SQUARES OF ELLIPSOIDAL SU11FACE HARMONIC FUNCTIONS. 127 



We have then 

 /*0W* = 2 2 . 3 3 . 5 3 (1 -f /8) (1 - ft - V/3 3 ) = 2 2 . 3 2 . 5* (1 - . /3 - V/8 2 )- 



Introducing this into the value of / 3 2 (cos), we find 



(44), 



agreeing with the result in p. 548 of" Harmonics" for i = 3, .s = 2, type OEC. 



(2) and (G) First Tesseral Cosine Harmonic and factorial Conine Harmonic. 

 These are defined thus : 



P 8 (p.) = (K* sin* - q*} (1 - * 2 sin 3 0)*, ] 



> . . . . (45), 

 </ (<) = cos $ (</'- K'- cos 3 (f,}, (s = l,B)J 



where <f = ^(1 + 2/c 2 ^ (1 K 3 + 4**)*), with upper .sign for .v = 1 and lower sign 

 for A- = 3, and </'~ := I q' 2 . 

 Writing ' 2 = *' 3 - (7' 2 , 



COS K + K COS 



= r / - /c /3 sin 3 < /c /3 - /c' 3 sin 3 



where /= 1,0= . 



It is clear that [_P-'(p.)(^ s s (<j>)J(n =1,3) has the same form as HM/'OO/ (<)?(* = - -) 

 when in the latter we interchange 9 with -.J-Tr <^>, and K with '. The interchange of 

 the variables of integration clearly makes no difference in the result, and therefore we 

 need only interchange K and K, and replace t hy t'. 



In the present instance 



This shows that t''~ is the same function of K- that ^' was of /<' 3 , but that /y 1 (cos) is 

 analogous with / 3 2 (cos), and 7 3 3 (cos) with /., (cos). Thus we may at once write down 

 the results by interchanging K and K' throughout. 



Let jy = (1 -- /c 2 + 4*c*)*. 



Then putting K''\f' 2 g~ = 1, we have by symmetry with (38) 



7 3 ! (cos) = '^f^- % D W - > + **) ^ - (1 - K)(l - ^ 

 / 3 3 (cos) = the same with the sign of D' changed. 



