the calculus of functions, 409 
From these n 4- 1 equations we may eliminate the n arbi- 
trary constants, and the resulting equation will be of the form 
0 — F | x, \kr, a x, . : ip v x j ( A ) 
In arriving at this equation, we have eliminated n arbitrary 
constants, and therefore it might possibly be inferred that the 
general solution of (A) is 
4 x = F j x, a, a, . . . a j. 
12 X 
But this is too hasty a conclusion , for it is evident, that we should 
equally have arrived at equation (A), if each of the constant 
quantities in (0) had been changed into a function of x so 
constituted that it should not alter by the substitution of ax, 
&X, &C. vX. 
It would now appear, that putting such values for the con- 
stant quantities, the result would be the general solution of (A). 
This reasoning is certainly plausible, and such a solution is 
undoubtedly a very general one ; still, however, there are 
reasons which incline me to believe, that other solutions exist 
of a yet more general nature. 
On functional equations of the second and higher orders . 
When we consider functional equations of an order supe- 
rior to the first, new difficulties present themselves ; the arti- 
fices which were used with success in the preceding part of 
this paper, are no longer of any avail. 
Those which we have now to consider seem to possess an 
entirely distinct character. 
3 G 
MDCCCXV. 
