772 Dr. J. W. Nicholson on 



which is more troublesome, but which factorizes ultimately to 

 (j3 + m +- n + 2)(/3 — m — n)(fi + m — n + l)(fi — m + n + 1) = 0, 

 and the possible values of /3 are 



— (772 + 92 + 2), ?nfw, m—n — 1, n—m — 1. 

 The second clearly will lead to Adams'' formula. 



The Ascending Series. 

 We may now write the identical relation in the form 



% a {r+ m-\- n + 2) (r — m — n) (r -\-m — n + l) (r — 7)2 + 72 + 1) 



X ¥r^l~dfT 

 = £ a (V + 772 + 72 + 1) (?* — 7» -n — 1) (r + m— w)(r + 7i— m) 



or its equivalent, 2r + 1 d P 



2 (?' + '/H + 71+2) (*' — 971 — 7l) (7' + 111 — 77 + 1) (r — 777 + 77+ 1) 



2r+l~^uT 



— 2 a (r + 7?7 +77 + 3) (7' — 777 — 7i+ 1) (r + ?72 — 72 + 2) 



X (r + n — m+%) , — ;— 



+ (« + 7?i + 77 + 1) (e — m — n— l)(a+m— ?l)(#+?i— ?rc) 



a a dP a -i 



2u + l dfju 



The last term, not under the sign of summation, vanishes if 

 a is chosen appropriately, as above, for an ascending series. 

 We then deduce, for any value of r equal to or greater 

 than «, 



a r +2 2r + 5 r + m + 72 + 2 r — m — n 



a r . 2r + 1 ' v + 7)2 + ?2 + 3 ' r — m — 72 + 1 



T +7)7— 72+1 V — 7)2 + 7) +1 

 " r + 772 — 72 + 2 ' V — 7)7 + 72 + 2 ' 



