SQUAUKS OF ELLIPSOIDAL SURFACE HARMONIC FUNCTIONS. lit) 



On squaring p^ we find A = 0, A\ = 1, and 



In " Harmonics" the definitions were the same, and therefore 



This agrees with the result on ]>. 548 of" Harmonics" with i = 1 , ,s- = 1 , type OOS. 



HARMONICS OF TUK SKCOND ORDI.;U. 

 [n these the only coefficients are A lt , A } , A*, and (2) hecomes, 



f , /COS \ _ 47I-F COS /3 COS 7 r , , __ ! 3 ,., .0,1 4 /| 4 3 



- ( ) * " ' 



with .v = 0, 1, 2. 



(1) W (4) The Zonal and Scctorial Cosine Harmonics. 

 These are defined thus, 



$,*( / x) = /c--sm-0- 7 -, 1 



^ ...... (15), 



(T/ (^) = 5 ' 3 - K'* cos 3 </,, (, = 0, 2) J 



where (/ = [! + /c 2 ^ (1 K'V~)*], with upper sign for .s = and lower for s = 2 ; 

 and (/'- = 1 */ 2 . 

 Writing 



OOO /.T /.T 1 I- O /.) - / , /.T\l-l 



t y = K~ <f = q~ K~ = jj IK" K ~ (1 K~K ~) 1 J, 



..... (15), 

 = (J . t K~ sin 2 </,), (,s = 0, 2) J 



where f=l,g=l. It may he noted that t- is a symmetrical function in K" and 

 Squaring ^ 2 * we find 



