776 J Dr. J. W. Nicholson on 



We shall denote this as Series C. It stops just before the 

 P's are of negative order. 



This series we must proceed to identify. From the order 

 in fju of the functions P which occur, it cannot involve P m P», 

 in which they are of orders m-\-n, m + n—2, ... or Q m Q ?l , in 

 which they are clearly of orders m+n + 2, m + n, ... . It is 

 therefore a linear combination of P m Q,i and P n Q m , and since 

 it is a terminating series, it must be of the type 



For this is the only such type which is polynomial. We 

 know from OhristoffePs formula that 



Q n = ±F n log- — - + X«_i, 

 i — fi 



where X n is a polynomial in ^ of degree n — 1. Thus 



= — 9— lo g'i — ~ +A,»-iP TO — log — c ~X m _iP M 



I °1 - /-t 2 ° 1— ft 



which is a polynomial whose degree cannot exceed n + m — 1. 

 No other such polynomial can be constructed, all other com- 

 binations being logarithmic. 



Now when n = 0, 1, 2, ... we find that the series y becomes 



__p J. im . zm — i) ^ 



1 2m. 2m -5 

 3 ' 2m-2.2m-l 



1 2m. 2m- 

 5 ' 2m — 4 . 2m — 1 



+ E • « r~^ 7 Pm-5+ • 



-P .lLiA 2m. 2m -7 p 



yi -r.^+ 2 _ 5 . 2m _ 4>2m _ 1 n l -4 



1.3.6 2m. 2m — 2.2m- 5. 2m -11 p 



2.5.7* 2m — 1 . 2m — 3 . 2m — 4 . 2m — 6 Fw " 6+ ' * '' 



_ p ,1.6 2m.2m-9 

 yt-f.-8+5-y • 2m . 6#2 m-l ^ l - 5+ * * ' ' 



These may be compared with the values of P Wi Q« — Q wl P„ 

 for the earlier values of n. We have, if P's of negative 



