774 Dr. J. W. Nicholson on 



The Descending Series. 



The identical relations can also be written 



(i3+m + n+2)( i 8-m.-w)(i3 + m-« + l) : ( i 8-m + »-(-l) 



ap dFfs +l 

 2ft + 1 dfi 



fl-3 



-f- Z a {r + m + n + 2)(r — m — w) (r + m — ?i + 1 )(r — m + r< + 1) 



Of,, a L r+l 



x 2^TT^r 



= S a (»• + m + n 4- 1) (r — m — n — ■ 1) (r + ??i — n) (?' -f » — in ) 



where the first term vanishes if ft is chosen suitably. This 

 is equivalent to 



(3 



X (r + wi + n) (f — m— n — 2) (r + ?n — n — 1) (r — m+n — 1) 



0l r —2 d± r -i 



X 2r~^Z~dfI~ 

 =2 (r+wi + n + l) (r— w— n— l)(r + m — w)(r + w — w) 



a r dP,._i 

 X 27+1^7"' 



The last term, r being an integer like ft?, n, involves -r-- 

 or zero, thus defining a. We find 



a^_2 27' — 3 r + w-t-n + lf-m-w-1 r + ??2 — ?i 



a r 2? 1 -f- 1 r + w* + ft 7' — wi — ?z — 2 r -f ??i — n — 1 



r + n — ?» 

 r + n — m — 1 ' 



and the series is, a* being arbitrary, proportional to 



Taking ft — m + n, we have 



-P / ui 2m . 2ft 2>;i + 2ft + l 2ro + 2n -3 



y-T r »+»W+ 2 • 2m -l 2n-l ' 2m + 2?i ' 2m + 2n+l 



x P» 1+r -2(a*)+ 



