the calculus of functions. 
22 9 
Substituting these values in the original equation, we have 
2, I, 
2 , 2 , 2 , 2 
2 , 2 
( I, 2, I, 2, X I, 2, I, 2, 2 
=° Ly = a Q 
This is an equation similar to that of Problem XXVIII., and 
may be solved by the same means. 
Mew methods of solving functional equations of the first order , and 
also differential functional equations. 
The new methods which I now propose to explain are only 
applicable to equations of the form 
F | W, if/ X, a X, ^ 0? sc, . . 4/ 0i n DC | =0 
where a must be such a function that a n + J x=x. By the me- 
thod of Prob. VII. Part I. all functional equations of the first 
order may be reduced to this form ; and although in many 
cases this reduction is very difficult, or even in the present 
state of analysis out of our power, yet it is theoretically pos- 
sible, and we shall therefore consider all equations as so 
reduced. There is this remarkable difference between the 
former methods and the present one : 
Those which I have already given always led to the general 
solution, and perhaps, in some cases, to the complete one ; 
these, on the contrary, which I shall now propose, always 
conduct us directly to a particular solution, which does not 
contain even an arbitrary constant . f It has, however, several 
advantages; it is the most direct method with which we are 
yet acquainted ; and if by any means we could introduce into 
these solutions an arbitrary constant, it would afford us gene- 
ral ones : this is a step which is wanting to connect it with 
the former methods. In the case of differential functional 
H ll2 
