Mr. Babbage’s essay towards 
Assume q> (x 9 y) = q> (/(#,y)» f(x, y)), then equation (2) 
1 z " 3 v y 
becomes 
? { / (f( x ’y)J (*,y)) j (f{v,y) J '(ay)) | = <P | /(K (ay ),’K(a?,jy)), 
This equation must be solved in the same manner as the for- 
mer by means of two particular solutions, and by continuing 
the same method, we shall find that the form of the function 
may be determined by means of 2 11 particular solutions of 
certain functional equations, when there are n pair of conditions 
assigned. A less general solution may, however, be found 
when we are only acquainted with n particular solutions. 
A similar method would lead us to the form of ■$/, whatever 
might be the number of variables. If, however, we are ac- 
quainted with any number of particular solutions which remain 
the same, in all the cases assigned by the conditions of the 
Problem, we may have the general solution by making 
'p =<?{/./..•/} 
12 f 
f, f, . .f being it i particular solutions. 
12 • 
ir 
Ex. Let it be required to find a symmetrical function of 
x , x, . x, the equations to be satisfied are 
12 n 
4/ [x, X , . . X, ) = \J/ lx, X, . . X, xj = ypfx, X . . X, X, x) = &C. 
\I2 n 1 \ a 3 n I / '34 n X 2 / 
or the whole of the conditions may be more concisely denoted 
by the expression 
< x, X , X . . x | 
t I 2 3 n J 
We may easily find n particular solutions which fulfill these 
equations : for in the first place it is evident that the sum of 
