VII. Astronomical Consequences of the Electrical Theory of Matter

Philosophical Magazine Series 6

ISSN: 1941-5982 (Print) 1941-5990 (Online) Journal homepage: http://www.tandfonline.com/loi/tphm17

VII. Astronomical consequences of the electrical theory of matter

Sir Oliver Lodge

To cite this article: Sir Oliver Lodge (1917) VII. Astronomical consequences of the electrical theory of matter , Philosophical Magazine Series 6, 34:200, 81-94, DOI: 10.1080/14786440808635680

To link to this article: http://dx.doi.org/10.1080/14786440808635680

Published online: 08 Apr 2009.

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Download by: [University of California Santa Barbara] Date: 21 June 2016, At: 01:12 THE LONDON, EDINBURGH, AND DUBLIN PHILOSOP[I[CAL MAGAZINE AND JOU RNAL OF SCIENCE.

[SIXTH s aI S.]

A UG US 7 1917.

VII. Astronomical CoJ~.~equencesof the Electrical Theory of Matter. By Sir OLIVER LODGE*. HE inertia of an electric charge was predicted by T Sir J. J. Thomson ill 1881 ; the dependence ot electric inertia on speed of motion through the rather was calculated by Oliver Heaviside in 1889, and confirmed by Thomson, who also isol',ted the ,,nit charge in 1899; while the experimental verification of inertia as a thnction of velocity, by Kaufmann, occurred in 1902. On the electrical theory of matter a body of any material moving at higil speed through the tether acquires extra or spurious or apparent inertia; and this inertia is presumably not subject to gravity, since, as in the case of" a solid moving through a fluid, it probably is an effect of pressure reaction and not a real increment of mass. All this is quite independent of the theory of relativity. The inertia factor for small speedst is (1--v~/c2)-t, or * Communicated by the Author. t The simple factor for inertia-increase m/me= (1-- v2/e~-)-~is supposed to be an approximation, thoueh a close one, to a more comphcated

Downloaded by [University of California Santa Barbara] at 01:12 21 June 2016 expression, such as m 3[., , 40 20-sin20 me - ~\-t sin20 sin20(-]~--~oY20))' where sin O=v/c. See my book on ' Electrons,' pp. 133 & 225. This becomes 1 when @ is small, slightly less than see 0 when 8 is moderate, and approaches infinity in this form, [~1~(~+1), as 0 approaches ~z'. For instance, if v came within a tenth of 1 per cent. of the velocity of light, so that sin 0='999, the ratio m/mo would be 20. Phil. Mug. S. 6. Vol. 34. No. 200. Aug. 1917. G 82 Sir Oliver Lodge on what is practically the same thing, 1 + v~/2c ~ ; where c is the characteristic velocity with which every known disturbance is propagated by free rather. It is just this uniformity of transmission which makes the connexion between rather and matter so ehlsive to experimental observation: measurement is foiled save when matter moves relatively to matter. That is the foundation, and I venture to think all the foundation, for the Theory of Relativity considered as a philosophical reality instead of only a more or less convenient summary of experimental results. An increase in inertia, without corresponding increase of gravitational control, cannot fail to have astronomical consequences, though they may be so small as to be barely observable. For inertia is a function of speed even when speed is but planetary, and if the force of gravitation is not correspondingly increased--a thing which we have no reason whatever to think likely, since eether is presumably the vehicle of gravitation and not subject to it; until contorted into the singular points called electrons,--then some small but cumulative effects may be caused in the more rapidly moving bodies. Professor Einstein's genius enabled him in 1915 to deduce astronomicaland optical consequences (some not ye: verified) from the Principle of Relativity. See an interesting account by Professor Eddington in '~ature,' vol. xcviii. p. 328 (28 Dec. 1916). I wish to show that one of them at least can be deduced without reference to that principle. In so far as the deduction is incompletely in accord with quanti- tative observation, there is something further to be considered ; but it is unlikely that a result of approximately the right order of magnitude can be devoid of significance. Consider the amount of the perturbation caused by extra inertia in the case of Mercury, whose orbital speed is approximately half as great again as that of the earth, or say 1"5 • 10 -4 times that of light. In one part of its orbit it will be travelling parallel to and in the same sense as that component of the sun's way which lies in the plane of the planet's orbit, which Downloaded by [University of California Santa Barbara] at 01:12 21 June 2016 we may call Mercury's ecliptic. In all this half of the orbit therefore its inertia will be slightly grea:er than the average

JSHstorical 57ote.--Heaviside's expression for the coefficientof }v2 in the value of kinetic energy for a charged sphere was first given in Phil. Mag. April 1889. J. J. Thomson's expression for the coefficient of v in the value of momentum is contained in his ' Recent Researches in ]~lectricity and Magnetism,' page 21, published in 1893. The above trigonometrical expression is intended to represent this. Astronomy and High-speed Inertia. 83 value ; whereas in the opposite half of the orbit, where it is travelling against the sun's way, the inertia will be less than the average. But in all cages the effective or apparent inertia oE the planet will be slightly larger than the mass on which the force of gravity acts. Hence we may expect the orbit to revolve in its own plane, or, in other words, the apses must slowly progress; for the effect will be much the same as if gravity were proportionally diminished, the inertia remaining constant. The theory of the apsldal angle in nearly circular orbits, under a central force varying as any power of the distance, is given by Newton with unexampled genius in Principia, Book I, Section IX, and is thrown into orthodox analytic form in Tait & Steele w 248. The result is that for a central orbit subject to a law of force as the nth power of distance, the angle swept through by the radius vector between two consecutive apses, i. e. between maximum and minimum radii, in orbits nearly circular, is

'71" ~/(3 + ~)" Newton himself remarks that for a direct distance law this angle will be a right angle; for a constant force, ~r/,./3; and for the exact law of inverse square, precisely 7r; thus giving a perfectly repeated orbit without any apsidal progression. But if the inverse square law were inexact, the apsidal angle would differ from ~- by a corresponding amount ; and the direction of motion of the apses would be progressive if gravity diminished faster than the inverse square law, i. e. it' the index n exceeds -2 numerically. To get tim perturbed rate of progression for Mercury, 41 or 43 seconds of arc per century,~which value was reckoned by Newcomb as the outstanding discordance from theory,--we have only to remember that the planet makes 4 revolutions per annum, or 8 journeys from apse to apse, so that in a century the discrepancy 7r/v/(3+n) --~r has accu- mulated 800 or more accurately 830 times; so, to give the Downloaded by [University of California Santa Barbara] at 01:12 21 June 2016 observed value, 1 43 v/(3+n ) 1= 830 X180 X 3600" Whence n = -- 2"00000016. This value was reckoned by Professor Asaph Hall in Astr. J. vol. xiv. page 49, Boston 1894:; but then th_ere is no other reason for supposing that the index is not exactly --2. G2 84 Sir Oliver Lodge on If, however, instead of a reduction in force, the mass were slightly greater than is proportional to weight, the result will be similar, and similar aspidal progression should occur. It is noteworthy that a discrepancy of electrical mass due to planetary speeds will be of tile order 10 -s, and that that is of the same order as the above imagined discrepancy in n. But to determine the amount of the extra or unexplained advance of perihelion properly, we must know the absolute speed el the planet through the rather. This involves a knowledge of the proper motion of the sun in our cosmic group of stars, and also an estimate of the unknown drift of that group itself as suggested by Kapteyn. Let the sun's true way be inclined to the plane of the planet's ecliptic by an angle s (latitude). Let the sun's gross velocity through the rather be w, meaning its motion towards Vega (which alone would make X about 60 ~ ('oln- blued with the unknown drift of the whole cosmic group of stars to which the sun belongs. Let the velocity of a planet in its orbit be v, and let 0 be the angle which its direction: makes with the projection on its orbit of the sun's true way; then, taking account of the normal component w sin X, as well as of the component in the plane of the orbit, w cos X, the resultant speed of the planet through the rather is v/(w ' + v~ + 2vw cos X cos O) ; .... (1). so that its apparent mass at any point in its orbit is: w ~ -t- v ~ + 2vu~ cos X cos 0) ~imes the gravitational mass. 1 + 2r 2 In nearly circular orbits this angle ~? may be taken as also representing the angle between the radius vector and the normal to the above line of reference; i. e. as the ~ of ordinary polar coordinates. }Cow writing the differential polar equation for an inverse square orbit as usual, with u the reciprocal of the radius vector, and h twice the rate of description of areas, cl2u F/m_ t~

Downloaded by [University of California Santa Barbara] at 01:12 21 June 2016 7tr 2 + u = lf~u ~ h-~, /, is the acceleration multiplied by the square of the distance, and is constant for constant mass. But /, contains the effective inertia in its denominator, and is therefore affected inversely by the mass factor just considered. So on the electrical theory of matter the equation becomes d~u i, (1 w~ + v' vw cos a, ) d~---~ +u= ~ 2c 2 c~ cos 8 , (2) Astronomy and High.speed Inertia. 85 which contains a slightly modified constant, and also a periodic term. It will suffice to take the case of orbits sufficiently circular to enable us to treat v as constant, so as to put a mean value in the small correction term instead of introducing av2=lx(2au--1). In that case also we shall have no trouble about the precise meaning of 0. In so far as the anglo between tangent and radius vector is not a right angle, i. e. in so far as the velocity t~ and the position 0 are not the same, the effect, for small excentrieities is to modify the solitary k in the denominator of equation (5) below into k([--2e/3i; but, as this is just the k which can be neglected, the slight complication will not be here attended to. Before solving (2), we may note that this equation has the form of the ordinary resonance equation, ~ + ~k + n~x = E cospt, with x equal to u minus the constant terms above, but with Che damping coefficient ~ zero and with the frequencies n and p equal. So it is "m equation which is liable to give infinite values; and even in practice it gives large accumu- lative amplitude when the damt3ing is small. It is the equation on which all ~uning or syntony in Wireless Tele- graphy is based. The ordinary particular integral for this case, E .v--~ n~_p: cospt,

gives an oscillation of infinite amplitmte, the main infinite part of which, however, may be got rid of by combining it with the supplementary part of the complete solution with arbitrary constants, A cos nt + B sin nt. For putting p--=n+z, and proceeding to the limit when z is zero, we get

Downloaded by [University of California Santa Barbara] at 01:12 21 June 2016 E cos (n + z'}t E cos nt + Ezt sin nt n ~- (n + z) ~ 2nz 2nz The first term is infinite, but when combined with the arbitrary term Aces nt it disappears, and the solution is left, x= t sin nt, which exhibits an amplitude steadily increasing with time. 86 Sir Oliver Lodge on So in like manner the astronomical solution for orbits of small excentricity, with v reasonably constant and both w and v small compared with c, is IX ( l W~ 4-v"- ) u= p 2c~ +e cos (0--a)-- ~,v2c cos2 X 0sin0 , (3)

instead of the usual

-= cos

Now 0 being defined with reference to the sun's way we cannot make ~ zero ; in fact a must be practically the angle between the major axis of the orbit and the line of reference; for, save for a minute correction term, it represents the value of 0 at the perihelion apse. The extra constant term in (3) only matters in cases where w or v is beginning to be comparable with c; but the progressive term containing 8, an angle which steadily increases with the time, is important. For in a century the whole angle swept through by Mercury's radius vector, at the rate of 4 revolutions per annum, is 800 ~r. The progressive term shows either that the constants of the orbit must change, or that the orbit must revolve in its own plane. To find its rate of revolution, consider the apses as places where du/dO-=O, and where O=a. Then in general

dOdu = -- h~ ~ e sin (0-- a) + vwco~c ~ X

So tanO= esina-kO ~ esin~ kO e cos ~ + k e cos ~, t.~ e cos a' (5)

Downloaded by [University of California Santa Barbara] at 01:12 21 June 2016 k being very small; so that if' 0o is the initial value of 0 at an apse, and 0~ its value after n revolutions, 27rnk tan ~'0 -- tall o~n -- r COS but d tan a=sec ~ a da,

SO 27r~g/c (;os oe '/rwt~w - cos x cos =. (6) ~0 -- Otn - e ~:'C2 " " Astronomy and High-speed Inertia. 8 T The sign shows that for progress the smfs effective motion must be in opposite sense to that of the planet at perihelion. Now cos 8cosk is the cosine of the angle between the sun's and the planet's motion, so cos a cos k is the cosine ot~ the angle het~een the lines of the sun's motion and of the planet's motion a~ an apse ; or say between the sun's way and the minor axis of the orbit ; c~dl this q~. So the ~Tsidal progression during n revolutions is d~- ~r.vw cos r ec2 , . .... (7)

and vw cos~b is a scalar, as it ought to be to compare wi~h c~. But the change in a during a century (i. e. the known progress of the perihelion not accounted for by orthodox gravitational perturbations) amounts to 40 or 43 seconds of arc in the case of Mercury; so this gives us, for the unknown motion of the sun, ec~d~ ec • 43 W COS (~ ~ vnrc 1"5 x 10 -4 x 400 • 180 x 3600 43ec ec 6 • 180 • 36 904" The excentricity of Mercury is given in Galbraith 8~ ]:Iaughton's ' Astronomy' as 20"56 per cent. ; so '2056 wcos~- (-~0~- c-~2"27x10"4c, (8)

or two and a quarter times the orbital speed of the earth. I hardly think that astronomers will regard that large velocity as quite unreasonable. From (7) it appears that a nearly circular orbit can be made to revolve very easily, though the revolution would be un- important, while an excentric orbit would be stiff. Another curious result would seem to be that if the major axis of

Downloaded by [University of California Santa Barbara] at 01:12 21 June 2016 the ellipse points along the projected component of the sun's way, this extra apsidal progession disappears; whereas if the minor axis lies along the sun's true way, the kind of apsidal motion now under consideration reaches a maximum. Can this be true ? In its favour we can say that the perturbation under consideration, though depending on velocity, is equivalent to a radial force varying sinuously with position, directed away from the sun during that half of the orbit where the motions 88 Sir Oliver Lodge on compound additively, and towards the sun during the other half where they compound subtractively. If the sun's motion concurs generally with that of the planet at aphelion, the virtual decrease in solar attraction near aphelion will cause that point to progress; ~hile the virtual increase of solar attraction at and near perihelion will cause that point also to progress. On the other hand, a coucurrence of the motions at perihelion would cause both apses to regress. So the line of apses steadily revolves either forwards or backwards according to the sense of the sun's proper motion in the direction of the latus rectmn. But if the sun's way is directed along the major axis, the varying inertia of the planet is equivalent to a radial force acting oppositely on the approaching and receding halves of the orbit; so one apse progresses while the other recedes, and there is no cumulative effect on the line of apses. Nevertheless there should in this case be a change in the excentricity. If the sun's proper motion concurs generally with the motion of the planet from aphelion to perihelion, this half of the orbit will be subject to a virtually diminished solar attraction, and so the excentricity will diminish. The other or receding half of the orbit will be subject to a virtually increased solar atraction, and that also will diminish the excentricity. The causes combine. A reversal of the sun's motion, so that the component velocities are added from perihelion to aphelion and sub- tracted on the reverse journey, will h~ve an opposite effect; and in that case, in both halves of the orbit, the excentricity will increase. Concerning any possible effect on the Moon:--astronomers appear satisfied that Dr. G. W. Hill and Professor E. W. Brown have settled the small residual discrepancy in the acceleration of the moon's mean motion, but inasmuch as the frequency of revolution in the case of the moon is considerable, and its speed through the ~ether is compounded of many causes, it may seem worth while to examine whether its fluctuations of inertia do not call for residual attention. True its monthly Downloaded by [University of California Santa Barbara] at 01:12 21 June 2016 speed is almost insignificant, being only about a thirtieth of the earth's orbital speed, but it shares in the motions of earth and sun, and so the w+v to be compounded with it is considerable. Any cumulative effect, however, can only be a slight, residual one, since its monthly orbit presents every aspect to the sun's way in the course of a year or a decade. As to the constant term in equation (3), it would seem that a modified/~ might affect the period of revolution, because T--- 27rv/(a3//~) ; unless there were a compensating effect on a. Astronomy and High-speed Inertia. 89 And a very minute acceleration would become important with lapse of time. But the modified/~ is no new thing, it has been there all the time, and so presumably it is only the fluctuations that we have to attend to. Moreover, the virtual force, whether variable or not, being always central, does not seem likely to affect h or T. In any case the action, being wholly in the plane of the orbit, has no effect upon the nodes.

Another way. To cheek the calculation of (7), take the expression for the reciprocal of the semi latus rectum, (3) with (0--a)=90 ~ a being the angle between the latus rectum and a projection on to the orbit of the sun's way,

a(l_e21 ) --/~(1]-~ w"+v~2c 2 § (t?--a) wv cos X t?sinO) 2C 2 and see how a must change to keep it constant although 0 increases by 2nv. Initially the variable part equals e cos 0 sin 0~ finally it equals e cos ( ~--de) -- k (2nTr + O) sin 0, so the difference, which is to be zero, gives e sin da = 2n~rk cos ~, or d,z = n.rvw cos )~ cos a c2e which is the same result as before.

Another check. Initially let O=a, so that the planet is at an apse and u is constant for an instant; then the terms in u e cos (O--a)--k8 sin 8,

Downloaded by [University of California Santa Barbara] at 01:12 21 June 2016 differentiated, become e(sin ~ cos a- cos 0 sin a) + k(sin ~ -F 6 cos 0) ----0. After n revolutions 0 has returned to its old value +27rn, but a has changed slightly to a I, such that e(sin 8 cos a'- cos 6 sin a +) + k(sin 0 + (0-4- 27rn) cos 0) = 0. So subtracting sin 8(cos ='-- cos =) -- e cos 8(sin mr- sin =) + k 2~rn cos 0-- 0; 90 Sir Oliver Lodge on but still 0 and a are practically equal, so e sin ~a da + e cos~ ~t da ~ 27rnk cos a, or da =- 27rnk cos a ) # with k=vweosX/2cL Again the same result as (7). Prof. Einstein's result, as quoted by Prof. Eddington ia ' :Nature' of Dee. 2~, 1916, is 24~r 3 a 2 -- radians per revolution. c: T2(l--e:) To make my result agree with this in appearance, we must gratuitously replace w cos ~b (which is quite foreign ta Prof. Einstein's ideas) by the following,

e r 6v. ]~e~ " 2a-- r or, what is much the same for nearly circular orbits, we caa without reason write 6ev instead of w cos ~b. But the differ- encos, both in reasoning and in result, are fundamental.

Problem for an orbit of greater excentricltS. The equations for an inverse square orbil whose excentricity is not small are dO~d2u+u= ~/(1 v~ ' w~+ 2vwe~ ,

cos 6= cos x cos 0', (9) sin (0' - 0) = e sin (~- 0% | v ~ = 2~u-/~/a, / where 0' is the angle between the tangent and the projectiovt of sun's way on the orbit, 0 is the angle between radius vector and the normal to

Downloaded by [University of California Santa Barbara] at 01:12 21 June 2016 that projection, a is the angle between major axis and the same normal. :But to get a cumulative result in a reasonable time the period of revolution should be not too great.

Other _Planets. Substituting, from (8), 2"27 • 10-tc for w cos ~b in (7), we can get the apsidal progression for any other planet of small Astronomy and High-speed I'ne~'tia. 91 excentricity (though only on tile assumption that tile aspect of its orbit to the sun's way does not differ numerically from that of Mercury). Its value for n revolutions of the planet is

"000227~rn y ...... (10)

For Mars e is variable but is about 9"33 per cent., v=10-4e/l"22, and the period of its revolution is 687 days. So in a century of earth years its perihelion would progress

2"'27 x 365 x lO-STr : 7 seconds of arc. 6"~7 • "0933 • 1"22 I do not know what the actual outstanding discrepancy is in the case of Mars: the longitude of its perihelion differs from that of Mercury by about 100 ~ As stated above, in order to rotate the orbit forward, i. e. to secure apsidal progress, the main solar motion which is to be compounded with that of the planet must have a component agreeing with the motion of the planet at aphelion: a component agreeing at perihelion would cause regress, and a component along the major axis would only modify the excentricity. I now point out, for what it is worth, that if the main solar-stellar-drift in plane of ecliptic were of magnitude 3 (meaning 3 • 10-%) and were directed towards longitude 29~ ~ it would have a component 2 in the direction of the aphelion motion of Mercury, 2 ,, ,, ,, ,, Mars, z Earth, _ z Venus ; thus suggesting that through a comparison of the outstanding discrepancies between theory and observation for different planets, if they were definite enough, it might be possible to get some indication of the direction as well as the magnitude of the sun's true motion through the rather of space. Downloaded by [University of California Santa Barbara] at 01:12 21 June 2016

The arguments are :-- 1. That motion of matter tbrough a3ther has a definite meaning, apart from relative motion with respect to other matter. 2. That an extra inertia duo to this motion is to be expected at high speeds, in accordance with the FitzGerald- Lorentz contraction. 92 Sir Oliver Lodge on 3. That this extra or high-speed inertia is not part of tho mass but is depend~nt on the ~ether and hence is not subject to gravity. 4. That from this reasonable hypothesis astronomical consequences follow which may be detected when cumulative. 5. That under certain specified conditions merely a small change in excentricity is to b~ expected as the chief result, in certain others an apsidal progress or regress is to be expected. 6. That the outstanding discrepancy in the theory of the perihelion1 of Mercury would be accounted for l~y attri- buting a certain value to a component of the true solar motion through the tether in the direction of the planet's aphelion path. 7. That using this value for the solar-plus-stellar drift, ),iz. two or three times the eartids orbital velocity, a result can be obtained for ~he perihelion of Mars, subject to a hypothesis about direction. 8. That by discussion of discordances in the elements of different planets an estimate may be formed of the magnitude and direction of the locomotion of the solar system in its invariable plane.

ADDE~DUMo I have inquired from Professor Eddington what is the outstanding discrepancy of Mars ; and he replies eS~= +0'"6t+__0'"35. This value (without the probable error) happens to agree exactly with what is reckoned above in equation (10), since it gives an apsidal progress 8~" of 7" per century.

.Note on the possible deflexion of Light. It becomes a question whether the gravitative deflexion of

Downloaded by [University of California Santa Barbara] at 01:12 21 June 2016 a ray of light, predicted by Einstein, also follows from sethor theory. A wave-front undoubtedly simulates some of the properties of matter. It conveys momentum, as Poynting has shown, and an advance wave-froni presumably has to sustain and convey the light-pressure until a target is struck. The mechanical stress exists ultimately between source and receiver ; but though one end acts on the source, all the tim% the other end of the stress has to react on the advancing wave-front until the receiver is reached, unless Newton's Astronomy and Tligh-speed Inertia. 93 Third Law meets with some exception. ]t is, therefore, not unnatural to associate a specific semi-material inertia with the travelling mtherial disturbance which occurs in light. The nature of the disturbance is presumably quite different from that accompanying high-speed locomotion of matter : there was nothing likely to simulate general material properties about that. So we are at liberty to ask, concerning the special kind of ~etherial inertia associated with light-pro- pagation,--Is this momentum-inertia likely to be subject to gravity? In other words, how far does the temporary or travelling rather disturbance in a wave-front correspond to the permanent rather disturbance constituting' an electron, which is so interlocked as not to need any locomotion for its existence ? Is it a sort of temporary matter, not permanently c(mstituted, but possessing material properties while it lasts ? Experiment must answer the question. An affirmative answer would be of the greatest interest. If it be assumed, as probable or possible, that gravitation is one of the properties of this imitation or temporary matter, though its gravitation constant may be q times the ordinary q~ (q being either greater or less than 1), the problem of deter- mining the gravitative deflexion of a ray becomes the easy one of reckoning the angle between the asymptotes of a small comet flying in hyperbolic orbit near the sun with immense velocity c. The angle is $ = 2q,,/M e.2r~ ...... (11)

where r0is its nearest approach to the central body of' mass 3I. This expression for the deflexion $ agrees with Einstein's predicted value if q is taken as 2 ; though the onL" reason I can see ~br that at present is the very speculative "one that its gravitational entanglement may depend on the maximum amplitude squared, while its inertia depends on the mean. The observational determination of this quantity q is clearly an important one. It may be anything from 2 to 0,

Downloaded by [University of California Santa Barbara] at 01:12 21 June 2016 though to me the value 1 seems more probable than anything except 0. Taking q as 1 for arithmetical purposes, and writing ~/M as gR ~, the value of the deflexion for grazing incidence is 2gR/c~; but ~/(gR) is the velocity in a grazing circular orbit, which for the earth is 5 miles a second and for the sun 50 times as 94 Lord Rayleigh on the Pressure developed much. So the deflexion for a ray of starlight grazing the sun is ( 250 .~2 2 x \185000] --3"6 x 10 -~, or 0'"74.

Longitudinal Possibility. The velocity of light issuing radially from a body might on this hypothesis also be affected, since it could be pulled back by a maximum amount represented by the free fall from infinity, i. e. by J(2q.qR) ; thollgh the ]ongitudimd q need no~ be the same as the transverse q concerned in deflexion. But this sort of action, if it can be imagined as likely to occur, and even if it caused a reduction in snnlight-velocity of 26 miles a second in the neighbourhood of the earth, would not yield any Doppler effect ; the waves wouht still be received at their emiLted frequency.

"VIII. On the Pressure developed in a Liquid during the Collapse of a Spherical Cavity. By Lord RAYLEIGIt, O.M., F.R.S.* HEN reading O. Reynolds's description of the sounds W emitted by water in a kettle as it comes to the boil, and their explanation as duo to tile partial or complete collapse of bubbles as they rise through cooler water, I proposed to myself a further consideration of the problem thus presented; but I had not gone far when I learned from Sir C. Parsons that he also was interested in the same question in connexion with cavitation behind screw-pro- pellers, and that at his instigation Mr. S. Cook, on the basis of an investigation by Besant, had calculated the pressure developed when the collapse is suddenly arrested by impact against a rigid concentric obstacle. During the collapse the fluid is reg;trded as incompressible. Downloaded by [University of California Santa Barbara] at 01:12 21 June 2016 In tile present note I have given a simpler derivation of Besant's results, and have extended the calculation to find the pressure in the interior of the fluid during tile collapse. It appears that before the cavity is closed these pressures may rise very high in the fluid near the inner boundary.

* Communicated by the Author.