Mr, Herschel on various points of Analysis, 453 
Ti"^. 6 ^‘_ (a-* 
(_i)^+‘.J 
I _ 5 Q^_ i±I 
“j“ &c. 
And after the substitution of o for or 1 for and its powers 
we obtain, 
(— 1 )*+» 
a = 
X 
1.2 
[■'-{ 
l-{* 
X(i+ v/— 
X A-+I -*^1^ 
x-\-i 
I 
X AT+ 1 X , (jr+ 1 ) . a: x 
- -I- . 2 -j- V -T -i — . 1 
1.2 
} + Ac.J 
h . . . . (16) 
The equation (12) will thus take the following form, 
s[(_,)-+‘. = Y, 9 + + . . . r- ; ( 17 ) 
where 
22a: -i) . (2 
2x vn2x tc/^n — zx — 1 ^ ^zn-—zx 
Y = - - 
2X — I 1.2 ... . t2.r) X 1-2 
, , . B .B ; 
( 2M 2X) 2X 1 2«-— 22 : — 1 ’ 
'2. Retaining the same form of the function /, and of course 
the same value of a , for n write 9 . 71 , and for Q .V HT, and 
the first member of (12) becomes 
And the second, 
. "L{2) - a^. “~'L(2) . e"+ . . . + (-1)''. °L(3) . 
Now since a == — collecting into one the coefficients of 
’ O V — 1 ^ ^ 
^”L( 2), viz.: V—i — 2^7^, we find their sum equal to 1, and of 
course, 
3N 
MDCCCXIV. 
