122 PROBLEMS IN THREE DIMENSIONS. [CHAP. V 



Since the latter series will now be convergent when # = 1, we see that 



#(****) 



becomes divergent as (1 x}^~ a ~^ ; 



more precisely, for values of x infinitely nearly equal to unity, we have 



* (to PI y,*) 



ultimately. 



For the critical case where y - a - 8 =0, 

 we may have recourse to the formula 



^ (a, fty, X)- (a+1, 

 which, with (ii), gives in the case supposed 



y, *)-(l~.)r? . ^(y-a, y-0, 



The last factor is now convergent when x=\, so that ^(o, 8, y, a;) is 

 ultimately divergent as log (! #) More precisely we have, for values of x 

 near this limit, 



86. Of the two series which occur in the general expression 

 Art. 85 (2) of a zonal harmonic, the former terminates when n is 

 an even, and the latter when n is an odd integer. For other 

 values of n both series are essentially convergent for values of /u, 

 between + 1, but since in each case we have 7 a /3 = 0, they 

 diverge at the limits fi=l, becoming infinite as log(l //r). 



It follows that the terminating series corresponding to integral 

 values of n are the only zonal surface-harmonics which are finite 

 over the unit sphere. If we reverse the series we find that both 

 these cases (n even, and n odd,) are included in the formula 



1 - 8 - 6 -&amp;lt; 2 - 1 &amp;gt; 



2.4.(2n-l)(2 )l -3) 





* For n even this corresponds to A = ( - ) n ~J - , B = ; whilst for n 



2i 4 . . . 7Z- 



odd we have 4 = 0, B = ( - )*&amp;lt;-i&amp;gt; ^ See Heine . * [ - PP- 12 &amp;gt; 147 - 



