Prof. Challis on the Dispersion of Light. 455 



formation gives 



d 2 . rcr , 1 /d 2 . rcr , d.rcr , A \ , n^rcr 



dr* 



, 1 /a 2 , rcr , d.rcr , n \ . n 2 rcr ~ 



Having regard to the ultimate application of this analysis, I shall 

 now assume the magnitude of the sphere to be so extremely small 

 that the distance from its centre to which any sensible effect on 

 the incident waves extends is very small compared to A. In that 



case, since -2 = t-2~> the last term of the above equation may be 



omitted. Then, to obtain a particular solution of this equation, 

 let it be assumed that 



d.rcr j. , . * 



f x being, in accordance with what is said above, a periodic func- 

 tion of the time, and yjr l being supposed to be a function 

 of r only. Hence by substitution, after differentiating the equa- 

 tion with respect to 6, the result will be 



/,©-»).«-», 



which is consistent with the supposition that ty x is a function of 

 r and constants only. By integration, 



Now if U be the velocity in the direction of r at any point whose 

 coordinates are r and 0, and if W be the velocity perpendicular 

 to r, and be supposed positive in the direction in which 6 increases, 

 we shall have 



aHa . dV A , aHa dW „ 

 — + - 5F =0,and 7 ^ + — = 0. 



d . rcr 

 But from the expression for ' above, it follows, after substi- 

 tuting the value of ^r l} that 



Hence by integration, 



*=£(>•, 0^/i(^+v) cos ». 



It is here to be remarked that this value of a contains a term 

 increasing indefinitely with r only because the analytical reason- 

 ing has been conducted approximately, as will appear from the 



