140 PROBLEMS IN THREE DIMENSIONS. [CHAP. V 



or, on substitution from (3), 



This may also be written 



101. If < be a finite function of p, and o> from /x = 1 to 

 fjb = + 1 and from co = to o> = 2?r, it may be expanded in a series 

 of surface harmonics of integral orders, of the types given by Art. 

 87 (6), where the coefficients are functions of f ; and it appears on 

 substitution in (4) that each term of the expansion must satisfy 

 the equation separately. Taking first the case of the zonal har- 

 monic, we write 



4> = P(p).Z ........................... (5), 



and on substitution we find, in virtue of Art. 85 (1), 



(6), 



which is of the same form as the equation referred to. We thus 

 obtain the solutions 



and (/> = Pn(f^) Qn(Z) (8), 



where 



df 



1.3. .. 



, 





2 . 4 (27i + 3) (2n + 5) .. 



The solution (7) is finite when f = 1, and is therefore adapted 

 to the space within an ellipsoid of revolution ; whilst (8) is infinite 

 for f=l, but vanishes for =oc, and is appropriate to the 



* Ferrers, c. v. ; Todhunter, c. vi.; Forsyth, Differential Equations, Arts. 96 99. 



