calculus of functions and other branches of analysis. 209 
It may be observed that the functional equation under con- 
sideration comprehends a large class of others which may 
easily be reduced to it, such as 
Fxpjr -f- A -j- B F\J/a 2 x -f- &c. -f* N F \pa n — l x -j- X = 0 
or more generally 
F ( X, yf/X ) 4* A F (oiX,ypaiX) -f- B F(ce s x, /a*x) =j~ &c. 
-f- NF(« k ”^, •^a n -‘ I x) -j- X=o 
both of which may be reduced to the same equations wanting 
the last term, these transformations apply with advantage to 
a great variety of other equations : the solution of the latter 
of these equations may be reduced to that of 
4«P + A-tyc&X -f- See. -j- l x = 0 
* * » * * 
and if we find ^x = Kx we shall have 
1 
$x = Fo-i( x , Kx) 
for the solution of the given equation. 
As an instance of the utility of the former equation we may 
take } + {'!' (-7 — #) } = 1 
whose solution will be found by that method to be 
tx=y/ v+ (2^ — v)%( x ’T— 
the same equation or the more general one 
{ | n -f [ $ux } 1 
where ux = x may be solved by another process which re- 
quires the aid of vanishing fractions, and we shall have 
