25 s Mr. Babbage's essay towards 
we shall have 
d n -4/ (a 4 @y) 
(dux)* 
which may always be reduced to the form 
d n (a 4 x, jS y) 
/(*) 
eta" 
If now in the original equation we substitute successively 
oloc, cdx, . . ctP— l x for a?, and (3y, /3 2 y, &c. fory, we shall 
have pq equations containing pq forms of the unknown func- 
tion and their differentials. By means of these pq equations 
and the differentials of them, we may eliminate all the diffe- 
rent forms of the function if/, except one : let the one which 
remains be if/ (a?,y), then we have an equation of partial diffe- 
rentials containing only x, y, if/ (a?, y) and their differentials : 
and from the solution of this equation if/ (x,y) may be found; 
a certain number of arbitrary functions will be contained in 
this integral ; these must all be determined so as to satisfy the 
original equation. 
Amongst the numerous questions to which the calculus 
of functions is applicable, I shall select a problem proposed by 
Euler in one of the volumes of the Acta Acad. Petrop. as it 
will offer an example of a mode of treating of functional equa- 
tions of a nature yet more general than those contained in 
this paper. 
