252 Methods for the Solution of Special Problems [CH. vm 



already investigated. Combining the values just obtained for H, H with the 

 value for < given by equation' (221), we obtain the general solution 



= 22 (A P% (f) + UQ ()} [A'P% (,) + BQK (,)J [Ccos m * + sin ></>). 



m n 



At infinity it is easily found that 



d* 



77 = oo , f = : = = cos 6, 



v # 2 + ro- 2 



while at the origin ?? = 1, f = 0. 



Thus in the space outside any spheroid of the series, the solution 



is finite everywhere, while, in the space inside, the finite solution is 



P 



Oblate Spheroids. 



299. For an oblate spheroid, a 2 6 2 is negative, so that in equation (218) 

 we replace 6 2 a 2 by 2 , so that /c = ic, and obtain, in place of equations (219) 

 and (220), 



x = Kir 



Replacing irj by f, we may take f, f and <^> as real orthogonal curvilinear 

 coordinates, connected with Cartesian coordinates by the relations 



x = 



We proceed to search for solutions of the type 



and find that H, ^ must satisfy the same equations as before, while Z must 

 satisfy 



The solution of this is 



Z-A'.P<t 



and the most general solution may now be written down as before. 



