408 SIR J. F. W. HERSCHEL ON THE ALGEBRAIC EXPRESSION OF THE 
But by equation (5.), 
=n{n-l)....{n-i+\) 0 "-, 
and therefore the foregoing- expression becomes 
X log(l + A) n/xy log(l + A) 
7- A ^ “2^; • A ® 
or, inverting the order of the terras, 
( n« r 1 Iog(l+A) /a^Y log (I + A) 
n /x 
■^2U 
■’ log(l+A) n{n—\)/'xy ^ log (1 + A) 
2,3 
&C.L 
Now if Bi, Bg, B5, &c. be the numbers of Bernoulli in their order (the even values 
B2, B4, &c. being severally =0), we have 
Iog(l+A) log(l+A) _ 1 log (1+Al p _ 1 
A 0—1; ^ 0—2’ A 0— i>i— g’ 
and so on. Consequently 
n{n — \) ^ fx 
273 ~ 
” ^ n{n—l){n — 2) 
+ 2 . 3.4 
B 
.(f)’" + &C.} (38.) 
the series on the right-hand side being continued to n -\- 1 terms. This is in fact no 
other than Euler’s expression for the sum of the series l”-{-2"-{-3”-l- &c. to a given 
CC X 
number of terms represented by only that in the case here under consideration - 
may be any fraction, while Euler’s demonstration of the series in question is essen- 
tially confined to a positive integer. 
(24.) If we make x= — 1 , the foregoing expression becomes 
1-(H-A)1 
1 / 1 \ 
«-)- 1 V s / 
1 /I 
) +0(7 — qB,.(7 
n{n — 1) 
2.3 
.B 2 
-f- &c. 
. (39.) 
continued to n-\-\ terms, inclusive of the vanishing ones having Bj, B 4 , &c. for co- 
efficients. 
