H. E. SoPBR, A. W. Young, B. M. Cave, A. Lee, K. Pearson 347 
Then, if 
V = log (cosh z - po^) = log (1 - Pq^) + a^'z^ + o^'z* + agZ^+ 
it follows that 
1 /d'^v\ , 1 /d*v\ 
«2 = 2 
and so on. 
sinh z / 1 ox • 1 
gz = coshz-po ^- (coshz-po^)^-^ = smh.. 
Apply Leibnitz's Theorem, differentiating {2s — 1) times, and we have 
. ^ dv ,^ (2s- 1) (2s -2) . , d^v 
sum z Y H- (2s — 1) cosh z~j-„ + - ~ sum ~ ' 
'dz ' ^ ' dz^ ' 2! dz^ 
+ (cosh z - po^) = cosh z. 
Hence when z = 0 
- ') - '^^-ii^^-^ (0)„ ..... (1 - (SDr 
Now put s in succession 1, 2, 3, etc. and there results 
etc., etc. 
These lead to 
^ 2 1-po'' 24(l-yOo')'' 
, _ 16 + 13po^ + , _ _ (272 + 297 po^ + 60^0^ + po«) 
^« 720 (l-po')^ ' 40320 (1 -po2)4 
, ^ 7936 + 10841po--^ + 3651po^ + 251po^ + p^^ 
3,628,800 (1 - po2)5 ^ etc. 
Accordingly we have, raising (xlviii) to the nth power and expanding the 
exponential after the term in a^'z^, 
(C0shz-po2)« (1 _p„2)« ^ 4 6 V 8 - 4 ; 
