296-298] Spheroidal Harmonics 251 



It is easily found that 



1 !- , 1 /-l 



from which Laplace's equation is obtained in the form 



298. Let us search for solutions of the form 



F=EHc|>, 



where 5, H, <I> are solutions solely of f, ?; and < respectively. On substituting 

 this tentative solution and simplifying, we obtain 



snn i3 ! *_ 

 + > a, }J + * 8<j>* ~ 



As in the theory of spherical harmonics, the only possible solution results 

 from taking 



where m 2 is a constant, and m must be an integer if the solution is to be 

 single valued. The solution is 



<|> = C cos ra</> + D sin m^ (221). 



We must now have 



m 2 m 2 



~ 1 _ 2 + ^1 



and this can only be satisfied by taking 

 i 



together with 



Equations (222) and (223) are identical with the equation already dis- 

 cussed in 273, 274. The solutions are known to be 



H= 



where s = n(n + l) and P% t Q% are the associated Legendrian functions 



