140 PROBLEMS IN THREE DIMENSIONS. [CHAP. V 



or, on substitution from (3), 



This may also be written 



101. If &amp;lt; be a finite function of //, and to from /z = 1 to 

 //, = + 1 and from o&amp;gt; = to w = 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 



&amp;lt;t&amp;gt; = P n (fi).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 



* = P n (/).P n (?) ........................ (7), 



and tf&amp;gt; = PG*).Q(?) ........................ (8), 



where 



rn \ 



i L. i . \ &quot; i - / \ i *- / if n g 



1.3. ..(2/1+1) I&quot; 2(2?z + 3) 



~* ~t&amp;gt; A. /Si i O\ /O ~i K\ y + 



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

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

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



* Ferrers, c. v. ; Todhunter, c. vi.; Forsyth, Differential Equations, Arts. 9699. 



