42 
ON LAGRANGE'S THEOREM. 
[8 
In the case of several variables, if 
x = u+hf{x, x 1 ...), x 1 = u 1 + hjfi (x, x 1 ...), &c (17), 
writing for shortness 
F, f, f.I... for F(u, %...), f(u, Mj...), /i(m, ... 
then the formula is 
d 
F (x, a*! 
...) 
(dyi(d\, 
{l-hf (a?)} {1 - 
K fi f) • • 
.} \duj \duj 
{where fix) is written to denote ^ fix, x\ ...), &c.} 
or the coefficient of h n h™' in the expansion of 
F{x, x! ) 
[1 - hf (j?)j {1 - hfY (a^)] 
(19) 
is 
1 / d \ n (df f 
1.2.../1.1.2...«! du \d^ 
Ff n f n - 
20). 
From the formula (18), a formula may be deduced for the expansion of Fix, ), 
in the same way as (13) was deduced from (14), but the result is not expressible in 
a simple form by this method. An apparently simple form has indeed been given for 
this expansion by Laplace, Mécanique Celeste, [Ed. 1, 1798] tom. I. p. 176 ; but the 
expression there given for the general term, requires first that certain differentiations should 
be performed, and then that certain changes should be made in the result, quantities 
z, z , are to be changed into z n , zY h ; in other words, the general term is 
not really expressed by known symbols of operation only. The formula (18) is probably 
known, but I have not met with it anywhere.