SQUARES OP ELLIPSOIDAL SURFACE HARMONIC FUNCTIONS. 129 



Therefore /VV 3 = - 3 -~ 6 "[l 4- *P + -ft-/? 2 ], and I find 



A _ 



(50), 



agreeing with the result on p. 549 of " Harmonics" with i = 3, /? = 1, type OOC. 

 In " Harmonics " I defined 



+ 30) ^Y 0*) + JV 



l+*0 + A0'-ntf(l+t/8-i|/8)] . (51). 



But P 3 3 GU) =/(! - * 2 sin 2 0)* ( 

 Therefore >* = - 15 (1 + ft - if/3 3 ), ./y 3 = - 15 (I + |/3 + 



*3 O 91 



XT i '-* / "> i ' ' / 1 i ^ ' /i", 



Now q~ = I - 22 K'- + afl K ' + ^ K' S ... 



Therefore q* = I - 1/8+ ?|/8 3 , and 



/=- 15(l+-V-/8+i)8 J ). 

 This also gives the correct value toftc. 



Again C :j 3 (^) = f /3 ( 1 + f /3) cos ^> + cos 3<, 



= cos ^,[4 cos 2 ./, -3 (I . ---i-^-3 3 3 -/8 2 )] 



But Cj 3 (<^)) = rX cos < (/c' J cos 2 </> - r/'-). 



Therefore (//c' = |, , and gr/c' . q'' 2 = 3 (1 - - ^ /3 2 ). 

 /c 



If we eliminate (JK', these equations give the correct value for <f 2 . 



Therefore 

 Hence we find 



7,3 (cos) = ^. 3GO (1+1/3+ f| /3 2 ) ...... (53), 



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

 VOL. com. A. s 



