[638 
26 ON A ^-FORMULA LEADING TO AN EXPRESSION FOR E x . 
so that 
qX' = q + 3 q 3 + 6q e + lOg 10 + &c., 
then 
X + 8qX' = 1 + 9q + 2 5q s + 49 q 6 + 81^ 10 + &c., 
so that the right-hand side of (B) is 
X + 8qX' 1 0 X' 
X ’ 1 + 8q X • 
But {Fund. Nova, p. 185, Ges. Werke, t. i., p. 237), 
g _ 1 — q*. 1 — q 4 .1 — q 6 ... 
so that 
I-# . 1 — 2 3 • 1 — 2* ... ’ 
X' - 22 _ 42 s _ 6q 5 
X 1 — 2 2 1 — 2 4 1 — 2 ti 
• + — + Jf +AA + 
1-2 1 - 2 3 1 - 2 5 
And the equation (B) intended to be proved thus becomes 
1+ sifi* + *£-_ AA + .J 
(1 + 2 1+2“ 1 + 2* J 
-16 
= 1+82 
1^5_ + ®2L+ .. ' 
(1 — 2 2 1 — 2 4 1 — 2^ 
( — 2q 4<q 3 Qq 5 
(1 — 2 2 1 — q 4 1 — 2 6 
1 32 2 5q 4 
+ ~, ... 
1 — 2 1 — 2 3 1 — 2 5 
viz. omitting the terms unity, dividing by 82, and transposing, this is 
1 
1 + 2 + 
1 + 2 2 
1 + 2* 1 
, 2 
Asl + 
6q 2 
1 -2 2 
i-2 4 
1 -2 6 
2 2 
+ T—^-„ + 
4 2 3 
i « + 
1-2- 
1 — q 4 
1 — 2 6 
1 
Sq 2 
oq 4 
! -2 
1 — 2 3 
1 - 2 s 
The second and third lines unite together, and the equation becomes 
1^2 2 32 2 42 3 
+ 
1+2 I + 2 2 1 + 2 3 1+2 4 
2 42 62 2 82 3 
1 — 2 1 + 2 2 1 — 2 s 1 + 2 4 ' 
1 _ Sq 2 _ bq 4 _ 7q 6 
1-2 I-2 3 1-2 5 1~2 7