12 Mr. G. G. Stokes on the Aberration of Light. 



one; but if ?f, v and ta are such that udx-\-vdij + iiodz is an 

 exact differential, we have 



dw _ du dw dv ^ 



d x~~ dz^ dy'~ dz* 



whence, denoting by the suffixes 1, 2 the values of the varia- 

 bles belonging to the first and second limits respectively, we 

 obtain 



«2 - «1 = - v" » ^2 - /3, = -^y-*. . . (6.) 



If the motion of the aether be such \h?i\.udx-^'ody -{-nadz 

 is an exafct differential for one system of rectangular axes, it 

 is easy to prove, by the transformation of co-ordinates, that it 

 is an exact differential for any other system. Hence the for- 

 mulae (6.) will hold good, not merely for light propagated in 

 the direction first considered, but for light propagated in any 

 direction, the direction of propagation being taken in each 

 case for the axis of ^. If we assume that udx -\-v dy +'wdz 

 is an exact differential for that part of the motion of the a3ther 

 which is due to the motions of translation of the earth and 

 planets, it does not therefore follow that the same is true for 

 that part which depends on their motions of rotation. More- 

 over, the diurnal aberration is too small to be detected by ob- 

 servation, or at least to be measured with any accuracy, and I 

 shall therefore neglect it. 



It is not difficult to show that the formulae (6.) lead to the 

 known law of aberration. In applying them to the case of a 

 star, if we begin the integrations in equations (5.) at a point 

 situated at such a distance from the earth that the motion of 

 the aether, and consequently the resulting change in the di- 

 rection of the light, is insensible, we shall have u^=0, Wi = 0; 

 and if, moreover, we take the plane xz to pass through the 

 direction of the earth's motion, we shall have 

 v^ = 0, /32 - /3i = 0, 



and «2 — «i=^; 



that is, the star will appear to be displaced towards the direc- 

 tion in which the earth is moving, through an angle equal to 

 the ratio of the velocity of the earth to that of light, multiplied 

 by the sine of the angle between the direction of the earth's 

 motion and the line joining the earth and the star. 



In considering the effect of aberration on a planet, it will 

 be convenient to divide the integrations in equation (5.) into 

 three parts, first integrating from the point considered on the 

 surface of the planet to a distance at which the motion of the 



