1Q2 Mr. Babbage’s essay tozvards 
some conclusions very analogous for functional equations of 
the form of those treated of in this Problem, but the length 
to which these enquiries would lead, render it sufficient merely 
to indicate them. 
In the equations solved in Problem I. and II. it is obviously 
immaterial whether we first put ax instead of x, and then in the 
result put (3y for y or conversely ; but in the equation of Pro- 
blems III. and IV. the case is different. If in the function^ (x, y) 
we put simultaneously a. (x,y) for x, and j G(x,y) for y the 
result will be different from that which would arise from 
first putting « (z,y) for x and then in the result putting 
(3 (x,y) fory, or from inverting this operation; the three results 
stand thus : 
^ (« (.r ,y), (3 ( X,y )) 
{“) 
■l/(«.(x,$(x,y)),&{x,y)) 
(b) 
4 i(a(x,y),m«,\x,y),y)) 
(0 
These three functions are evidently different, and in the solu- 
tions of the Problems, regard was only had to the first of 
them which may be called the simultaneous function. Those* 
however, of the second and third class might occur, and it 
becomes necessary to point out the means of solution which 
are applicable to them. 
According to the notation laid down, these functions may 
be thus expressed 
0(x,y)) (a) 
d {x,y)) (b) 
V’\u(x,y),p(x,y)') (c) 
But to avoid the trouble of indices I shall show how those of 
