

SQUARES OF ELLIPSOIDAL SURFACE HARMONIC FUNCTIONS. 115 



and by symmetry 



{ 2n }' = - f- " 2 (K- - x'~) {2 - 2]' + *"_ ' 3 K~K'~ \2n - 4}'. 



Multiplying the first of these by {2m'/ and the second by ( )-'" (2m| and 

 adding together we have 



^, ,,) 



'" - ir-i T sttv Ufa /o\ PA i \ ~\ i -j/fc ^~ > o /.i t-,^ . ~. -\ 



I 2n, 2m] = (K" /c -) [2n -2, 2m] + K-K - \_2it 4, 2m]. 



By successive applications of this formula we may reduce any function [2, 2m] 

 until it depends on [2, 0], but the result becomes very complicated after a few 

 successive reductions. 



Now 



*-*. w'-ff 



. o A . o I 



{2j - = 



cos- 



sn 



Then 



o A Jo r 



= KV + F'F - FF'. 





But it is well known tliat this combination of the complete elliptic integrals with 

 moduli K and K is ^TT.* 



Hence [2, 0] = \TT. 



It seems unnecessary to reproduce the simple algebra involved in the successive 

 reductions, and I therefore merely give the results, as follows: 



[0, 2] = [2, 0]=^ 

 -[0, 4] = [4, 0] = |(^-/<'-)47r 



2, 4 | = | 4, 2]=4/<V~.^7 

 -[2, GJ= 6, 2]=.', (ic 2 -^)^ 3 . I-TT 



[G, 8] = [8,Gj = -l/cV (1 47r 



These are the only functions of this kind which are needed for the evaluations of 

 integrals in this paper. 



* See for example DUIIEGE'S ' Theorie der Elliptischen Functionen,' p. 293. 



