H. E. SoPER, A. W. Young, 
B. M. Cave, A. Lee, K. Pearson 339 
By (xxv) 
V 
' n 
i + 
w + 1 2 
- rn 
C n 
1 \ TO 
[n 
n 
1 
2 2 
n \ 
n 4- 
-12 + - 
1 . 32 ... (2s - 1)2 p2s 
' ' s! (« + 3) (n + 5) - (to + 2s + 1) 2 
12. 32 ... (2s - 1)2 
+ ■■■ (to + 1)(to -t- 3) ... (to + 2s- 1) 2« 
+ 3 2 ^ ■■■ 
_ P . 32 ... (2s - 1)2 p2s 
Since 
...), 
s!(TO + 3)(n + 5) ... (TO + 2s + 1) 2^ ' ■") 
12 . 32 ... (2s - 1)2 p^s x| 
+ s! (to + 1) (to + 3) ... (to + 2s - 1) + ■■■jj ' 
_ TO 
+ 
+ 
g'n+i n - 1 ■ 
The general term is therefore 
1 qn±^ ^-Z^ ( n'{n+ 1) _ 2 _ \ 
(to - 1) (to - 2) p TO Vto + 2s + 1 7 
12. 32 ... (2s - 1)2 p2« 
^ s ! (to + 1) (TO + 3) ... (to + 2s - 1) 2* ' 
Now in (xxvi) we have seen that 
32.52... (2s + 1)2 p2>' ) 
s! (TO + 3) (to + 5) ... (TO + 2s + 1) 2^^ + 
and the general term is 
^„+2TO-2 / 32.52 ...(2S+ 1)2 
^ TO \s ! (to + 3) (to + 5) . . . (to + 2s + 1) 
2 . 32 . 52 ... (2s - 1)2 
(s - 1) ! (to + 3) (to + 5) ... (to + 2s - 1) 2' 
TO - 2 / (2s + 1)2 _ \ 12. 32.52 ... (2s- 1)2 p2s 
TO U + 2s + 1 s ! (TO + 3) (to + 5) ... (to + 2s - 1) 2-' 
TO - 2 (« +J)jl_~ 2* (" " 12.32.52... (2s- 1)2 p 
^1 TO + 2s + 1 sr(TO 4- 1) (to + 3) ... (to + 2y^I) 
2s 
9,8 
= ?«+2 { n"" (to + 1) _ 1) 
TO Vto + 2s + 1 ^ ' 
12. 32. 52 ... ( 2s- 1)2 p2s 
^ s ! (to + ly (to + 3)Xto + 5)TT7(TOr+ 2s +T) 2^ 
= (to - 1) (to - 2) (r„+2 - ^n)- 
Hence r„+2 - r„ = (^/^ l)^{n - 2) (xxviii). 
