SQUARES OF ELLIPSOIDAL SURFACE HARMONIC FUNCTIONS. 131 



Developing these expressions in powers of r' 2 , reintroducing the factor 

 and reverting to the notation of " Harmonics," I find 



rj (sin) = 



(sin) = 



In " Harmonics " I defined 



To make our former definition agree with this we must take 



A. = (! +10 -A/8 8 ); >-r=;Hi+t/3-ii/3 2 ). 



Hence /AC = (l+/8 -Htm 



It will l)e found that </ e ^-(1 I /3 -\- -/',. /3 C ), and tliat //c. ry- lias tlie above form. 

 Again I defined 



S, ^ = sn < /3 1 - fyS sin 3<, 



= sin^[1 --- I 8 + M/3 C + P(l --P)siu 3 ^] . . . (58). 



To make our former definition aree with this we must take 





cV - 1 . t- = 1 - i 8 5-y8 + 1I/8-, 



It will be found that t~ = f (1 - ^-j3 + W/S 2 ). 



Whence .^K'^/ 1=1(1+^/8 H/S 2 ), and (7/c'v/ 1 . /c' 2 has the correct form. 



Therefore 



y^V - i - 3 (i + W + W 



- 



Wlience 



(59), 



agreeing with the result on p. 548 of " Harmonics" with i = 3, s = 1, type OOS. 



s 2 



