

SQUARES OF ELLIPSOIDAL SURFACE HARMONIC FUNCTIONS. 123 



<> 

 Therefore /~(/W a = 3 a (I + ft), and on multiplying (28) by this factor we have 



/ 9 i (sin) = ^M. 3(1 + 0), ........ (30) 



agreeing with the result on p. 549 of ' : Harmonics " with i = 2, s = 1, type EOS. 



(5) The Sectorial Sine Harmonic. 

 This is defined thus : 



= sn (> cos <. 



j 



If in the last integral we had written \ir for (, and |-TT < for 9, and K' for *, 

 l^., 1 would have become , 2 S , and S.>' would have become P.,'-'. Therefore the result (28) 

 gives what is needed by merely interchanging K and K. 



Therefore 



/ 8 >in) = 4 ^ 5 ^|?? . i ....... (32). 



For the purpose of comparison I must put 



3 2 (<A)=0- v/' - 1 *' sin </. (^ - *'- sin 2 ^)*, (31) 



TO-/ \ vr cos /3 cos y / i ,., .^ ,,\ /.. \ 



and / a s (sin)= 1 r (~ i/W^*) .... (33). 



J olll ^J 



In " Harmonics" the definition was 



= sin 2(> =2 sn cos <>. 



In order to make the two definitions agree we must take 



Thus /Y/cV* = 2 2 . 3 s ILJ ^; introducing this in (33) we have 



2(l+2^ + 2n . . . (35) 



agreeing with the result on p. 548 of " Harmonics" with i = 2, s = 2, type EES. 



B 2 



