the calculus of functions. igj 
any number of quantities is symmetrical with respect to them, 
therefore 
f (x> x, . . x ) -fi x + &c. + x = S (x) 
1^12. k ia « 1 
, Again the sum of their products two by two is also sym- 
metrical, therefore 
f(x,x,.. x') == XX + a?# + + &c. = S fxx) 
1 X * » 12- 23 13 \ia/ 
and similarly the sums of their products three by three, four 
by four, &c. are symmetrical. We may, therefore, find n dif- 
ferent particular solutions, and the general solution will be 
any arbitrary function of all these particular solutions, or 
\! ) (x. X, . , x) — p f S(*),S( XX ) , . . . S ( XX . . X ) r 
V I a U 1 1 1 * I 2, » J 
Instead of taking for particular solutions the sum of all the 
quantities, the sum of all the products by two’s, the sum of all 
the products by three’s, &c. &c. we might have employed the 
sum of all the quantities, the sum of their squares, the sum of 
their cubes, &c. but the solution thus deduced would not be 
essentially different from the former. 
On functional equations of the second and higher orders involving 
two or more variables. 
The notation to be employed in these enquiries has already 
been sufficiently explained, and the different species of second 
functions have been noticed. Preserving the same symbols, 
let it be required to solve the. following Problem. 
Problem VIII. 
Given the equation 
^ 1 (^y) = x 
This equation though apparently involving two variables 
D d 2 
