calculus of functions and other branches of analysis. 213 
tions, varying of course in a certain degree from the difference 
of the objects to be obtained : the theory which lam now to 
explain will show that this suspicion is not without founda- 
tion, and will at the same time unfold one of the most beau- 
tiful parallels between the integral and functional calculus 
which I have yet observed. It has been already shown, that 
when an equation is symmetrical relative to the different 
forms of the unknown function, as 
F | X, tyx, -tyoiOC, . . . j = 0 
the method of solution by elimination apparently becomes 
illusory, but that the general solution of the equation may be 
deduced from the vanishing fraction to which this method 
then leads. If therefore we can make any functional equa- 
tion symmetrical, relative to the different forms of the 
unknown functions, we can then obtain its general solu- 
tion. Now this may be effected by multiplying the equation 
by some factor, and determining this factor, so that the result 
shall be symmetrical. The discovery of this factor (as in 
the case of a differential equation) requires the solution of a 
functional equation of several variables, but fortunately the 
class of equations to which they belong are not of very great 
difficulty. 
To begin with a very simple case, let us consider the equa- 
tion -tyx -\-fx . §ax —fx where ofx = x 
z 
multiplying by q>x we have 
<px . ^<2? -f- (poe .fx . -fyxx = $x .fx 
l 
now the first side of this equation will be symmetrical rela- 
tive to -tyx and \pax, if we make 
q>x =fax . (potx 
