Electric Waves round a Large Sphere, 519 



But <y x p is a special case of a symmetrical solution, and 

 must therefore have a similar form. Introducing the 

 function 



K«(3)= o * { €-*""** -Jz)-e* mw JJz)\, . (4) 

 2 sin mir <- J v v 



finite at infinity, then since when r < rj we have 



TT = (,ri)_i 2 " (*t+l>**"K»(*^)J-(*r>P'-W 



(where m = 2n + l), it follows that when ?'<r, 



W>=/» J^ 2; 2m (rr,) ~* e* m «K m (fa*) J.(*r)P»M • 

 But B " - / - S 9 



>h-*-*t'i-*& 



and after some reduction, if 



^(r 1 )=in- i «*" r {e*"<» +u K. +i +«*" ( "- 1) K.-i(*ri)} 



dp 



yip = r*sm 2 2g 1 £r 1 )J m (kr)-~, ... (5) 



a formula given by Macdonald*. 



But there is an important point to be noticed at this stage, 

 which formed the real subject of Poincare's criticism of 

 Macdonald' s investigation. For the oscillator is afterwards 

 placed on the sphere, and when the effect at a point on the 

 sphere is sought, we are involved in the case r = r x . It 

 requires to be proved, therefore, that the formulae for r>r x 

 and r<zri are continuous at r = r x , not only in themselves, 

 but in their derivates, which are used in determining the 

 electric force afterwards. That no difficulty is introduced 

 may be shown by the following method, which indicates that 

 Poincare's criticism, though sound in principle, is immaterial 

 in practice, and that the series are convergent in the proper 

 way. 



The formula for <?~'* R /R, with r>r lt is found to be 



C^- = K)"* r 2mJ m (Ar 1 ) K w (*r) P.GO, 



XV o 



* Loc. cit. 



