SQUARES OF ELLIPSOIDAL SURFACE HARMONIC FUNCTIONS 133 



Squaring P 3 2 we have 



A =0, ^, = KV 2 , A z = **-,<>*, A,= -l. 

 Therefore 

 / 3 (cos) = 



A 



Reducing this expression and putting / 2 </W 4 = I, we have 



47r& cos i6 cos y 



7 



In " Harmonics" I defined 

 ~ 



. (65). 



1 a 2 ~ 



To make our former definition agree with this we must take 



Again I defined 



S 3 2 (<) = (1 - /3 cos 2<)* sin 2<^ = 2 (1 + ^p (l - 2 ^ cos 2 ^ )' sin </> cos f (6(5). 



To make the former definition agree with this we must take 



^V- 1 = 2(1 + )*. 

 Therefore 



/<//cV 2 y - 1 = 2 . 3 . 5 j [t |1 :V nd 



/V*cV* = 2 2 . 3 3 . 5 2 ^ + ^' = 2 2 . 3 2 . 5 s (1 + 3/8 + 4/3 2 ). 

 Hence in the notation of " Harmonics" 



/ 3 2 (sin) = 47 " M .3.4.5(1+3/8+ 4/8 2 ), ..... (67) 

 agreeing with the result on p. 548 of " Harmonics " with i = 3, s = 2, type OES. 



