534 Mr. E. T. Glazebrook on the Application of the 



y ■ ■ (45) 



d 2 U .dhi aTT ,dU 



p *w- p dt* = - aV -vw J 



The solution of equations practically the same as these has 

 been given by Ketteler*. It will be useful to have it here for 

 the sake of completeness, and also because the notation is 

 different. 



Let the solutions be given by 



u ~ u o e n (46) 



Then, on substituting, we have 



Uo {„ 2 ( p - B r2 )_B(^i-^)}=u »v^ (47) 



U { n 2 p 2 — oc 2 + y^cn} = u^pp' . 

 Thus 



{ y t^"T0" B (^"^)} K/ ° 2 " a2+72m}=nV2;(48) 

 - \ ^-T*) + B * 2 } (»"*-«') + ^v^ =nV2; (49) 



7 *{n*( P -^)+Bk*}-^(n*p 2 -* 2 ) = 0. . . (50) 



. 1 _**-P _ 2(n> 2 -* 2 )£ 

 ' ' V 2 n 2 B yW ^° ij 



^ 2 ^U^2-« 2 ) 2 + ^7l=^V 2 (52) 



Tlms M = r 2 A 3 , w 



nV [^v 2 -« 2 } 2 +A 2 ]b ^ 06) 



Y 2 n 2 B [(^-a 2 ) 2 + n » 7 4]B * * &*) 



Let a* = ^2. 



Then 2k 7 y g n 3 



nV~[ / o 2 (^- J / 2 ) + 7 V]B ^ 00) 



1 __ ^_ P_ p' 2 p^{n 2 -v^) . 



V 2 n 2 B [^2(^-^)2 + 7 V]B * ' ^ ; 



* Ketteler, Theoretische Optik, § 42, and various papers alrady re- 

 ferred to. 



t The symbols k, n, X, ft, v, have no longer the same signification as 

 above, pp. 524 seq. 



