84-85] SPHERICAL HARMONICS. 121 



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

 disappears, and putting cos 6 /JL we get 



the differential equation of 'zonal' harmonics*. This equation, 

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

 for integration by series. We thus obtain 



1 



n(n + I) . (n-2)n(n + l)(n + S) ) 



1.2 * ~~iT27374T ^ "j 



172.3.4.5 





The series which here present themselves are of the kind 

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



CC. 



1.2.8.7.7+1.7+2 

 we have 



+ '" W 



n, 1 + K f , /t 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>0. 

 In this case we have 



where n (z) is in Gauss's notation the equivalent of Euler's r (z+ 1). 

 The degree of divergence of the series (3) when 



y-a-/3<0, 

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



^(o, ft-y, x} = (l-xy-*-*F(y-a, y-8, y, x) ............ (H)|. 



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

 unit sphere into parallel belts. 

 t 1. c. ante p. 113. 

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



