4° 
Mr. Herschel on the 
But the coefficient of t 2 x 1 in tan. 
is 
(-i) x 
( z %x — l). 
I. 2 ...... ( 2.z) ^ 2X— I 
where B here denotes the x th in order of the numbers of 
ZX — I 
Bernouilli. Equating these two values, we find 
B =- 
( —I ) x . zx 
ZX-l 
it C®); 
We will now proceed to consider the developement of any 
function of the form 
u —f{t\ i i", &c.) 
t , t t" , See. being any number of independent variables. The 
coefficient of t x . t'K t" x . Sec. being denoted by A 
have 
j^+3'4- &c - M 
, we 
A 
Tj y> z > i. 2 ,rx i.. . .y X &c. x dA dl! y . Sec . 
Now, regarding w as a function of /, we have 
£i=/(i+A, &c.) o" 
Again, considering this as a function of we obtain 
_ f( l+A, 1+A', £ '", &c.) o*. cfl. 
dt*. dt 'y J v 
(the accents over the A, and o, indicating, as before, the 
application of the symbols) — and so on. Thus we find 
d x +y+z+Scc. u _ i-UA', Sec.) o". o'y. o"\ Sec. 
it*. di'y. dt"*. Sec. J K 
(®7-) 
di 
and of course, 
A /(* + A , 1 + Ah I + A", &c.) o*. p'r o Sec. 
x,y,z, Sec.~~~ i .... x X i .... y X I &c. 
Laplace has shown,* that, in any function u x x ^ &Cm of x, 
y, See. if x be made to vary by a,y by (3, See. simultaneously, 
the following equation, analogous to ( a ) will hold good : 
* Theorie Analytique des Probabilites, p. 70. 
