developement of exponential functions, &c. 33 
and so on, we shall have no difficulty in convincing ourselves 
that ’ . 
A/=(— O* 12 (x + S {~r,} 
All the constants vanishing but that added at the first inte- 
gration which is equal to 1.2. .. . n. When / = 1, the expres- 
sion for - reduces itself to A , and therefore the co- 
rf* x + * 
efficient of A* will become 
( — 1 ) x . I . : 2 .. .* . n "’ l Sf — -d — 1 
We are thus conducted to the following value of 
% = {^ 1 . *~'s {± } - ^ • -'s{^} 
+ ..... ...^.”-'5{ 77 ^ ri } (9) 
The cases where n = 1 and 11 = 2 are the only ones of suffi- 
cient importance to merit a more particular consideration. In 
the former, we have already in our equation (7) given the 
expression for X B or . Its alternate, even values (the 
signs alone excepted) are those numbers so well known in 
analysis by the name of the “ Numbers of Bernouilli/’ and 
among the variety of expressions they admit, I know of none 
so compendious, or so readily computed arithmetically. In- 
deed, to compute the higher numbers of Bernouilli directly 
has always been attended with some labour. If we examine 
the values of B . B , B , &c., we shall observe that all the 
0 12 
odd ones (with the exception of B^ = ~) vanish : as indeed 
may easily be shown a priori from the nature of the function 
t 
*>■ 1 
s’— 1 
A considerable simplification of the latter case takes place 
owing to this circumstance : the alternate values of 2 B^ being 
MDCCCXVI, F 
