Rev. J. Challis on the Aberration of Light. 325 



form velocity of propagation of the wave througli the aether. 

 Then, since the point a with regard to the direction of motion 

 of the aether is m advance of the point b, the velocity of the 

 aether is less at a than at b. For we are here considering a 

 position in that part of the {ether which, with regard to the 

 direction of the earth's motion, is in advance of the earth's 

 centre; and it is plain that if we trace at any time a line of 

 motion in that part, beginning at the earth's surface and pro- 

 ceeding in the direction of the motion of the particles through 

 which it passes, the velocity will be less the further we advance 

 along this line. Let the velocities V^ and V/ continue uniform 

 during the small time dt, and let the small straight lines ae^ 

 b d he respectively the spaces through which the points a and 

 b of the waves are carried by the composition of these veloci - 

 ties with the uniform velocity V. Join de. Draw ac equal 

 and parallel to bd and join c d. It is easy to see that as the 

 motion of propagation is less opposed by the motion of the 

 aether at a than at 6, « ^ is greater than b d, and inclined to- 

 wards bd. The general effect of the aether's motion is, there- 

 fore, to throw the normal to the front of the wave more and 

 more in the direction towards which the aether is moving; and 

 this effect would be similarly found to take place if we consi- 

 dered a position on the other side of the earth's centre. 



I proceed now to calculate the amount of deviation of the 

 normal, and to determine the planes in which it takes place. 

 For this purpose let us resolve the velocity Vy into w along a c, 

 u along a b, which we may suppose to be in a given direction 

 perpendicular to ac, and w along a straight line through a 

 perpendicular to a b and a c, and let us first consider the ef- 

 fect of u and w, abstracting from v. The point e will thus be 

 in the same plane as a b d c, and if v! be the resolved part of 

 V/ in a by we have cd — de, which is very nearly c o, equal to 

 {u' — u) 8 1. Hence if a c = 8 5, it follows that the angle cao 



iu' —II) It ^ ^T Is , 1-1 



=.- — _ But V = ^ very nearly, neglectmg the ratio 



of Vy to V. Hence the angle cao = — ^^, which is the angle 



of deviation of the normal in the plane cab. The sum of all 



such angles for the whole course of the wave may be found 



by taking, since 85 is perfectly arbitrary, ac of such a length 



that u' at c is the same as u' at b; or, which is the same thing, 



d u 

 supposing u' — u = -J- ds. Hence if Uj be the value of u at 



the earth's surface, and Uq at any distant point of the course, 



the whole deviation in the plane of u and w = • ' ^ - - . At a 



