406 
Then will 
Mr. Babbage's essay towards 
4 X = (pff . . . f a (fi y . . v X 
satisfy the conditions. 
We now possess the means of solving a much more general 
problem than we have yet attempted, it is the general solu- 
tion of any functional equation of the first order. 
having given a particular one, containing one or more arbi- 
trary constants. 
Find from Problem VI. an arbitrary function, such that it 
shall remain the same, when ax , /3jt, &c. vx, are substituted 
for x, call this cpx, and let the particular solution be 
As these arbitrary constants are supposed not to be contained 
in the given equation F j x, 4 x, 4 a x, . . 4 v x j = o, when the 
particular solution is substituted, they must destroy each other, 
whatever be their value. 
If therefore, instead of a , 6, c, &c. we put q>x , (px , <px, & c. 
(which are arbitrary functions fulfillin g this condition, viz. 
that they remain identical when ax, fix, See. are put for x:) it 
is evident that these arbitrary functions must also destroy 
each other, therefore 
Problem VIII. 
Required the general solution of the equation 
2 
I 2 
is a general solution containing as many arbitrary functions 
as the particular solution did constant quantities 
