84-85] SPHERICAL HARMONICS. 121 



85. In the case of symmetry about the axis of x, the term 

 d 2 S n /dco 2 disappears, and putting cos 6 = p, we get 



the differential equation of zonal harmonics*. This equation, 

 containing only terms of two different dimensions in //,, is adapted 

 for integration by series. We thus obtain 



; i = ^{ 1 _ H^^ + (- 2 l( + lK + 3)^_^J 



The series which here present themselves are of the kind 

 called hypergeometric ; viz. if we write, after Gauss -f*, 



..... 



1.2.8.7.7+1.7+2 

 we have 



S n = AF(- n, i + in, 1 ^) + ^^(J- - f/i, 1 + \n, f , A6 2 )...(4). 



The series (3) is of course essentially convergent when x lies between 

 and 1 ; but when ^=1 it is convergent if, and only if 



y-a-/3&amp;gt;0. 

 In this case we have 



F(n 8 ^ n 



1 (a ^ ^ 1 )= 



where n (z) is in Gauss s notation the equivalent of Euler s r (+!) 

 The degree of divergence of the series (3) when 



y-a-/3&amp;lt;0, 

 as x approaches the value 1, is given by the theorem 



F(o, fty, ^ = (1-^^^(7-0, y-8, y, x) 



* So called by Thomson and Tait, because the nodal lines (S n = 0) divide the 

 unit sphere into parallel belts. 

 t I.e. ante p. 113. 

 Forsyth, Differential Equations, London, 1885, c. vi. 



