34 
Mr. Herschel on the 
susceptible of an expression by means of those of *B : In 
fact, the odd values of B^ vanishing (except B ), we have 
= + + B 2 + &c .)Y 
and, comparing the coefficients of t 2X+1 in the two members 
of this equation, we obtain 
“ “ ( 2X + 1 )- B 2 .r * 
Hence this remarkable theorem, 
( 10 ) 
which may also be regarded as affording another general ex- 
pression for the numbers of Bernouilli. 
Laplace has shown that the developement of the function 
may be derived from that of and that, if the coeffi- 
cient of t" in the developement of the latter be represented 
by a , it will be — in that of the former. Now, by the 
application of our equation (2), we find that 
<»>. 
Making then n=i, we find for the value of a x — 1 
_ j z -^- 2 A - 2^-3 a * + ....... ± A *- 1 | 0 
&X—1 
1 . 2 . ..... ( X 1 ). 2 X 
and consequently the coefficient of t* in will be 
| A -** -3 A a + ± A * -1 } o*~ l 
1.2 
. . . z*. (2*— x) 
Dr. Brinkley has arrived at the same result. 
(12) 
The computation of the functions n l S j —■ J > n j & c . 
is attended with very little difficulty ; for, if we multiply toge- 
ther successively the terms 1+ it, 2 + %, 3 4 " & c * an< ^ ca ^ 
