is of glass and of radius
r s = oo . Further, for the steel we may take ^ = 8'2 x 10", and for glass
^=24x10", and thus
P = 330 x 10" . e
a being in cm. and P in dynes. It will be convenient for our purpose to
See Love's Math. Theory of Elatticity , 139.
1908] FURTHER MEASUREMENTS OF WAVELENGTHS 441
reckon a in wavelengths (equal say to 6 x 10~ 5 cm.) and P in kilograms,
taking the dyne as equal to a milligram weight. On this understanding
signifying that to cause an approach of one wavelength the force required is
15 kilogram. If P and a undergo small variations,
dP = f (15) c$da = f (15)*plda,
dP/dot being somewhat dependent upon the total pressure P.
For the purposes of experiment a springbalance was mounted upon
the frame of the apparatus (carrying the distancepieces) so as to diminish
the pressure exerted over one of the distancepieces, that is to diminish the
pressure by which one of the plates was held up to one of the distancepieces.
Starting from perfect parallelism of the plates and keeping the eye carefully
fixed so as to receive the light from the centre of the plates, it was observed
that to cause a shift of one band (helium yellow) the springbalance needed
to exert a pull of "78 kilo. At this stage the plates were of course no longer
parallel and a moderate shift of the eye would cause a displacement in
dependently of any change in the springbalance. At the same time the
rings lost their sharpness. On this account it is hardly practicable to use
a shift of more than one band, and indeed a smaller shift of half a band was
usually preferred. The total force required to compensate the spring of the
apparatus, and so to relieve the compression of the distancepiece on this
side, was 2'4 kilos. This is what is represented by P in the above formula,
while dP =  78.
In order now to compare theory and observation we must remember that
the one band (corresponding to half a wavelength) observed at the centre
implies three times as great a shift at the particular distancepiece where
alone the force was varied. Thus the observed dP corresponds to da =  f .
For this da, the calculated dP is
dP =  f (15)3 (24)1 = _ 85 kilo.
The agreement with the observed '78 is certainly as good as could have
been expected.
In considering what differences of distance are to be expected when the
plates are adjusted to parallelism under different pressures, we must re
member that the above calculation and observation relates to the compression
which may occur at the contact of a single distancepiece with a single plate.
There are in all six contacts of this kind, and we may conclude that when no
special pains are taken to regulate the absolute pressures employed, a shift of
six bands or more on remounting need not cause surprise.
328.
ON THE ABERRATION OF SLOPED LENSES AND ON THEIR
ADAPTATION TO TELESCOPES OF UNEQUAL MAGNIFYING
POWER IN PERPENDICULAR DIRECTIONS.
[Proceedings of the Royal Society, A, Vol. LXXXI. pp. 2640, 1908.]
THE present paper consists of two parts to a large extent independent.
The reader who does not care to follow the details of calculation may take
the results relative to unsymmetrical aberration for granted. The subject
of the second part is somewhat larger than the title. It treats of the
advantage which often attends a magnification unequal in different directions
and of the methods available for attaining it. Among these is the method
of the sloped objectlens. Such sloping introduces in general unsymmetrical
aberration. The intention of the first part is largely to show how this may
be minimised so as to become unimportant.
PART I.
Before proceeding to actual calculations of the aberrations of a sloped
lens, it may be well to consider briefly the general character of a pencil of
rays affected with unsymmetrical aberration.
The axis of the pencil being taken as axis of z, let the equation of the
wavesurface, to which all rays are normal, be
/ Sy*+ (1)
2p 2p
The principal focal lengths, measured from z = 0, are p and p. In the
case of symmetry about the axis, p and p are equal, and the coefficients of
the terms of the third order vanish. The aberration then depends upon
terms of the fourth order in x and y, and even these are made to vanish in
the formulae for the objectglasses of telescopes by the selection of suitable
curvatures. In the theory of imperfectly constructed spectroscopes and of
1908] ON THE ABERRATION OF SLOPED LENSES, ETC. 443
sloped lenses it is necessary to retain the terms of the third order, but we
may assume a plane of symmetry y = Q, which is then spoken of as the
primary plane. The equation of the wavesurface thus reduces to
v >
terms of higher order being omitted. In (2) p is the primary and p the
secondary focal length.
The equation of the normal at the point x, y, z is
 t= %~ x __ = *>y ...(5)
es/p + Seta? I 7i/ 2 y/p + fyxy ' ' '
and its intersection with the plane f = p occurs at the point determined
approximately by
= / 3(3a^ + 72 / 2 ), r ) =^^y2py X y > ............ (6)
terms of the third order being omitted.
According to geometrical optics, the thickness of the image of a luminous
line (parallel to y) at the primary focus is determined by the extreme value
of , and for good definition it is necessary to reduce this thickness as much
as possible. To this end it is necessary in general that both a and 7
be small.
We will now examine more closely the character of the image at the
primary focus in the case of a pencil originally of circular section. Unless
p' = p, the second term in the value of ij in (6) may be neglected. The
rays proceeding from the circle a; 2 + ;z/ 2 = r 2 intersect the plane =p in the
parabola
and the various parabolas corresponding to different values of r differ from one
another only in being shifted along the axis of (. To find out how much of
the parabolic arcs is included, we observe that for any given value of r the
value of r) is greatest when x = 0. Hence the rays starting in the secondary
plane give the remainder of the boundary of the image. Its equation,
formed from (6) after putting # = 0, is
*? ..............................
and represents a parabola touching the axis of 17 at the origin. The whole
of the image is included between this parabola and the parabola of form (7)
corresponding to the maximum value of r.
The width of the image when rj = is 3o/? r 2 , and vanishes when a = 0,
i.e., when there is no aberration for rays in the primary plane. In this case
444 ON THE ABERRATION OF SLOPED LENSES, ETC.
the two parabolic boundaries coincide, and the image is reduced to a linear
arc. If, further, 7 = 0, this arc becomes straight, and then the image of
a short luminous line (parallel to y} is perfect to this order of approximation
at the primary focus. In general, if 7 = 0, the parabola (8) reduces to the
straight line f = 0; that is to say, the rays which start in the secondary
plane remain in that plane.
We will now consider the image formed at the secondary focus. Putting
= p' in (5), we obtain
(9)
If 7 = 0, the secondary focal line is formed without aberration, but
not otherwise. In general, the curve traced out by the rays for which
00)
in the form of a figure of 8 symmetrical with respect to both axes. The
rays starting either in the primary or in the secondary plane pass through
the axis of f, the thickness of the image being due to the rays for which
Or if in order to find the intersection of the ray with the primary plane
we put 17 = in (5), we have approximately
p I Ip' + Zyx '
showing that f is constant only when 7 = 0.
The calculation of aberration for rays in the primary plane is carried out
in the paper cited for the case of a thin lens sloped through a finite angle.
If the curvature of the first surface be 1/r and of the second 1/s, and if p be
the refractive index, the focal length /i in the primary plane is given by
7i" * (r~)'
and the condition that there shall be no aberration is
11 S 7*
Here u is the distance of the radiant point from the lens, <j> the obliquity
of the incident ray, <f>' of the refracted ray, c = cos<, c / = cos<^ / , and
/*' = /* cos <f>'/cos<f>.
The above is taken from my " Investigations in Optics," Phil. Mag. 1879; Scientific Papers,
Vol. i. p. 441, and following. Some errata may be noted : p. 441, line 9, insert y as factor in
the first term of , ; p. 443, line 9, for (7) read (8), line 10, for r, read .
1908] ON THE ABERRATION OF SLOPED LENSES, ETC. 445
A result, accordant with (12), but applicable only when <f> is small, was
given in another form by Mr Dennis Taylor in Astron. Soc. Monthly Notices,
Ap., 1893.
If the incident rays be parallel, u = oo , and the condition of freedom from
aberration is
As appears from (11), opposite signs for r and s indicate that both surfaces
are convex.
If <f> = 0, p = fj,, so that (13) gives, in this case,
Thus, if fi = 1*5, the aberration vanishes for small obliquities when
s =. : 9r. This means a double convex lens, the curvature of the hind surface
being oneninth of that of the front surface. If s = oo , that is, if the lens be
planoconvex with curvature turned towards the parallel rays,
l + /4/* 2 = 0, .............................. (15)
or fi = (1 + V5) = 1618.
Returning to finite obliquity, we see from (13) that whatever may be the
index and obliquity of the lens, it is possible so to choose its form that the
aberration shall vanish. If the form be planoconvex, the condition of no
aberration is
1 + //V 2 = 0, .............................. (16)
or /*' = fi cos <'/cos <f> = 1618.
Here cos <f>' > cos tj>, and the ratio of the two cosines increases with
obliquity from unity to infinity. Hence if /* > 1*618, there can be no
freedom from aberration at any angle. When /u. = 1*618, the aberration
vanishes, as we have seen, when <f> = 0. If fj, be less than 1*618, the
aberration vanishes at some finite angle. For example, if /u, = 1*5, this occurs
when <f> = 29.
In many cases the aberration of rays in the secondary plane is quite as
important as that in the primary plane. In my former paper I gave a result
applicable to a planoconvex lens, on the curved face of which parallel light
falls. It was found that the secondary aberration vanished when the relation
between obliquity and refractive index was such that
< 17 >
For small values of <j> this gives the same index as before (15), inasmuch as
446 ON THE ABERRATION OF SLOPED LENSES, ETC. [328
I inferred that for a planoconvex lens of index T618 neither kind of
aberration is important at moderate slopes.
Having no note or recollection of the method by which (17) was obtained,
and wishing to confirm and extend it, I have lately undertaken a fresh
investigation, still limiting myself, however, to parallel incident rays. For
simplicity, the lens may be supposed to come to a sharp circular edge, the
plane containing this edge being that of X Y. The centre of the circle is the
origin, and the axis of Z is the axis of the lens. The incident rays are parallel
to the plane ZX, and make an angle $> with OZ\ so
that 4> is the angle of incidence for the ray which
meets the first surface of the lens at its central point.
Everything is symmetrical with respect to the primary
plane y = 0. It will suffice to consider the course of
the rays which meet the lens close to its edge, of
which the equation is a? + y 2 = R*, if 2 R be the
diameter.
In order to carry out the calculation conveniently,
we require general formulas connecting the directioncosines of the refracted
ray with those of the incident ray and of the normal to the surface. If we
take lengths AP, AQ along the incident and refracted rays proportional to
p, fi, the indices of the medium in which the rays travel, and drop per
pendiculars PM, QATupon the normal MAN, then by the law of refraction the
lines PM, NQ are equal and parallel ; and the projection of PA + AMon any
axis is equal to the projection of NA + AQ on the same axis. Thus if I, m, n
are the directioncosines of the incident ray, V, m, n of the refracted ray,
p, q, r of the normal taken in the direction from the medium in which the
light is incident, <f>, </>' the angles of incidence and refraction,
fjd /JL cos <f> . p = fj.' cos <' . p + fi'l'
and two similar equations. Hence
(/*' I' fjd)/p = (n'm  nm)/q = (pri /m)/r = /t' cos </>'  p cos <j>. ...(18)*
Also cos< = Ip + mq + nr, (19)
and </>' is given by ^ sin $ = p sin <j> (20)
For our purpose there is no need to retain the two refractive indices, and
for brevity we will suppose that the index outside the lens is unity and
inside it equal to /A; so that in the above formulae we are to write p,= 1,
fi = p. Hence
0^'  0/P = (pm'  m)lq = (pri  n)/r = p cos <'  cos <f>. ...(21)
Equation (19) remains as before, while (20) becomes
p sin <' = sin <j> (22)
* See Herman's Geometrical Optics, Cambridge, 1900, p. 22.
1908] ON THE ABERRATION OF SLOPED LENSES, ETC. 447
For the first refraction at the point x, y, we have
I sin <E>, m = 0, n = cos < ;
and if Xi be the angle which the normal to the first surface at the edge of
the lens makes with the axis,
P = sin %i . x/R, q = sin X i y/R, r = cos Xi ;
so that
ul' sin ^> am' an' cos <I>
^7^ =  = nf = = a cos <f> cos <f> = (7, , (23)
x/R . sm Xi ylR sin %i cos Xi
and cos< = sin <I> sin Xi x IR + cos ^ c s%i ................ (24)
In like manner if I", m", n" be the directioncosines of the twice
refracted ray, p, q, r' those of the second normal, we may take
I" id' m" urn' n" tin' . ,
"
/p D
x/R . sin x? y/R sm %2 cos %2
if i/r, $>' be respectively the angles of incidence and refraction at the second
surface.
Here cos i/r = I'p' + m'q' + rir' ......................... (26)
Eliminating I', m, n between (23) and (25), we get
I" = sin <I> + ((7i sin Xi + C* sin %z) */^B,
m" =(C l sin Xi + C 2 sin Xz) y/R>
n" = cos <& + G l cos Xi + ^2 cos % 2 .
The equation of the ray after passage through the lens is
4 ...(27)
The aberration in the secondary plane (depending on 7) is most simply
investigated by inquiring where the ray (27) meets the primary plane ij = 0.
For the coordinates of the point of intersection,
J'y_
"~
^_^ l  2 ^
m" d sin Xi + @2 si n X*
In interpreting (28), (29) we must remember that is now measured
parallel to the axis of the lens and not, as in the preliminary discussion,
along the principal ray. Freedom from aberration requires that the line
determined by varying x and y in (28), (29) should be perpendicular to the
principal ray, or that f cos $ + sin < should be constant. And
1 + (fi cos %i + C* cos % a ) cos 3> ^
(7j sin Xi + Ca 8 i n %a
44S ON THE ABERRATION OF SLOPED LENSES, ETC.
Before proceeding further it may be well to compare (30) with known
results when the aberration is neglected. For a first approximation we may
identify </> and ^>' with 4> and 4>', and also ^ and i/r' with 4>' and <1> respec
tively. Thus
C l =  C, = fji, cos 4>'  cos <t> ......................... (31)
Again, if r, s be the radii of the surfaces, we have, neglecting %*,
Xl  X , = R/rR/ S  ........................... (32)
and thence, from (30),
t ^^ = (/iCos4>'cos4))fiiV ...(33)
fcos4> + :sm<I> ' \r sj'
the usual formula for the secondary focal length. The reckoning is such that
the signs of r and s are opposite in the case of a doubly convex lens. We
have now to proceed to a second approximation and inquire under what
conditions (30) is independent of the particular ray chosen. In the
numerator it is sufficient to retain the first power of ^ 1( ^ 2 , so that we may
take cos%i, cos^ 2 equal to unity; but in the denominator, which is already
a small quantity of the first order, we must retain the terms of the second
order in Xl , ^ 2 . It is not necessary, however, to distinguish between the
sines of Xl , ^ 2 and the angles themselves. The first step is to determine
corrections to the approximate values of d and (7 2 expressed in (31).
For cos (f> itself we have, from (24),
cos <f> = cos 3> + Xi x /R sin <I> ;
and again
, , sin <l> cos <t> v,a;
M cos 4>=V{/* 2 l+ cos 2 <}=/* cos <fr'+
so that C^faeQBVcoBtylr ................ (34)
In like manner, for C 3 in (26),
p'
so that
whence
Thus, if we write p = p cos O'/cos 4>,
1908] ON THE ABERRATION OF SLOPED LENSES, ETC. 449
and C t
= cos '
...(37)
in which fttf  (//  1 ) %, Xa  Xl 2 = ( %2  %i) (A* # + %,).
Accordingly, cos 3> + sin 4>
^i^X^i^'y^^^^"**^
...(38)
and the condition of no aberration is
................ (39)
Since %i, % 2 are inversely proportional to r and s, we may write (39) in the
form
~ (X 8  X) cos 2 < _
~ =U ' ............ V 4 '
where /*' = ft cos 4>'/cos ^ ........................... (W
If s = QO , so that the second surface is flat, we have as the special form
of (40)
l_(^2_ At ') CO s 2 <I) = 0; ........................ (42)
or in the case where 3> = 0,
l+yu/t 2 = 0, .............................. (43)
the same condition as that (15) required to give zero aberration in the
primary plane for small obliquities. In the case of finite obliquities we
may write (42) in terms of //,,
/A cos <&cos <E>' = p? cos 2 <!>' 1, ..................... (44)
or if we take the square of both sides of the equation,
/i 2 (1  sin 2 <I>) (1  sin 2 <*>') = (p? cos 3>'  1) J .
Of this the lefthand side may be equated to
(1  sin 2 3>) (/A 2  sin 2 3>) = p?  (p? + 1) sin 2 $> + sin 4 3>,
while on the right we have
(p?  1  sin 2 3>) 2 = (/A 2  I) 2  2 (fj?  1) sin 2 3> + sin 4 <I> ;
so that sin 2 ^ = '~_r.
as formerly found (see (17)).
In interpreting (45), which we may also write in the form
Af*.Pl>W + ll\ .................. (46)
we must bear in mind that it covers not only the necessary equation (44;,
but also the equation derived from (44) by changing the sign of one of the
R. v. 29
450 ON THE ABERRATION OF SLOPED LENSES, ETC. [328
members. For instance, if we put /* = 1 in (46), we derive sin 2 3> = , or
4> = 45 ; but on referring back we see that these values satisfy, not (44),
but
 p cos 4> cos <' = /x 2 cos 3 4>'  1.
The transition occurs when cos 3> = 0, or <E> = 90, when (45) gives
/*' = 2, or p = 14142. For smaller values of JJL there is no solution of (44).
Onwards from this point, as JJL increases, <t> diminishes. For example,
when n = 1'5, sin" <J> = , whence 4>= 73. The diminution of 3> continues
until i& /A 1 = 0, or /A=1'618, when <J> = 0, so that this is the value
suitable for a planoconvex lens at small obliquities. After this value of fi
is exceeded, sin 2 <I> in (46) is negative until /* 2 = 3, or /i = l'732. When this
point is passed, sin 2 < becomes positive, but a real value of < is not again
reached. We infer that in the case of a planoconvex lens (curved face
presented to parallel rays) there can be no freedom from secondary aberration
unless ft, lies between the rather narrow limits T414 and 1*618.
If the planoconvex lens be so turned as to present its plane face to the
parallel rays, r = oo ; and (40) requires that
fi + (p f *  p!) cos 2 4> = 0,
which cannot be satisfied, since /A' > 1.
Leaving now the particular case of the planoconvex lens, let us suppose in
the general formula (40) that 3> = 0. We have
.(47)
from which we see that, whatever may be the value of /z, compensation may
be attained by a suitable choice of the ratio r : s. If ft < 1618, r and s have
opposite signs, that is, the lens is double convex ; while if /j, > T618, r and s
have the same sign, or the lens is of the meniscus form. For example, if
/* = To, (47) gives s =  9r, so that the lens is double convex, the hind
surface having oneninth the curvature of the front surface.
We have seen that the aberrations in both the primary and the secondary
planes are eliminated for small obliquities in the case of a planoconvex lens if
p = 1'618. The question arises whether this double elimination is possible at
finite obliquities if we leave both the form of the lens and the refractive
index arbitrary. It appears that this can not be done. The necessary
condition is by (13), (40)
_S_ (Ji* \l + (ft 3 //) COS 2 $
/*' ft (ft 1) sin 8 <fr
1 + fjk pf* ~ 1 + ^  p* + (JL*  p!) sin 2 4> '
whence (/' I)sin 2 4> = 0, (48)
which can be satisfied only by <l> = 0, since p > 1.
1908]
ON THE ABERRATION OF SLOPED LENSES, ETC.
451
Since it is not possible to destroy both the primary and secondary
aberrations when the angle of incidence is finite, it only remains to consider
a little further in detail one or two special cases.
We have already spoken of the planoconvex lens ; but for a more detailed
calculation it may be well to form the equation for absence of primary
aberration analogous to (45). From (16),
fj, cos <' cos <f> = p?  I, (49)
whence, if we square both sides,
sin 4 <  (p? + 1) sin 2 < + 3/* 2  ^  1 = 0,
giving
so that
2
.(50)
sin 2 = 1618034 0618034 /*, .................. (51)
the other root being excluded if p > 1. It may be remarked that there is
no distinction between </> here and <J> in (45).
The following table will give an idea of the values of <f> from (51) and (45)
for which the planoconvex lens of variable index is free from aberration in
the primary and secondary planes respectively.
M
Primary plane
<t>
Secondary plane
10000
90 Of
14142
38 11'
90 0'
15000
28 29'
73 13'
15500
21 24'
58 37'
15900
13 38'
39 45'
16000
10 55'
32 25'
16100
7 16'
22 r
16180
V
0'
In the above the curved face is supposed to be presented to the parallel
rays. If the lens be turned the other way, r = <x> , and (13) gives //=0,
an equation which cannot be satisfied. In this case neither the primary
nor the secondary aberration can be destroyed at any angle.
Next suppose that the lens is equiconvex, so that s = r. In this case
(13) gives
/**iA*'i = 0, (52)
whence //.'=!, or \, of which the latter has no significance. Also from
(40) we get // = 1. It appears that neither aberration can vanish for an
equiconvex lens, unless in the extreme case /& = 1, < = 0, when the lens
produces no effect at all.
292
452 ON THE ABERRATION OF SLOPED LENSES, ETC. [328
PART II.
It is a common experience in optical work to find the illumination
deficient when an otherwise desirable magnification is introduced. Some
times there is no remedy except to augment the intensity of the original
source of light, if this be possible. But in other cases the defect may
largely depend upon the manner in which the magnification is effected.
With the usual arrangements magnifying takes place equally in the two
perpendicular directions, though perhaps it may only be required in one
direction. For example, in observations upon the spectrum, or upon inter
ference bands, there is often no need to magnify much, or perhaps at all, in
the direction parallel to the lines or bands. If, nevertheless, we magnify
equally in both directions, there may be an unnecessary and often very serious
loss of light.
In discussing this matter there is another distinction to be borne in mind.
Sometimes it is not necessary or advantageous that there should exist a
pointtopoint correspondence between the object and the image. It suffices
that a point in the object be represented in the image by a narrow line
This happens, for example, in the use of Rowland's concave gratings.
A conspicuous instance occurs in the refractometer which I described in
connection with observations upon argon and helium*. Here while the
objectglass of the telescope was as usual, a very high magnification in one
direction was secured by the use, as sole eyepiece, of a cylindrical lens
taking the form of a glass rod 4 mm. in diameter. An equal magnification
in both directions, such as would have been afforded by the usual spherical
eyepieces, would have so reduced the light as to make the observations
impossible.
Whenever the field of view varies only in one dimension, there is usually
no loss, and there may even be gain in the presence of astigmatism. In
other cases a pointtopoint correspondence between image and object is
desirable or necessary, and the question arises how it may best be attained
otherwise than by the use of a common telescope, which limits the
magnification in the two directions to equality. I had occasion to consider
this problem in connection with observations upon Haidinger's rings as
observed with a Fabry and Perot apparatus. Here the field is symmetrical
about an axis, and all the advantage that magnification can give is secured
though it take place only in one direction. At the same time light is usually
saved by abstaining from magnifying in the second direction also. In this
way the circular rings assume an elongated elliptical form a transformation
which in no way prejudices observation by simple inspection. The question
* Roy. Soc. Proc. Vol. LIX. p. 198 (1896) ; Scientific Papers, Vol. iv. pp. 218, 364.
1908] ON THE ABERRATION OF SLOPED LENSES, ETC. 453
whether light is saved, as compared with symmetrical magnification, depends
of course upon the aperture available in the two directions. In a Fabry and
Perot apparatus this is usually somewhat restricted.
One simple solution of the problem, available when the light is homo
geneous, may be found in the use of a magnifying prism, that is a prism so
held that the emergence is more nearly grazing than the incidence. In this
way we may obtain a moderate magnification in one direction combined with
none at all in the second direction. A magnification equal in both directions
may then be superposed with the aid of a common telescope. This method
would probably answer well in certain cases, but it has its limitations. More
over, the accompanying deviation of the rays through a large angle would
often be inconvenient.
If we are allowed the use of cylindrical lenses, or of lenses whose curvature
though finite is different in the two planes, we may attain our object with a
construction analogous to that of a common telescope. Suppose that the
eyepiece is constituted of a spherical and a contiguous cylindrical convex
lens. In one plane the power of the eyepiece is greater than in the other
perpendicular plane. Thus, if the objectglass be composed of spherical
lenses only, there cannot be complete focusing. With the spherical lens or
lenses of the objectglass, mounted as usual, it is necessary to combine a
cylindrical lens of comparatively feeble power, which may be either convex or
concave. All that is necessary to constitute a telescope in the full sense of
the word, that is an apparatus capable of converting incident parallel rays
into emergent parallel rays, is that the usual condition connecting the focal
lengths of objectglass and eyepiece should be satisfied for the two principal
planes taken separately. The magnifying powers in the two planes may thus
be chosen at pleasure ; and since there is symmetry with respect to both
planes the apparatus is free from the unsymmetrical aberration expressed
in (1).
When the magnifying desired is considerable in both planes, there is but
little for the cylindrical component of the objectglass to do, and it occurred
to me that it might be dispensed with, provided a moderate slope were given
to the single (spherical) lens. In the earlier experiments the objectglass
was a nearly equiconvex lens of 14 inches focus. The eyepiece was a com
bination of a spherical lens of 6 inches focus with a cylindrical lens of
2 inches focus, so that the focal lengths of the combination were about
2 inches and 6 inches in the principal planes, giving a ratio of magnifications
as three to one. With the above objectlens the actual magnifications would
be about 2 and 6. During the observations the axis of the telescope was
horizontal and that of the cylindrical lens vertical, so that the higher
magnification was in the horizontal direction of the field. During the
adjustments it is convenient to examine a cross formed by horizontal and
454 ON THE ABERRATION OF SLOPED LENSES, ETC. [328
vertical lines, ruled upon paper well illuminated and placed at a sufficient
distance.
When the objectlens stands square, there is, of course, no position of the
compound eyepiece which allows both constituents of the cross to be seen in
focus together. If we wish to pass from the focus for the horizontal to that
necessary for the vertical line, we must push the eyepiece in. In order to
focus both at once we must slope the objectlens. And since while both the
primary and secondary focal lengths are diminished by obliquity the former
is the more diminished, it follows that the sloping required is in the
vertical plane, the lens being rotated about its horizontal diameter. If we
introduce obliquity by stages, we find that the displacement of the eyepiece
required to pass from one focus to the other gradually diminishes until an
obliquity is reached which allows both lines of the cross to be in focus simul
taneously. At a still higher obliquity the relative situation of the two foci
is reversed. In the actual experiment with the 14inch objectlens, the
critical obliquity was roughly estimated at about 30.
The above apparatus worked fairly well when tried upon interference
rings from a thallium vacuum tube. But it was evident that the image
suffered somewhat from aberration. A better result ensued when the
magnification in both directions was increased by the substitution of an
objectlens of 24 inches focus, although this also was equiconvex.
Being desirous of testing the method of the sloped lens under more
favourable conditions, I procured from Messrs Watson a lens of baryta
crown glass of index for mean rays 1*59, and of pianoconvex form. The
aperture was about If inch, and the focal length 24 inches. When this was
combined with the compound eyepiece already described, the performance
was very good, if, in accordance with the indications of theory, the curved
face of the objectlens was presented to the incident light. The test may be
made either upon a cross or upon a system of concentric circles drawn upon
paper. The angle of slope giving the best effect was now very sharply
defined. When, however, the objectlens was reversed, so as to present its
plane face to the incident rays, no good result could be attained, evidently in
consequence of aberration. The change in the character of the image was
now very apparent when the eye was moved up and down, the rings
appearing more elliptical as the eye moved in the direction of the nearest
part of the edge of the sloped lens. Next to nothing of this effect could be
observed when the objectlens was used in the proper position. It is scarcely
necessary to say that care must be taken to ensure that the axis, about which
the lens is turned, is truly perpendicular to the axis of the cylindrical com
ponent of the eyepiece.
Altogether it appears that the combination of sloped objectlens with
compound cylindrical eyepiece constitutes a satisfactory solution of the
1908] ON THE ABERRATION OF SLOPED LENSES, ETC. 455
problem. I believe that it may be applied with advantage in the many
cases which arise in the laboratory where an unsymmetrical magnifying best
meets the conditions. The question as to the precise index to be chosen for
the planoconvex lens remains to some extent open. Possibly a somewhat
higher index, e.g. 1'60, or even T61, might be preferred to that which I have
used.
With the view to the design of future instruments, it may be convenient to
set out the formula giving the distance between the primary and secondary
foci of the objectlens as dependent upon the obliquity <f>. If/j,/ 2 be the
primary and secondary focal lengths, it is known (compare (33)) that
/2 /* 1
/ being the focal length corresponding to </> = ; so that
/ a /i = (/*l)sin 2 <ft ^
/ //, cos <'  cos < " '
In this
fi cos <f>  cos </> = *J(p?  sin 2 <)  V(l  sin 2 <) = (/*  1) jl + ^^j
approximately. Hence
(56)
from which the required obliquity is readily calculated when the nature of
the eyepiece and the focal length of the objectlens are given.
P.S., June 6. From von Rohr's excellent Theorie und GeschicKte des
Photographischen Objectivs, Berlin, 1899, 1 learn that Rudolf and, at a still
earlier date (1884), Lippich had proposed a different method of obtaining a
diverse magnification, and one that I had overlooked. This consists in the use
of an eyepiece formed by crossing two cylindrical lenses of different powers.
The two lenses are mounted, not close together, but at such distances from
the image as to render parallel the rays diverging from it in the two planes
separately. In this method the objectlens remains square to the axis of the
instrument. Lippich had the same object in view as that which guided me.
I have tried his method with success, obtaining an image as good, or nearly
as good, as that afforded by the sloped lens. I understand that Professor
S. P. Thompson also has used a similar device.
329.
HAMILTON'S PRINCIPLE AND THE FIVE ABERRATIONS
OF VON SEIDEL.
[Philosophical Magazine, Vol. xv. pp. 677687, 1908.]
LARGELY owing to the fact that the work of Hamilton, and it may be
added of Coddington, remained unknown in Germany and that of v. Seidel in
England, it has scarcely been recognized until recently how easily v. Seidel's
general theorems relating to optical systems of revolution may be deduced
from Hamilton's principle. The omission has been supplied in an able
discussion by Schwarzschild, who expresses Hamilton's function in terms of
the variables employed by Seidel, thus arriving at a form to which he gives
the name of Seidel's Eikonal*. It is not probable that Schwarzschild's
investigation can be improved upon when the object is to calculate complete
formulae applicable to specified combinations of lenses ; but I have thought
that it might be worth while to show How the number and nature of the five
constants of aberration can be deduced almost instantaneously from Hamilton's
principle, at any rate if employed in a somewhat modified form.
When we speak, as I think we may conveniently do, of five constants of
aberration, there are two things which we should remember. The first is
that the five constants do not stand upon the same level. By this I mean,
not merely that some of them are more important in one instrument and
some in another, but rather that the nature of the errors is different. In
earlier writings the term aberration was, I think, limited to imperfect
focusing of rays which, issuing from one point, converge upon another.
Three of the five aberrations are of this character ; but the remaining two
relate, not to imperfections of focusing, but to the position of the focus. It
is, in truth, something of an accident that, e.g. in photography, we desire to
focus distant objects upon a plane. The second thing to which I wish to
refer is that, although Seidel did much, four out of the five aberrations were
pretty fully discussed by Airy and Coddington before his time. To these
The word Eikonal was introduced by Bruns.
1908] HAMILTON'S PRINCIPLE AND THE FIVE ABERRATIONS OF VON SEIDEL 457
authors is due the rule relating to the curvature of images, generally named
after Petzval, so far, at any rate, as it refers to combinations of thin lenses.
Some remarks are appended having reference to systems of less highly
developed symmetry.
According to Hamilton's original definition of the characteristic function
V, it represents the time taken by light to pass from an initial point (x, i/, /)
to a final point (x, y, z), and it may be taken to be I p.ds, where fi is the
refractive index and the integration is along the course of the ray which
connects the two points. If the path be varied, the integral is a minimum,
for the actual ray ; and from this it readily follows that
l = dVldx, m = dV/dy, n = dV/dz, (1)
l'=dVldx', m'=dV/dy', n'=dV/dz > , (2)
where I, m, n, I', m', n' are the directioncosines of the ray at the end and
beginning of its course, the terminal points being situated in a part of the
system where the refractive index is unity.
In his communication to the British Association (B. A. Report, Cambridge
1833, p. 360) Hamilton transforms these equations. As his work is so little
known, it may be of interest to quote in full the principal paragraph, with a
slight difference of notation : " When we wish to study the properties of any
objectglass, or eyeglass, or other instrument in vacuo, symmetric in all
respects, about one axis of revolution, we may take this for the axis of z, and
we shall have the equations (1), (2), the characteristic function V being now
a function of the five quantities, a? + y 2 , xx + yy', x'* + y' 2 , z, z, involving
also, in general, the colour, and having its form determined by the properties
of the instrument of revolution. Reciprocally, these properties of the instru
ment are included in the form of the characteristic function V, or in the form
of this other connected function,
T=lx + my + nzl'x'm'y'riz'V, (3)
which may be considered as depending on only three independent variables
besides the colour ; namely, on the inclinations of the final and initial portions
of a luminous path to each other and to the axis of the instrument. Alge
braically, T is in general a function of the colour and of the three quantities,
P + m 2 , II' + mm', I' 2 + m' 2 ; and it may usually (though not in every case)
be developed according to ascending powers, positive and integer, of these
three latter quantities, which in most applications are small, of the order of
the squares of the inclinations. We may therefore in most cases confine
ourselves to an approximate expression of the form
, (4)
458 ON HAMILTON'S PRINCIPLE AND [329
in which T w is independent of the inclinations ; T w is small of the second
order, if those inclinations be small, and is of the form
T< 2 > = P (I 9 + m*) + P, (II' + mm') + F (V* + m' 2 ) ; (5)
and T (4} is small of the fourth order, and of the form
T* = Q (I* + w 2 ) 2 + Q, (I 3 + w 2 ) (IV + mm')
+ Q' (P + w 8 ) (J' + m' 2 ) + Q n (IV + mm')*
the nine coefficients, P P l P' Q Q l Q' Q n Q,' Q", being either constant, or at
least only functions of the colour. The optical properties of the instrument,
to a great degree of approximation, depend usually on these nine coefficients
and on their chromatic variations, because the function T may in most cases
be very approximately expressed by them, and because the fundamental
equations (1), (2) may rigorously be thus transformed;
I dT m dT
x = 77, y z = j>
n dl . n dm
, I' , dT , m' , dT
x . z = ffT . y . z 5 > !
n dl n dm
The first three coefficients, P Pj P / , which enter by (5) into the expression of
the term jT (2) , are those on which the focal lengths, the magnifying powers,
and the chromatic aberrations depend : the spherical aberrations, whether for
direct or inclined rays, from a near or distant object, at either side of the
instrument (but not too far from the axis), depend on the six other coefficients,
Q Qi Q' Qn Qi Q'> in the expression of the term T w . Here, then, we have
already a new and remarkable property of objectglasses, and eyeglasses, and
other optical instruments of revolution ; namely, that all the circumstances of
their spherical aberrations, however varied by distance and inclination, depend
(usually) on the values of six RADICAL CONSTANTS OF ABERRATION, and may
be deduced from these six numbers by uniform and general processes. And
as, by employing general symbols to denote the constant coefficients or
elements of an elliptic orbit, it is possible to deduce results extending to all
such orbits, which can afterwards be particularized for each ; so, by employing
general symbols for the six constants of aberration, suggested by the fore
going theory, it is possible to deduce general results respecting the aber
rational properties of optical instruments of revolution, and to combine these
results afterwards with the peculiarities of each particular instrument by
substituting the numerical values of its own particular constants."
Equations (7) are easily deduced. So far as it depends upon the
unaccented letters, the total variation of T is
dT = Idas + mdy + ndz + xdl + ydm + zdn j dx j dy r~ dz,
1908] THE FIVE ABERRATIONS OF VON SEIDEL 459
or regard being paid to (1),
dT=x
in which Idl + mdm
dT Iz dT
so that ^7 = x  . 
dl n dm
and in like manner by varying the accented letters the second pair of
equations (7) follows.
If we agree to neglect the cubes of the inclinations, we may identify n, n'
with unity, and (7) becomes
2P') I', y' =  P.m + (/  2P') m,
determining x, x' in terms of z, z', I, I' supposed known, or conversely I, I' in
terms of z, z', x, x' supposed known. The case of special interest is that in
which x, y, z and x', y', z are conjugate points, i.e. images of one another in
the optical system. The ratio x : x' must then be independent of the special
values ascribed to I, I'. In order that this may be possible, i.e. in order that
z, z' may be conjugate planes, the condition is
(z + 2P}(z f  2P') + P 1 2 = 0, ........................ (8)
?}
giving the magnification.
Equations (8), (9) express the theory of a symmetrical instrument to a
first approximation. In order to proceed further we should have not only to
include the terms in (7) arising from T w , but also to introduce a closer
approximation for n. Thus even though T w = 0, we should have additional
terms in the expressions for x, x' equal respectively to
i^(/ 2 + m 2 ) and \l'z'(V+ m' 2 ).
If the object is merely to express the aberrations for a single pair of conjugate
planes, we may attain it more simply by a modification of Hamilton's process.
Supposing that the conjugate planes are z = 0, / = 0, we have V a function
of the coordinates of the initial point x', y', and of the final point x, y. And
if as before I, m, n, V, m', n' are the directioncosines of the terminal portions
of the ray, we still have
l = dVjdx, m = dV/dy, ..................... (10)
l' = dV/dx', m'=d7/dy' ................... (11)
But now instead of transforming to a function of I, m, I', m', from which
as', y', x, y are eliminated, we retain x, y' as independent variables, eliminating
460 ON HAMILTON'S PRINCIPLE AND [329
only x, y, the coordinates of the final or image point*. For this purpose we
assume
U=lx + my V. ........................... (12)
The total variation of U is given by
dV, dV, dV , dV,
dU= xdl + Idx + ydm + may y dx j dy TT dx  j ,dy ,
or with regard to (10), (11)
dU^xdl + ydm + l'dx' + m'dy', .................. (13)
from which it appears that V is in reality a function of x, y', I, m. As
equivalent to (13), we have
y = dU/dm, ..................... (14)
m' = dUjdy' ...................... (15)
So far U appears as a function of the four variables x', y', I, m ; but from
its nature, as dependent upon Ix + my and V, and from the axial symmetry, it
must be in fact a function of the three variables
x'^ + i/ 2 , / 2 + ra 2 , and lx' + my',
the latter determining the angle between the directions of x', y' and I, m.
When these quantities are small, we may take
U= 7<> + U+ UM + ..., ..................... (16)
where U w is constant and
U* = $L(P+m*) + M(x'l + y'm) + $N(x'* + y' 2 ), ...'. ..... (17)
L, M, N being constants. If we stop at U (V , equations (14) give
x=Ll + Mx', y = Lm + My', .................. (18)
determining x, y as functions of x, y', I, m. We have next to introduce the
supposition that x, y is conjugate to x', y'. Hence L = 0, for to this approxi
mation x, y must be determined by x', y' independently of I, m. Accordingly,
x = Mx, y = My' ............................ (19)
We are now prepared to proceed to the next approximation. In order to
correspond, as far as may be, with the notation of Seidelf we will write
7 + B (/' + m') (lx' + my')
D) (lx r + myj +$D(l*+ w') (x'* + y' 2 )
my')(x'* + y'*) + F(x* + y t *y>, ........................ (20)
which is the most general admissible function of the fourth degree.
* Compare Bouth's Elementary Rigid Dynamics, 418.
t Finsterwalder, Miinchen. Sitz. Ber. Vol. xxvii. p. 408 (1897).
1908J THE FIVE ABERRATIONS OF VON SEIDEL 461
From (20) we obtain by use of (14) the additional terms in x and y
dependent on U {i> . No generality is lost if at this stage we suppose, for the
sake of brevity, y = 0. Accordingly,
............ (21)
(22)
In order to complete the value of x we must add the expressions in (19)
and (21).
Since F disappears from the values of x and y, we see that there are Jive
effective constants of aberration of this order, as specified by Seidel. The
evanescence of A is the Eulerian condition for the absence of spherical
aberration in the narrower sense, i.e. as affecting the definition of points lying
upon the axis (a/=0). If the Eulerian condition be satisfied, B=0 is identical
with what Seidel calls the Fraunhofer condition*. The theoretical investi
gation of this kind of aberration was one of Seidel's most important contributions
to the subject, inasmuch as neither Airy nor Coddington appears to have
contemplated it. The conditions A = 0, 5 = are those which it is most
important to satisfy in the case of the astronomical telescope.
To this order of approximation B = is identical with the more general
sine condition of Abbe, which prescribes that, in order to the good definition
of points just off the axis, a certain relation must be satisfied between the
terminal inclinations of the rays forming the image of a point situated on the
axis. The connexion follows very simply from the equations already found.
By (15), (16), (17), (20), with m = 0,
I' = M I + Bl 3 + terms vanishing with x', y' ;
so that for the conjugate points situated upon the axis
l' = Ml + Bl* ............................... (23)
The condition B = is thus equivalent to a constant value of the ratio I' /I,
that is the ratio of the sines of the terminal inclinations of a ray with the
axis. And this is altogether independent of the value of A.
On the supposition that the two first conditions A 0, B = are satisfied,
we have next to consider the significance of the terms multiplied by C and D.
Since
dxjdl = Gx'\ dy/dm = Dx\
* If A be not equal to zero, it can be shown that the best focusing of points just off the axis
requires that
where Z is the value of I for the principal ray. For example, if the optical system reduces to a
combination of thin lenses close together, lo=xff, where / is the distance of the lenses from the
image plane. Since by (19), x = Mx', the condition may be written
AM+Bf=0.
462 ON HAMILTON'S PRINCIPLE AND [329
we see that C and D represent departures of the primary and secondary foci
from the proper plane. In fact if I//),, l//> 2 be the curvatures of the images,
as formed by rays in the two planes,
1/^ = 2(7, l/p 2 =2D (24)
The condition of astigmatism is then
C = D; (25)
but unless both constants vanish the image is curved.
Finally the term containing E represents distortion.
If we impose no restriction upon the values of the constants of aberration,
we have in general from (21), (22)
dxfdl = A (3J + m 2 ) + QBx'l + Gx'\
dy/dm = A(P + 3m 8 ) + ZBas'l + Dx\
These equations may be applied to find the curvatures of the image as formed
by rays infinitely close to given rays, as for example when the aperture is
limited by a narrow stop placed centrally on the axis, but otherwise
arbitrarily. The principal ray is then characterized by the condition m = 0,
and we have
dxfdl = 3AP + QBx'l + O' 2 = 3H + K, (26)
dy/dm = Al*+2Ba;'l + Dx ; '* =H+K, (27)
equations which determine the curvatures of the images as formed by rays in
the neighbourhood of the given one, and deviating from it in the primary and
secondary planes respectively.
According to (26), (27),
(28)
(29)
The requirement of flatness in both images is thus satisfied if H=Q, K=Q.
The former is the condition of astigmatism, and it involves the ratio of x' : I,
which is dependent upon the position of the stop ; but the latter does not
depend on this ratio. It corresponds to the condition formulated by
Coddington and later by Petzval. From (28), (29) we may of course fall back
upon the conditions already laid down for the case where A = 0, B = 0.
The further pursuit of this subject requires a more particular examination
of what occurs when light is refracted at spherical surfaces. Reference may
be made to Schwarzschild*, who uses Hamilton's methods as applied to a
special form of the characteristic function designated as Seidel's Eikonal.
A concise derivation of the CoddingtonPetzval condition by elementary
methods will be found in Whittaker's tract f.
* Odttingen Abh. VoL iv. 1905.
t Theory of Optical Irutrwnentt, Cambridge, 1907. The optical invariants, introduced by
Abbe, are there employed.
1908] THE FIVE ABERRATIONS OF VON SEIDEL 463
Before leaving systems symmetrical about an axis to which all the rays
are inclined at small angles, we may remark that, as U (4} contains 6 constants,
in like manner U (6} contains 10 constants* and U {s} 15 constants, of which in
each case one is ineffective.
The angle embraced by some modern photographic lenses is so extensive
that a theory which treats the inclinations as small can be but a rough guide.
It remains true, of course, that an absolutely flat field requires the fulfilment
of the CoddingtonPetzval condition; but in practice some compromise has to
be allowed, and this involves a sacrifice of complete flatness at the centre of
the image. It will be best to fulfil the conditions dxjdl = 0, dy/dm = 0, or,
what are equivalent,
not when I is very small but when it attains some finite specified value. If
we suppose y' = 0, U is a function of a;"*, l z + m?, and lx', or sa