calculus of functions and other branches of analysis . 215 
from which the form of <p must be determined, one value is 
<p ( X, xX, tyx, i\>ax) = F| ax, X , ipoiX, tyx j 
hence the equation 
F j X, OCX , ^x, tyxx | x F | ax, x, \pctX, ypx ] = 0 
is symmetrical, and its general solution may therefore be 
found. It must however be observed, that all the solutions of 
this latter equation are not necessarily solutions of the former 
one, and it may be a matter of some difficulty to ascertain 
what solutions ought to be excluded. There is no part of 
this process which limits it to the particular case we have 
considered, and if the given equation were 
F j x, •i/X, ipaX, . . *x J = 0 
we might take the equation 
<p | F | a:, fa, -\xx, . . \pf~ * l x j , F j xx, ipxx, . . ipx j , . . 
• • • • • F | & 1 x, -tyf 1 1 x , . . . -tyx n z x 1 1 = 0 
which is symmetrical relative to 4/x, $xx, &c. i&x n ~ l x: 
I shall not at present enter into any farther details respect- 
ing this mode of solving functional equations, as it forms the 
subject of some investigations on which I am now engaged, 
and which are as yet incomplete. 
In the preceding pages I have endeavoured to point out 
some of the more prominent points in which the calculus of 
functions resembles common algebra or the integral calculus: 
it ought however to be observed, that several of the methods 
which I have applied to the solution of functional equations, 
directly resulted from pursuing this analogy : for when I had 
ascertained the remarkable similitude which exists between 
Ff 
MDCCCXVII. 
