456 Prof. 1). N. Mallik on Magnetic 



.*. since x 2 if 'z 2 _ 



A 2 (r 2 -1) T h 2 (f-l) "*" AV ' 



r=constant is a set of confocal prolate spheroids. 



Now 



^2 ^ ^-COS 2 fl d ^ + ^ _ cQg2 ^ ^ + ^ _ ^ gin2 ^ ^ 



.*. transforming V 2 ^ r to ?'? ^ <f> coordinates, we have 



j>_ f( r 2 - cos 2 0) (?- 2 - 1) sin 2 fl -l *av 

 ^ [_ r 2 — cos 2 J ^/ j 



_9_ nV 2 -cos 2 0) ^ 2 -l)si n 2 6>-|^V 

 B6>L r 2 -l r- 2 -cos 2 J 30' 



_a p 2 ~cos 2 6> 7- 2 -cos 2 6> yav_ n 



+ B</>L r 2 -l (r 2 -l)sm 2 0J ~d<\> ' 

 or 



^^ n^ V ^ X 3/- z)BV\ , r»-cos 2 1 d 2 V 

 If V is independent of <j> } we have simply 



Putting Y =u n ~P n (Gos 6)j we have, since 



u n , given by 



u»=AP»(r) + BQ II (r), 



or V=S{AP w (cos 0)P n (r) + BP„(cos 0)Q„<V)}. 



Where P's and Q's are Legendre's coefficients of the] first 

 and second kind. 



2. It would be useful to expand (distance) -1 in these 

 harmonics. 



For this we have, writing 



pr- = x 2 + y 2 + ~ 2 

 =/*V-sin 2 0). 



D 2 = p 2 + p' 2 — 2/op'[cos a cos a'+ sin a sin a' cos cf)—<f)'], 



