•L 
(— + 
Jf'* 
45^ Mr. Herschel on various points of Analysis. 
I -f jf) -f (— 1 )". "L (I + _ ^ 1 .( 2 ) . (log. xY ^ (2) . (log. x) 
1.2 .... n 
n — 2 
See, 
1.2 .. . (?i 2 } 
continued as far as it will go without involving negative 
powers of log, x. Supposing then F (^) = ”L ( 1 + / ), and 
writing a® for h we obtain 
lx"' 2 3 
D"-^ + &c.) (/ log- : log ^ ; ( n ) 
A very remarkable case of this equation is when ^ = 1 , or 
log t =z 0, for then ^ = the coefficient of V in 
1.2 
the developement of / log"~* [t) or of /(e ). If then we 
suppose 
f ^ 6^) = a^ “h <^2 • • • • -j- &c. 
we shall have the following equation : 
1 —“L (2) -aj” + (2) . + &c.; (i 
The second member being continued so long as it does not 
involve negative powers of 9 . 
With regard to the functions T(2), 'L(2), &c. we have, as 
is well known 
*L(2)==i — 1+1 — 1 + &c. = ~ , and 
(2' 
“L( 2 ) 
,2J:— I 2X 
— I) 7 T . ]3 
1.2 
{2X) 
ZX 1 
^2.r— 1 number of Bernouilli. 
The equation (12) by assigning specific forms f ( t) 
affords an indefinite variety of interesting results, of which we 
shall only notice a few, the most remarkable. 
1 . Let f(t)== . and for n write 2u — 1 , and 9 \^ — 1 for 9 * 
^ ‘ V— I + t 
