the calculus of functions. 
217 
and generally 
V’ m (*>y) = §f n > m ( <P*» $y ) 
If there are more variables than two, the proper substitution 
to be made is 
4/ (a?, ec , . . .r ) = (p /*’ *' &c - ((pa?, <p.r, . . <]Xe) 
12 i i » » 
and there would result generally 
w, p, &c _ j n, ?n, &c. 
+ ( ci/ ^ cX j • • oi/ )— - Cp / ( (par, (par, . . . (par ) 
x 2. i x a s 
By such substitutions all simple functional equations of every 
order and of any number of variables, may be reduced to 
those of the first order : but the difficulty is not then over- 
come, the resulting equations are by no means easy to solve, 
and in a variety of cases it appears, that they are contradic- 
tory or impossible. 
Let us apply this substitution to the equation of this Pro- 
blem, then since (ar, y) = (p 1 / ((par, (py) we have 
?'/*’’ (<p*» = ('?•*> W) 
Performing the operation denoted by <p l on both sides it 
becomes / 2 > J (<par, $ y) =f 2 >'f(px, <py ) 
Put <p ar for ar and $ y for y then it becomes 
Z 3 * 1 (x,y) ==/“>*( j;,y) 
which is nothing more than the original equation ; from it, 
however we learn, that if we can find one particular solution, 
we can always deduce from it the general one, which sup- 
posing/ a particular case, will be 
tfay) = ffiv*, <py) 
After repeated endeavours I have been unable to find any 
particular case which will satisfy the equation 
V* y) = V* y) 
