SQUARES OF ELLIPSOIDAL SURFACE HARMONIC FUNCTIONS. 121 



Whence 



Introducing this I find 



7 cos = 4 5 



7 2 (cos) = 4?rM (l + 3/3 + 3/3*) ....... (20), 



agreeing with the result on p. 549 of " Harmonics" with i = 2, a = 0, type EEC. 

 Again in " Harmonics " 



3 ~ M = 3 ftP, (p.) + 7V (/.) = 3/8 + 3(1- f/3) cos- 0, ] 



f . . (21). 

 &8 + cos 2( = 1 + l/S - 2 sin 3 J 



In order to make the two definitions agree we must take 



or 



So that /V : = 2' . 3* (1 + 8 - IF). 



Introducing this in (18) we have 



/ 3 a (cos) = ^M. 12(1 - + /3 2 ) ....... (22), 



agreeing with the result on p. 548 of " Harmonics " witli i = 2, * = 2, type EEC. 



(2) The Cosine Tesseral Harmonic. 

 These are defined thus, 



P., 1 (p.) = sin 0(l- K * sin- ff)* = f. (/c e - /c e cos- 6>) 4 (K'- + /c 2 cos 3 0) 4 , 



(23) 



C 3 l (<) = COS <^ ( I - K* cos --' ^)1 = g ( K 3 + K '2 S i U - ^)i ( K '- - K'" sill 3 ^), 



, 1 1 



where / = - , / = ^ 



Squaring P.2 1 we find 



A = JK"\ A, = (K~ - /c' 2 ), A, = - 1. 



On substituting in the formula, I find, on putting f-g'icK' 1 = 1 and reducing, 



^ ....... (24). 



5 SIB 3 

 VOL. CCIII. A. R 



