770 Dr. J. W. Nicholson on 



This is the required equation of the fourth order. In 

 this form, it is much more convenient than when expanded 

 in full. 



Solution in Series of P Functions. 



A series solution in powers of yj cannot be obtained in a 

 simple form, but one solution in zonal harmonics of integral 

 order is readily possible, as we know from Adams' expres- 

 sion for P n (//,) P m (yu,), and it must have a simple general 

 term. We may anticipate the existence of other series 

 solutions of this type representing such solutions as 



Write _/» _ . . 



the summation being for integral values of r, and the limits 

 oc and /3 being at present unknown. 



Then 



Write 



R = r{r + 1) 3 

 and we find that 



% a { (M + N-R) 2 -4MN} arPr(fl) 



Quoting the recurrence formulae, 



(2r + l)fiPr= (r + l)P r+ i + rP r _ lf 



( 2r + 1 ) p -= -dfT-^r' 



we may change this to the form, 



