122 PROBLEMS IN THREE DIMENSIONS. [CHAP. V 



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



JX%A) 



becomes divergent as (1 - x) v ~ a ~P ; 



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



. Il(y-l). n(a + )3-y 1) ,, NV-a-8 /\ 



r ^^*^-a^h^^r^-^ ......... (m) ' 



ultimately. 



For the critical case where y a /3 =0, 

 we may have recourse to the formula 



-f F(a,fty,.r) = ^ 

 dx y 



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



The last factor is now convergent when x = l, so that F(a t ft y, x) is 

 ultimately divergent as log (\-x). 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 termioates 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 ^ 

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

 diverge at the limits /*= + !, becoming infinite as log(l /u, 2 ). 



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 



n(n-l)(n-2)(n-3) 

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



* For n even this corresponds to A = (-)l* ' ' '"'" - , 5 = 0; whilst for n 



> . 4. ..n 



odd we have /I = 0, B = ( - )i<n-D ' '" n . See Heine, t. i. , pp. 12, 147. 



^ . *...(- 1) 



