3 s 
Mr. Herschel on the 
and in general, if 
= + .2.'+ &c. 
we shall have 
B 
| lQ g- ( )\ n 
A 
}V; (8). 
The coefficient of A* in the function j A) j n f deve- 
loped in powers of A, is evidently 
d x + n. ( log. t) n 
1.2 ( x + n ). dt x + n 
t being made = l after the differentiations Now we easily 
find, that the expression 
^i 0 g ,t) n 
77*- h* ’ 
After executing the operations indicated must take the form 
A -j - 1 A .log. £-f n *A . (log. t) 
XX X 
t x + n 
n— i 
and the equations which determine A , &c. are 
X 
‘ A *+i — — (*+») - n ~'K 
n— i 
n — 2 
A,, , = -(*+«) ,"-*A x +{n~i ).»-> a 
X 
& x+1 = ~ (*+n). A^-f i, 
The integration of these equations is attended with no diffi- 
culty, and gives for the value of A^ (the only one wanted) 
as follow#: 
( l )* i .2 (x -\-n — l). 2 — 1 2 — l — 2 2 1 
x-t-n x+n X -\- n 
where there are (n—i ) signs of integration ; a constant being 
included under each. If now we suppose 
f + T+ It}’ 
i^ + r 3 + r 3 + = 8 <s {t} 
