2. ON THE A B E R R A T I O N OF LIGHT*

G.G. STOKES , ,

The general expanation of the phenomenon of aberration is sosimple, and the coincidence of the valu of the velocity of l ightthence deduced with that derived from the observations of theeclipses of Jupiter's satellites so remarkable, as to leave no doubton the mind as to the t ru th of the expanation. But vvhcn vveex-amine the cause of the phenomenon more closcly, it is far frombeing so simple as it appears at first sight. On the theory of emis-sions, indeed, there is l i t t le difficulty; and it would seem that themore particular expanation of the cause of aberration usuallygiven, which depends on the consideraron of the motion of a tele-scope as l ight passes from its object-glass to its cross vvires. hasreference especially to this theory; for i t does not apply to thetheory of undulations, utiless we make the rather start l ing hypothe-sis that the luminiferous ether passes freely through the sides of thetelescope and through the eatth itself. The undu la to ry theory oflight, however, explains so simply and so bcaut i fu l ly the mostcomplicated phenomena, that we are natural ly led to regard aber-ration as a phenomenon unexplained by it, but not incompat iblewith it.

* Reprinted from G. G. Stokes, Maih. and Phys. Papers, 1, 134-40. Thefirst part is unchanged from original (1845) publication in Pll. Afag., 27,9; The "Additional Note" was substituted in 1880 as an unprovement pf the1845 argument to the same end.

136

VSTOKES: ON THE ABERRATION OF LIGHT 137

The object of the present communication is to attempt an expla-nation of the cause of aberration which shall be in accordancewith the theory of undulations. I shall suppose that the earthand the planets carry a portion of the ether along with them sothat the ether cise to their surfaces is at rest relatively to thosesurfaces, while its velocity alters as we recede from the surface,till, at no great distance, it is at rest in space. According to theundulatory theory, the direction in which a heavenly body is seenis normal to the fronts of the waves which have emanated fromit, and have reached the neighbourhood of the observer, the ethernear him being supposed to be at rest relatively to him. If theether in space were at rest, the front of a wave of light at any instantbeing given, its front at any future t ime could be found by themethod explained in Airy's tracts. If the ether were in motion,and the velocity of propagation of light were infinitely small, thewave's front would be displaced as a surface of particles of theether. Neither of these suppositions is, however, rue, for the ethermoves while light is propagated through it. In the fol lowing inves-tigation I suppose that the displacements of a wave's front in anelementary portion of t ime due to the two causes jus t consideredtake place independent ly .

Let w, v, w be the resolved parts along the rectangular axes ofx, y, z, of the velocity of the particle of ether whose co-ordinatesare x, y, z, and let V be the velocity of l ight supposing the etherat rest. In consequence'of the distance of the heavenly bodies, itwill be quite unnecessary to consicler any waves except thosewhich are plae, except IT so lar as they are distorted by the mo-tion of the ether. Let the axis of z be taken in, or nearly in thedirection of propagation of the wave considered, so that the equa-tion of a wave's front at any time will be: V . . ' . - - - .', z = c+K/+t, " , ( i)C being a constant, / the time, and f a small quantity, a function10

138 NINETr.ENTH-CENTURY AETHER THEORIES

of x, y and /. Since u, v, w and f are of the order of the aberration,their squares and producs may be neglected.

Denoting by a, /?, y the anglcs which the normal to the wave'sfront at the point (x, y, z) makes with the axes, we have, to thefirst order of approximation,

. jyeos a = j- , eos R = -^ - , eos y 1 ; (2)

7X /V

and if we take a length V dt along this normal, the co-ordinatesof its cxtremity vvill be

,' x-~Vdt, y~Vdt, z+Vdt.dx dy%

If the ehcr were at rest, the locus of these extremilies would betiie wave's front at the time t + d, but since it is in motion, theco-ordinates of those extremities must be further increased by u dt,vdl, wd(. Denoting then by x', v', z the co-ordinates of the pointof the wave's front at the time t-\-dt which corresponds to thepoint (x, ;', z) at the time /, we have

-y dt, y' =\ dx) -

and eliminating x, y and z from these equations and (1), and de-noting C by/(x, >, /), we havefor the equation to the wave's frontat the time t + dt,

z'-(w-\-V)dt = C4-K/

or, expanding, neglecjtinjg dt- and the square of the aberration, andsuppressing the accents of x, y and z, "

, , z = C+Vt + t + (w+ y) di. (3)

STOKESI ON THE ABERRATION OF LIGHT 139

But from the definition of C it follows that the equation to thewave's front at the time -\-dt will be got from (1) by putting + dl for /, and we have therefore for this equation

(4)

Comparing the identical equations (3) and (4), we have

= w.

fThis equation gives C = ' vv//; but in thesmall term C we may*/

replace w dt by wdz -r V: this comes to taking the approxi-

mate valu of zgiven by the equation z = C+Vt instead of /for the parameter of the system of surfaces formed by the wave'sfront in its successive positions. Henee equation (1) becomes

z = C+Vt-tJLfv) \v dz.- Combining the valu of C just found with equations (2), we get,to a first approximation,

7t I dw 1 'dw (5)

equations which might very easily be proved directly in a moregeometrical manner.

If random vales are assigned to u, v and v, the law of aberrationresulting from these equations will be a complicated one; but ifw, v and u> are such thut'ifdx+vdy + wdz is an exact differential,we have,

dw __ du dw dv dx dz ' dv ~ dz

140 NINETEENTH-CENTURY AETHER THEORIES

whence, denoting by the suffixes 1, 2 the vales of the variablesbelonging to the frst and second limits respectively, we obtain

If the motion of the ether be such that u dx -f v dy -I- w dz is anexact di fTerent ia l for one system of rec tangular axes, it is easy toprove, by the transformaron of co-ordinates, that it is an exactdifTerent ia for any other system. Henee the formula; (6) wi l lhold good, Tiot merely for light propagated in the direction frstconsidered, but for l ght propagated in any direction, the direc-tion of propagation being taken in each case for the axis.of z. Ifwe assume that u dx 4- v dy+ w dz is an exact difTerential for thatpart of the motion of the ether-which is due t'o the motion of trans-lation of the earth and planets, it does not therefore follow that thesame is truc for that part which depends on thcir motions of rota-tion. Moreover, the diurnal aberration is too small to the detectedby observation, or at least to be measured with any accuracy, andI shall therefore neglect it.

It is not difficult to shew that the formula? (6) lead to the knownlaw of aberration. In applying therrt to the case of a star, if we beginthe integrations in equations (5) at a point situated at such a dis-tance from the earth that the motion of the ether, and conse-

quently the resul t ing change in the direction of the light, is insens-ible, we shall have z/i = O, Vi = 0; and if, moreover, we take theplae xz to pass through the direction of the earth's motion, weshall have

t>2 = 0, 'Pz-Pi = 0,

j - 2and

STOKES: ON THE ABERRATION OF LIGHT 141

the'velocity of the earth to that of light, multiplied by the sine ofthe angle between the direction of the earth's motion and the linejoining the earth and the star.

;

Additional Note[In what precedes waves of light are alone considered, and the

course of a ray is not-investigated, the investigaron not beingrequired. Therc follows iri the original paper an investigaronhaving for object to shew that in the case of a body like the moonor a planet vvhich is itself in motion, the effect of the distortion ofthe waves in the neighbourhood of the body in altering the apparentplace of the body as determined by observation is insensible. Forthis, the orthogonal trajectory of the wave in its successive positionsfrom the body to the observer is considered, a trajectory which inits main part will be a straight line, from which it will not differexcept in the immediate neighbourhood of the body and of theearth, where the ether is distorted by their respective motions.The perpendicular distance of the further exremity of the trajectoryfrom the prolongation of the straight line which jt forms in theintervening quiescent ether is shewn to subtend at the earth anangle which, though not actually O, is so small that it may be dis-regarded.

. * -

The orthogonal trajectory of a wave in its successive positionsdoes not however represent the course of a,ray, as it would do ifthe ether were at rest. Some remarks made by Professor Challisin the course of discussion suggested to me the examination ofthe path of a ray, which in the case in which udx + vdy-\~ wdz is anexact difTerential proved to be a straight line, a result which I hadnot foreseen when I wrote the above paper, which I may mentionwas read before the Cambridge Philosophical Society on the ISthof May, 1845 (see Philosophical Magazine, vol. xxix, p. 62). Therectilinearity of the path of a ray in this case, though not expresslymentioned by Professor Challis, is v i r tua l ly contained in what he

142 NINETCENTH-CENTURY AETHER THEOR1ES

wrote. The problem is rather simplificd by introclucing the consid-eration of rays, and may be treatccl from the bcginning in thefollowing manner.

The notat ion in other respects bcing as befor, let a', /?' be'thesmall angles by which the direction of the wave-normal at thepoint (,v, y, z) deviates from that of Oz tovvards Ox, Oy, respect-ively, so that a', fl' are the complements of a, /?, and let a,, /?, bethe inclinations to Oz of the course of a ray at the same poinl.By compounding the velocity of propagaton through the etherwith the velocity of the ether we easily see that

,='+, /.=/*'+.s i

Let us now trace the.changes of a,, fl, during the time dt. Thesedepend first on the changes of a', /?', and secn el ly on those ofw, v.

As regards the change in the direction of the wave-normal, \venotice that the seat of a small element of the wave in its successivepositions is in a succession of planes of particles nearly parallel

to the plae of x, y. Consequently the direction of the elementof the wave will be altered during the time di by the motion of theether as much as a plae of particles of the ether parallel to theplae of the wave, or, which is the same to the order of smallquantities retained, parallel to the plae xy. Now if we considera particle of ether at the time / having for co-ordinates x, y. r,another at a distance dx parallel to the axis of A% and a third at adistance dy parallel to the axis of v, we see that the displacementsof these three particles parallel to the axis of z du r ing the time dtwill be

j , , j j jwdl, {w + -r- dx] dt, [w+-rdy \ dt\

and dividing the relative displacements by the relation distances,we have dwd.\-df, dw/dy-clt for the small angles by which the

STOKCS: ON THE ABERRATION OF LIGHT 143

normal is displaced, in the planes of xzt yz, from the axes x, v, sothat

, / dw , ,.,, dw ,/a = r- dt, dp = -T- dt.dx dy

We have seen already that the changes of / / , r are du/dz. V dt,dv/dz. Vdi, so that -

_ l^u dw\ _ (du dw\' \dz dx) ' \dz dy]

Henee, provided the motion of the e ther be such that

.... , f udx+ vdy-{-wdz .... ,

is an exact differential, the change of direction of a ray as it travelsalong is ////, and therefore the course of a ray is a straight linenotwi ths tanding the motion of the ether. The rect i l ineari ty ofpropagation of a ray of l ight , which a priori would seem very l ikelyto be interfered with by the motion of the ether produced by theearth or heavenly body moving through it, is the tacit assumptionmade in the explanation of aberration given in treatises of Astro-nomy, and provided that be accounted for ihe rest follows asusual.* It follows fur ther that the angle subtended at the ear thby the perpendicular distance o'f the point where a ray leavesa heavenly body from the straight line prolonged which representsits course through the in t e rven ing quiescent ether, is not merelytoo small to be observed, but actually ////.]

t To make this explanation quite complete, vve should properly, as ProfessorChallis remarks, consider the light coming from the wires of the observingtelescope, in company with the light from the heavenly body.