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‘“‘Meditationis est perscrutari occulta; contemplationis est admirari perspicua .... Admiratio generat queestionem, queestio investigationem, investigatio inventionem.”—Hugo de S. Victore.

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JUN 9 1904

NUMBER XXXVI

Prof. A. Schuster on a Simple Explanation of Talbot’s Bands. Dr. H. A. Bumstead on the Variation of Entropy as treated peer Willard GDS wo. sion 19 oR. cep ely Ble AK sb eislewia Rey. O. Fisher on Deflexions of the Plumb-line in India.... Mr. A. H. Pfund: A Study of the Selenium Cell.......... Dr. C. Chree on the Bending of Magnetometer Deflexion-Bars. Dr. G. E. Allan on the Magnetism of Basalt and the Magnetic Behaviour of Basaltic Bars when Heated in Air. (Plates oh Pg AIR 4 ee ney ee Mr. P. E. B. Jourdain on the Transfinite Cardinal Numbers eee) -orderedeAmoregates. | ).). saree teers wd Soil: sie Prof. J. D. Everett on Borgnet’s Method of Dividing an ener in Arbitrary malt 5c. obialpe sa.5/~ & he <ielapsreyaiel Notices respecting New Books :— Charles Nordmann: Essai sur le rdle des Ondes Hertzi- panes en Aspronomie de Nysique wige.. os%s. sess s- Proceedings of the Geological Suciety :— Mr. E. E. Waiker on the Garnet-bearing and Associated Rocks of the Borrowdale Volcanic Series .......... Prof. J. W. Gregory on the Glacial Geology of Tasmania.

NUMBER XXXVIIL—FEBRUARY.

Mr. R. K. McClung on the Effect of Temperature on the Jonization produced in Gases by the Action of Rontgen Rea NE eis OS PR gS ore cia 2 SOE GR ICL. 8 Reh acs

Dr. E. Weintraub: Investigation of the Arc in Metallic Vapours in an Exhausted Space. (Plates IL[I.-X1.)

Mr. C. A. Chant on the Variation of Potential along the Transmitting Antenna in Wireless Telegraphy. (Plates 1S SUP | ES IS en he, ee TD RS 2

Mr. 8S. J. Allan on the Radioactivity of os Atmosphere. abe: Re ie, St SA eS a tale ose UW ote'y vols emia

SERIES). ) Wh

79 80

81 95

140

lv CONTENTS OF VOL. VII.—SIXTH SERIES. Page Prof. A. Schuster on the Number of Electrons conveying the Conduction’Ourrents in Metals. \........20e5 eee 151 Prof. E. Hagen and Prof. H. Rubens on some Relations between the Optical and the Electrical Qualities of Metals. 157 Mr. Franz Leininger on the Relation of the Electric Charges transported by Cathode and Canal Rays to the Exciting RULPeNb ase ee be eee ae be ae oO 180 Prof. Karl Pearson on a Novel Instrument for Drawing Parabolas.., (Plate XVi)'t... 2a, ....7.0 5a 200 Profs. E. Rutherford and H. T. Barnes on the Heating Effect ot the Radium: Emanation ..........2...%40—— 202 Contribution by Lord Kelvin to the Discussion on the Nature of the Emanations from Radium which was -opened by Professor E. Rutherford at the Meeting of the British Association last September °.... J... 0.0 [07 220 Rev. P. J. Kirkby on the Effect of the Passage of Electricity through a Mixture of Oxygen and Hydrogen at Low Pressures. .00200. 00 UR SON 223 Helitorial Note - 2.5 (4 (OUae PhS Ce aos 233 Notices respecting New Books :— Dr. J. A. Fleming’s Handbook for the Electrical Laboratory and Testing Room, Volj iin... oe 233 Prof. A. A. Michelson’s Light Waves and Their Uses .. 235 Prof. E. Mathias’s Le Point Critique des Corps Purs .. 235 Prof. A. Classen’s Ausgewihlte Methoden der Analyt- ischen Chemie, -Vol. TT. (02) .20 082 a ee ee 236 Annuaire pour lan 1904) 2.0 22)... eee 236

NUMBER XXXIX.— MARCH.

Prof. J. J. Thomson on the Structure of the Atom: an Investigation of the Stability and Periods of Oscillation of a number of Corpuscles arranged at equal intervals around the Circumference of a Circle; with Application of the

results to the Theory of Atomic Structure ............ 237 Mr. O. W. Richardson on the Solubility and Diffusion in

Solution of Dissociated Gases .......,.+.2.2.) «en 266 Mr. J. RK. Cotter on an Instrument for Drawing Conics .... 274 Prof. J. 8. Townsend on the Charges on Ions ..........-. 276 Mr. R: Appleyard on the Conduetometer >... 722-0: .. see 281 Mr. T. C. Porter on a Method of Mechanically Reinforcing

Sounds: : (Plate: AVa.): «230 Penn oe 283

Mr. S. Skinner on the Photographic Action of Radium Rays. 288 Major 8. G. Burrard on Deflexions of the Plumb-line in

Pda ek ae A, RR cee), eee 292 Mr. P. E. B. Jourdain on the Transfinite Cardinal Numbers of Number-Classes in General*-:...¢62..<....) see eee 294

Dr. J. Joly on the Motion of Radium in the Electric Field .. 3038

CONTENTS OF VOL. VII.—SIXTH SERIES. Vv

Page Notices respecting New Books :— ¥:

Jean Perrin’s Traite de Chimie Physique ............ 307 Dr. Max Planck’s Treatise on Thermodynamics ...... 308 Prof. W. Ostwald’s Die Schule der Chemie .......... 309

J. M. Pernter’s Allerlei Methoden, das Wetter zu Pprophecdien” Lakes & Rey suite oon 4 ns Mech tecatd: «gee 310 Prof. H. Poincaré’s La Science et l’Hypothése ........ 310

Proceedings of the Geological Society :-— Profs. ©. Ti. Morgan & 8. H. Reynolds on the Igneous Rocks associated with the Carboniferous Limestone

Bees one Beisiob Picgiien tte as. cach. a4 52's vee 3 owt 310 Mr. A. R. Short on the Rhetic Beds of England . 311 Mr. & Mrs. Clement Reid on a Paleolithic Floor at Pr ah peemecean Coriwall 17 oom ane as se Oe Je reeset ss 312 Mr. W.S. Boulton on the Igneous Rocks at EBnte Cove, Hea Weston-swper-Mare. 1 Sy ei i ee bas se seas 313 Prof. H. N. Rau on a Deep-Sea Deposit from an Artesian Boring at Kilacheri, near Madras ...............- Shs Prof. S. H. Reynolds and Mr. A. Vaughan on the Rheetic Beds of the South-Wales Direct Line ............ 314 Intelligence and Miscellaneous Articles :— Memes 77 toy te, Onareres’ 230726. ¢ ede cs 315 Siainary Notices ‘Dr. William Wrancis.).....-...2...0.... 315

NUMBER XL.—APRIL. Prof. D. B. Brace on Double Refraction in Matter moving

OG Sea Oe 9 ee ee 317 Mr. 8. Skinner on the Occurrence of Cavitation in Lubrication.

Beene ONL i ers S ee S SU GAR Ds ae 329 Prof. W. M°F. Orr on the Radiation from an Alternating

Speer Piechrie Cumann. Fo. 4) owt de oda e dines 306 Messrs. Simmance and Abady on the Simmance-Abady

Stheker” Photometer: -(Plate KX.)e. sie 8.20. d41 Prof. F. T. Trouton and Mr. E. 8. Andrews on the Viscosity

ee eM TEED UUBRANCES | 3 120121215 a See aed ged O47

Prof. J. A. McClelland on the Emanation given off by Radium. 305

Prof. J. A. McClelland on the Comparison of Capacities in Electrical Work ; an Application ot Radioactive Substances. 362

Prof. R. Threlfall on a New Form of Sensitive Hot-Wire

nS riers Ate) be APR bs FIO a doe elle oe oil Prof. Rk. W. Wood on some New Cases of Interference and

reractumeane mie eX Ah opie vk eins 5) oda 376 Messrs. L. Holborn and L. W. Austin on Disintegration of

the Platinum Metals in Different Guses................ 388 Dr. W. Watson on a Quartz-Thread Vertical Force Magneto-

pees, “(mame mera ONE) oe BE 4 ow De 393

Mr. G. W. Walker on Stresses in a Magneto-static Field .. 399 Mr. W. Sutherland on the Dielectric Capacity of Atoms.... 402

vi CONTENTS OF VOL. VII.—SIXTH SERIES. Page Mr. W. Sutherland on the Crémieu-Pender Discovery ...... 405 Mr. H. Darwin on an Electric Thermostat. (Plate XXIV.) . 408 Notices respecting New Books :— Dr. Johannes Stark’s Die Dissoziierung und Umwandl- wag Chemischer Atome *........ ~./c0 gmieteeieeeen 414 Proceedings of the Geological Society :— Lieut.-Col. T. English on Eocene and Later Formations

surrounding the Dardanelles... ..... 408 ae 414 Dr. C. Davison on the Derby Earthquakes of March 24th and May 3rd, 1903" ..... 24.4... 00- 6 (poe 416

NUMBER XLI.—MAY.

Mr. W. Sutherland on the Electric Origin of Rigidity and Consequences . .. sj: sidag - ees eee dine & pie aagnuae sc ee Prof. H. Nagaoka on ue Kinetics of a System of Particles illustrating the Line and the Band Spectrum and the Pheno-

meéna of Radwactivity 2-2 yada de ko eee 4435 Mr. KR, A. Houstoun’s Spectroscopic Notes”... 1... .:6.e ee 456 Mr. 8. H. Burbury on the Kinetic Theory of Gases ...... 467 Mr. R. Hosking on the Electrical Conductivity and Fluidity

of Solutions .. | .=\...cega sek eelare eee ee eee 469 Mr. J. Barnes on the Analysis of Bright Spectrum Lines.

(Plates XX-V. di XX Val) Rae eal. ee Aedes ee 485 Dr. C. Chree on the Whirling and Transverse Vibrations of

Rotating Shafts. :) tie) Oa do eee eer 504

Mr.C.G. Barkla on the Energy of Secondary Rontgen Radiation 543 Mr. 8. C. Laws on the Thomson Effect in Ailoys ot Bismuth

Nav led br) cae 8 eee be 4 RE 560 Prof. J. Larmor on the Intensity of the Natural Radiation from Moving Bodies and its Mechanical Reaction........ 578

Prof. J. A. Fleming on the Measurement of Small Inductances and Capacities, and on a Standard of Small Inductance .. 586 Prof. J. A. Fleming on a Hot-Wire Ammeter for the Measure-

ment of very small Alternating Currents .......2. 026 595 Notices respecting New Books :— Mr. E. T. Whittaker’s Course of Modern Analysis .... 605 Mr. H. Hilton’s Mathematical Crystallography and the Theory of Groups of Movements ie )@ 2/2)... (4. «+ eae 605 Prof. P. Duhem’s Thermodynamics and Chemistry .... 606 Prof. 8S. Young’s Fractional Distillation.............. 607

Intelligence and Miscellaneous Articles :— An Instrument for Drawing Conics, by Mr. J. R. Cotter. 608

CONTENTS OF VOL. VII.—SIXTH SERIES. vu

NUMBER XLIL—JUNE. Page Lord Kelvin on Deep-water Two-dimensional Waves produced by any given Initiating Disturbance .................. 609 Prof. J. Larmor on the ascertained Absence of Effects of Motion through the Ather, in relation to the Constitution of Matter, and on the FitzGerald-Lorentz Hypothesis .. 621 Dr. E. P. Harrison on the Temperature-Variation of the Coefficient of Expansion of Pure Nickel................ 626 Prof. J. A. Pollock: A Comparison of the Periods of the Electrical Vibrations associated with Simple Circuits. With a, Anpendm by J.C. Clase’... 2.25 8s hieweleae. oes sy. 635 Mr. O. U. Vonwiller: Contribution to the Study of the Dielectric Constant of Water at Low Temperatures...... 655 Mr. E. W. Morley on the Vapour-Pressure of Mercury at Seeitaty VeMPeCranres: os. iw. ole oe ojeepiwid be eee at 662 Prof. M. Smoluchowski-Smolan on the Principles of Aero- dynamics and their Application, by the Method of Dynamical Similarity, to some special Problems .............0.055 667 Mr. C. T. R. Wilson on the Condensation Method of Demon- strating the Ionisation of Air under Normal Conditions .. 681 Dr. G. Johnstone Stoney on Escape of Gases from Atmo- ALES SSE Se ae rane or eae 690 Mr. W. Bennett on Non-homocentric Pencils, and the Shadows produced by them.—I. An Elementary Treat- ment of the Standard Astigmatic Pencil..............., 700 Mr. W. Bennett on Non-homocentric Pencils, and the Shadows produced by them.—II. Shadows produced by Axially Symmetrical Pencils possessing Spherical Aberra- Bee OE ALG Nay A in ky os aisle Ne eee 706

I a OS te yn tel a ne A Pt 0 lear a 716

I. & HH:

IIL-XI. XII. & XIII.

XIV.

XV.

XVI. XVIT-XIX. XX,

XXI.

PLATES.

Illustrative of Dr. G. E. Allan’s Paper on the Magnetism of Basalt and the Magnetic Behaviour of Basaltic Bars when Heated in Air.

Illustrative of Dr. E. Weintraub’s Paper on the Arc in Metallic Vapours in an Exhausted Space.

Illustrative of Mr. C. A. Chant’s Paper on the Variation - of Potential along the Transmitting Antenna in Wire- less Telegraphy.

Illustrative of My. ‘8. J. Allan’s Paper on the Radioactivity of the Atmosphere.

Illustrative of Prof. Karl Pearson’s Paper on a Novel

. Instrument for Drawing Parabolas.

Illustrative of Mr. T. C. Porter’s Paper on a Method of Mechanically Reinforcing Sounds.

Illustrative of Mr. S. Skinner’s Paper on the Occurrence of Cavitation in Lubrication.

Illustrative of Messrs. Simmance and Abady’s Paper on the Simmance-Abady Flicker” Photometer.

Illustrative of Prof. R. W. Wood’s Paper on some new Cases of Interference and Diffraction.

XXII. & XXIII. Illustrative of Dr. W. Watson’s Paper on a Quartz-

XXIV.

Thread Vertical Force Magnetograph. Illustrative of Mr. H. Darwin’s Paper on an Electric Thermostat.

XXV.& XXVI. Illustrative of Mr. J. Barnes’s Paper on the Analysis

XX VII.

of Bright Spectrum Lines. | 4 Illustrative of Mr. W. Bennett’s Paper on Non-homocentric Pencils, and the Shadows produced by them.

ERRATUM.

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TH i

LONDON, EDINBURGH, ann DUBLIN

PHILOSOPHICAL MAGAZINE

AND JOURNAT, OF SCIENCE.

{/ f

[SIXTH SERTES.] | ~ {

JANUARY 1904. \

\ 4 g

I. A Simple Explanation of Talbot’s Banils. By ArtHur Scuuster, &.RS* |

4. HESE bands are observed in a spectrum when_half

the aperture of the pupil is covered with a thin plate of mica or glass, provided that the plate be inserted on that side on which the blue of the spectrum appears. The explanation of these bands which has been given by Airy and Stokes involves a rather elaborate mathematical process which, though convincing, does not leave the mind com- pletely satistied. The essential reason for the want of sym- metry which causes the bands to appear only when the plate is introduced on one side, ought to be capable of being rendered obvious in a more simple manner. This I propose to do in the present communication.

As the bands are seen with “white light,’ a single luminous impulse should be sufficient to produce them, and as the distribution of intensity in the spectrum is clearly not an essential factor in the ease, we may choose the shape and duration of the impulse as we like.

Let an indefinitely short impulse spread out from a distant point, and strike a plane grating GG’ normally. This grating may be imagined to be made up of a series of narrow reflecting parallel strips A;, Ay, &c., separated by intervals which refiect no light. A lens SS’ having its tocus at F receives the luminous disturbance. The impulsive velocity spreading from A, reaches F sooner than that reflected from

* Communicated by the Author.

Phil. Mag. 8. 6. Vol. 7. No. 37. Jan. 1904. B

a\\ \

2 Prof. A. Schuster on a Sample

A,, and so on as the optical distances of the reflecting strips gradually increase from A, to A,. Hence the disturbance at F consists of a series of impulses, following each other at intervals equal to the period of a homogeneous wave which, starting from the same lumi- nous point, and reflected by the grating, would have its first principal maximum at F. All this, of course, is simply the elementary illustration, first given, believe, by Lord Rayleigh, of the ordinary action of a grating when analysing white light. The s <a question now is: How can the im-

pulses which succeed each other at

F be made to interfere? Clearly only

by retarding those which first reach

F, or accelerating those which reach

|

= | |

J

that point last. A plate of appro- priate thickness introduced from the left-hand side of the figure as it is drawn, can be made to answer the purpose. If, on the contrary, the same plate were introduced on the right-hand side, it would only retard those impuises which already arrive late, and therefore no inter- ference could take place. This is really all that need be said in explanation of the bands; but a more detaiied consi- deration of this view of the problem leads easily to a clearer expression for the calculation of the best thickness of the interposed plate than the more elaborate calculations of previous investigations.

The best thickness is secured when the whole series of impulses is divided into two equal portions, the impulses. arriving in pairs simultaneously at F. If A, be the central line of the grating, the retardation ought to be such that the impulses coming from A, and from A, reach F at the same time. If N be the total number of lines of the grating, the best retardation is therefore $NA, and the plate should be pushed sufficiently far into the beam to affect halfits width. The wave-length \ here means the wave-length of that homo- geneous train of waves which has its first principal maximum at F’, so that the retardation of each impulse compared with the nextis A. If the retardation is either greater or smaller, some of the impulses arrive too soon or too late to overlap others, and the bands are less clear. If the retardation has more than twice its best value, the series of impulses from A, to

FE

Explanation of Talbot’s Bands, 3

A, pass through F later than those coming from Ag, A,41, Kc., and hence there cannot be any interference.

If at a certain point of the spectrum corresponding to a wave-length > there is a maximum of light, the relative retardation of the two interfering impulses must be equal to mA, m being an integer; the next adjoining band towards the violet will appear at a wave-length such that

mrX=(m+1)r.

Hence for the distance between the bands

=v) _ 1 iy > Seas with the best thickness of interposed plate, m=4N, and hence ee ee in De

where 2’ in the denominator may with sufficient accuracy be replaced by A. If A” be that wave-length nearest to » at which there is a minimum of light, it follows that

A ae a i

r N

If a linear homogeneous source of light of wave-length » be examined by means of a grating, the central image extends

to a wave-length A, such that <iopbtiales Ge r

where N, as before, is the total number of lines on the grating. Hence the following proposition :—If, in observing Talbot’s bands, that thickness of retarding-plate be chosen which reduces the minimum illumination of the dark spaces to zero, the distance between each maximum and the nearest minimum is equal to the distance between the central maximum and | the first minimum of the diffractive image of homogeneous light, observed in the same region of the spectrum with the same optical arrangement. This proposition holds for all orders of spectra; but the appropriate thickness of the retarding plate increases in the same proportion as the order.

Lord Rayleigh’s remark * that “the thickness of the plate must not exceed a certain limit, however pure the spectrum may be,” requires the qualification that for infinite purity the limiting thickness also becomes infinite.

To examine the casein which the thickness of the retarding

y)

Al ee

* ‘Encyclopedia Britannica’ and Collected Works,’ vol. iii. p. 133. B2

4 Prof. A. Schuster on a Simple

plate is not that which gives no light at the minimum, con- sider a certain retardation A,B, introduced into half the beam. The effect is the same as if the impulses, instead of starting from A,, A,,...As, were sent off simultaneously from 5,, B.,.. B;, where A,B,=A,B,=,,. .A, Beane If B,A, and B,A, are lines drawn parallel to the principal plane of the focussing-lens so that the optical dis- tances of B, and A, and of B, and A, to the focus are the same, the impulses starting from points between By and B, may interfere with those starting from points between A, and Ag; but the parts of the grating A,A, and A,A, are not effective as regards interference, the impulses from A,A, arriving too, soon and those from A,A, too late.

If the useless portions are cut off by means of screens, the bands become black again, and we return to the best retarda- tion, but with a reduced aperture, and consequently a reduced resolving power. ‘The introduction of these screens does not alter the width of the bands; but these will be further apart than for the best retardation on account of the reduced cross-section of the effective portion of the beam. If the retarding plate is thicker than that which gives the most distinct bands, it may similarly be shown that the blackness of the darkest portions may be restored by screening off the central portions of the beam; but the bands will be closer together in this case. The point at which the bands disappear altogether is that at which each homogeneous component of the impulse gives a central diffraction-image equal in width to the distance between the bands. In each case the width of the bands is easily seen to be the same as that given by the interference of two points of light separated laterally in the beam by a distance equal to that of any two of the pulses which reach the focus simultaneously.

Powell’s experiment in which bands are obtained by intro- ducing plates into part of the beam traversing a hollow prism filled with a refracting substance is only a modification of Talbot’s, and is explained in a similar manner.

2. An objection might be raised to the above explanation in so far as it is not perhaps at once obvious how darkness may result by the interference of a succession. of impulses which are all in the same direction. The answer to the objection is, that if the impulses belonging to the first half of the beam are retarded in such a way as to fit in exactly half-way between those belonging to the second half, the

Eeplanation of Talbot’s Bands. 5)

disturbance at F would be a succession of impulses corre- sponding to the half period of the original impulse. _ There is light at F, but it is light which belongs to the overlapping spectrum of the second order. As regards the wave-length AX under consideration, there is darkness. The difficulty, if it is still felt to be one, may be avoided by considering a grating giving rise to the “corrugated waves” of Lord Rayleigh*. I have called these gratings simple gratings,” as all other gratings may be imagined to be made up of superposed simple gratings. The light of a simple grating is concentrated into the two spectra of the first ordery. Tt may readily be shown that any device which gets rid of the spectra of different orders will change impulses which were originally in one direction into disturbances which are alternately in one and the other direction, so that no further question can arise as to the way in which, according to the view here adopted, the dark bands are formed in Talbot’s experiment.

3. The proposition proved in § 1 allows us to extend the investigation to the case where the spectrum is produced by a prism. Inthe immediate neighbourhood of a given wave- length, the spectrum may be taken to be a normal one, and there can be no intrinsic difference between the bands seen in this case and those observed when a grating of the same resolving power is used. It has been shown by Lord Rayleigh that in all questions relating to resolving power the number of lines in the grating has to be replaced in the dp. dn’ thickness of the prisms, » the refractive index, and A the wave-length. It follows at once that the retardation which - gives black bands is for prisms

ease of prisms by ¢ where ¢ is the aggregate effective

14 pM.

4, It is interesting to follow out the modus operandi of a prism when an impulse is transmitted through it. For the sake of simplicity we may confine ourselves to the case that the law of refraction is such that the group velocity is independent of the wave-length. If the impulse be con- fined to the wave-front W F (fig. 3) before entering a refracting substance, it will at a given time in its passage through it

* ‘Encyclopedia Britannica,’ Wave Theory,” and ‘Collected Works.’ + Phil. Mag. xxxvii. p. 509 (1894).

6 Prof. A. Schuster on a Simple

lie near some surface G G’ which moves forward with the group velocity. I have shown, in a paper communicated to the recent meeting of the British Association, that the shape of the impulse changes periodically and alternately passes through its original shape and one exactly equal

but opposite in direction. If now the Fig. 3. impulse has passed through the prism li

and a wave-front for a homogeneous wave of length A would lie in the direction RS, the “impulses”? will be confined at toa region immediately surrounding a plane HS, the position of which may be calculated by the ordinary law of re- fraction, substituting the group velocity for the wave velocity. But on HS the

impulsive motion is not uniform, but H

alters periodically from the original type

to that which is equal and opposite to it. 7“

Hence if the emergent beam be received s

by a lens, the disturbance at the focus

of the lens consists of a periodic motion which is the more homogeneous the greater the resolving power of the prism. It will be noticed that this explanation of the modus ope- randt of a prism differs materially from that given by Dr. J. Larmor (‘ Aither and Matter,’ p. 248); but as we may imagine continuous media of such elastic properties as to give dispersion, the true explanation must be independent of the sympathetic vibrations which Dr. Larmor calls to his aid. To calculate the angle between RS and HS, we note that H BR is equal to the space passed over in air in the time equal to the difference between that necessary to traverse the thickness ¢ of the prism when the velocity is that of a homogeneous wave and when it is that of the group. Hence U being the group velocity, V the wave velocity in vacuo, and V’ the wave velocity in the prism,

t t Vi 7 ye ee and if w= V/V’ be the refractive index,

j = (V’—U).

~!]

Explanation of Talbot's Bands.

But & being the frequency,

dkV! a ae

epee ok hg a mae dx Ton ay dX

hence MVE dp

“Uy ae

We may put with sufficient accuracy V'=U in this ex- pression. To observe Talbot’s bands, the retarding plate must be brought in on the side of the thin edge of the prism and the best thickness, according to $ 1, is that in which that half of the beam which is nearest the thin end of the prism is retarded through half the distance RH. The appropriate

Lew bo

4 Ne a= : ; thickness is therefore “P in accordance with the results of

2 dh the previous paragraph.

5. The previous investigation gives the retardation which the plate should produce if the best effect is to be observed. If we wish to determine in an actual case the best thickness of plate, we must remember that as we have been dealing with impulses the group velocity comes into play. Hence the usual expression (“—Lje for the retardation, the thickness being e, 1s not quite accurate.

If U be the group velocity, and V’ the velocity of light zn

. . . . 1 at . vacuo, the retardation in time is la ee a ;_ this corresponds to a distance in air of

(y-1):

or if V be the velocity in the substance of the plate, the

retardation is Vy Lior Ul

uw’ being the refractive index of the material of the retarding plate.

We obtain the right result by adding to (u—1l)e the distance through which the group has fallen behind the wave ; this corresponds to the quantity RH calculated as above, if for the thickness of the prism we substitute ¢ and write p’

io”)

Dr. H. A. Bumstead on the for the refractive index of the plate. This gives for the

retardation dp’ aes —= é (1 il a ),

and for the best thickness of the plate this must be equal to

d ae EN) or $M a according as a grating or prism is used to

dr produce the spectrum.

6. We are so much accustomed to regard the homogeneous wave as the simplest element into which all wayve-motions may be resolved that we sometimes overlook the fact that the phenomena of white light may all be reproduced by a single disturbance of short duration. ‘There are cases, and the phe- nomenon of Talbot’s bands may serve as a conspicuous example, where the consideration of the combined group yields to a simpler treatment than the resolution into homogeneous waves. I have shown in my paper on Interference Phenomena” how group velocities may be used to determine the conditions of achromatism. Considerations similar to those used in that paper may perhaps be usefully employed to simplify the treat- ment of achromatized interference-bands.

II. On the Variation of Entropy as treated by Prof. Willard Gibbs. By H. A. Bumsrgzap, Ph.D., Assistant Professor of Physics, Yale University *.

iB the August number of the Philosophical Magazine Mr. 8. H. Burbury has discussed certain difficulties which present themselves in Chapter XII. of the Principles of Statistical Mechanics,” by the late Prof. J. Willard Gibbs. Unless I have misunderstood Mr. Burbury’s statement, I believe these difficulties may be surmounted, and shall en- deavour to give, as briefly as possible, my reasons for this belief. For the sake of brevity I shall assume that the reader has Mr. Burbury’s paper before him, and shall refrain from quoting from it unless it seems necessary for clearness of statement. |

The first difficulty (which may be more conveniently dis- cussed in terms of the hydrodynamical analogue than in terms of the ensemble of systems) is in regard to Prof. Gibbs’s statement that ‘‘one may perhaps be allowed to say that a finite amount of stirring will not affect the mean square of the density of the colouring matter, but an infinite amount of stirring may be regarded as producing a condition

* Communicated by the Author.

Variation of Entropy. a

in which the mean square of the density has its minimum value, and the density is uniform.” It seems to me that this statement may be justified as follows :—If, after a certain amount of stirring, we should determine the density of the colouring matter in ‘the liquid, using finite elements of volume sufficiently large, we should find the density sensibly constant in the ditferent elements; but if the elements were chosen small enough (but still finite) some of them would be entirely within the coloured portions and some in the uncoloured portions, and the density in such an estimate would no longer appear to be uniform. If now the stirring were continued, a time would come when these smaller elements would all have the same average density, and so on indefinitely ; and no system of finite elements, however small, could be as- signed in advance in which the average density could not be made the same for all by a sufficient amount of stirring. In other words, if we are allowed to stir as long as we please, we may use elements (in the estimate of density) as small as we please. That this is what Prof. Gibbs means by his somewhat guarded statement about an infinite amount of stirring, seems plain in the light of the preceding paragraphs in which he discusses the effect of altering the order in which

limits are taken. This latter consideration was one of which he not infrequently made use; I recall that he once em- ployed it to reconcile conflicting views as to the interpretation of Fourier’s series in a discussion which arose in the columns of Nature’ (vol. lix. p. 200).

Although it seems to me that the statement can be thus justified, | nevertheless must agree with Mr. Burbury that the other way of escaping the difficulty, viz. by defining the density by finite elements of volume (or of extension-in- phase) is preferable. If I understand the matter correctly, this is not because there is anything in the structure of the ensemble of systems corresponding to a molecular structure in the liquid, for a system of n degrees of freedom occupies no finite extension in the 2n-fold space in which its possible phases lie; but it is because we are unable, owing to the finiteness of our perceptions, to recognize very small dif- ferences of phase, just as we are unable to recognize very small differences of position in the analogous case of the liquid. And it is certainly nearer the truth to base the doctrine of the increase of entropy upon the finiteness of our perceptions rather than upon the infiniteness of time. That this was also Prof. Gibbs’ opinion I believe is evidenced by the sentence following the one quoted (“ one may perhaps be allowed,” &c.) in which he says, ‘‘ We may certainly say that a sensibly

16 Dr. H. A. Bumstead on the

uniform density of the coloured component may be produced by stirrmg.” And it is really this latter form which he uses when he comes to apply the principle, as in the third paragraph on page 154, in which the qualification, “if very small differences of phase are neglected,” is of course equi- valent to taking finite elements of extension-in-phase. The infinite time idea was, I believe, introduced merely as an alternative (and nota preferable) way of regarding the subject.

Admitting then the possibility of variation of the density- in-phase D in the finite elements through which a moving system passes, Mr. Burbury finds a difficulty in the definition of 7 in the expression D=Ne’. He says, “so long as 9 remains constant for the same system, we may define 7 to be the entropy which that system has. ... But being now supposed to be variable for the same system. we require a definition.” Here I think the whole trend and spirit of Prof. Gibbs’ method has been misapprehended ; unless I have mistaken his position, Prof. Gibbs would not have admitted that the 9 for a single system, although exactly determined, corresponds to what we call entropy in bodies met with in nature. So far as he applies his results to ther- modynamies, he regards the bodies of nature as corresponding, not to a definite system, but to a system chosen at random out of a properly distributed ensemble; so that it is certain average properties of the ensemble which we observe ex- perimentally, and not the properties of a single system. The average value of 7 for the whole ensemble (taken with the negative sign) corresponds to the entropy of any body which the ensemble is capable of representing, and we are no more concerned about the 7 of a single system (except in so far as a knowledge of it may be necessary to get a correct average) than we are concerned with the exact configuration of the system. But it is evident that we may get a sufficiently close value of the average for the whole ensemble by adding, not the » for every system, but the mean values ot for each one of a set of finite elements of extension-in-phase taken sufficiently small. Thus all we are concerned to know is the mean value of » in an element, and hence the equation D=WNe" may still serve as the definition of 9, for all necessary purposes, even though D is no longer the exact density at a point but is the mean density throughout an element. Prof. Gibbs’ statement in a succeeding paragraph (p. 148) that 4 “is an arbitrary function of the phase,” which Mr. Burbury takes to be a new definition, is, I think, not a defini- tion at all, but the statement of an assumed initial condition in the particular problem which he is then considering.

Variation of Entropy. Lk

It is this problem (p. 148) which gives rise to Mr. Bur- bury’s final difficulty, and which leads him to the conclusion that the hypotheses made by Prof. Gibbs concerning the mechanical systems are not sufficient to serve as a basis of rational thermodynamics. Perhaps I may best show why Prof. Gibbs’ conclusions seem to me legitimate by restating his demonstration, as I understand it, in slightly different form and with special reference to the objections which have been brought against it. Let us consider an ensemble of systems not in statistical equilibrium. An ensemble is in statistical equilibrium if, during any interval of time, as many systems enter any fixed element of extension-in-phase as leave it during the same interval; the density-in-phase in any fixed element (finite or infinitesimal) does not change with the time. But when the ensemble is not in statistical equilibrium the density in fixed elements ot extension-in- phase will vary with the time, and therefore the value of » associated with a fixed finite element (as explained above) will obviously vary. ‘The question is whether the average value, 7, for the whole ensemble, as determined by the use of these elements, will increase or decrease with the time. At a certain initial instant t’ let the density be distributed in any arbitrary manner throughout the extension-in-phase, that is, at this instant, we may consider D (or 7) to be given as ‘an arbitrary function of the phase.” Later, of course, it will be a function of the phase and of the elapsed time. With this initial distribution given we may now choose a system of fixed finite elements of extension-in-phase, DV, small enough so that the density may be regarded as sensibly constant throughout any one of them. At a later time ¢” (anless the motion in phase is of a highly special and rela- tively improbable kind) the systems which were together in one element at ¢’ will not all be in a single element. Thus some of the systems which were at ¢’ in the element DV’ may now be in the element DV”, but they will have mixed with them systems which, at ¢’, occupied other elements, DV,’, DV.’, &. If, now, we ascertain the average density in DV” and take its logarithm (7), thus assuming that 7 has a constant value for all the systems in the element, we shall, by Theorem IX. of Chapter XI., get a less value than if we took the actual values of 7 which the separate systems have. But the actual values of 9 for the separate systems are those which they have brought with them into DV”, and are the same which they had in their scattered condition in DV/’, DV,/, &c. at the instant ¢’. Theretore the value of » which we have obtained by averaging the density in the element

12 Dr. H. A. Bumstead on the

DV” at ¢” is less than that based on the same systems as they were at ¢’; and, as this is true for every element, the average

value of y for the whole ensemble so deteonatede is less at Y! than at ¢’.. This diminution in the average is wholly the result of mixing, in the elements, systems each of which preserves a constant value of 7. Tt may be well illustrated by the hydrodynamical case which Prof. Gibbs uses earlier in the same chapter (p. 146), in which a cylindrical mass of liquid is imagined, one sector of 90° being black and the rest white. If it is given a motion of rotation about the axis of the cylinder, in which the