the calculus of functions. 227 
take as a particular case of the equation <p 2 > 2 > 1 (a?,y, z) = z 
q>Qjc,y, z)= —z, then the equation of the Problem becomes 
^ 2 >fx,y') = or y^ 2 ’ 1 (®>y) 
this is the equation solved in Prob. XXL therefore all its solu- 
tions are also solutions of this equation. This however is, 
comparatively speaking, but a very limited answer: every 
different solution of the equation q > 2 > 2 > 1 (x, y, z') = z furnishes 
a new solution of our Problem, containing one or more arbi- 
trary functions; each of these may very justly be called a 
general solution ; but to investigate the number and nature of 
the arbitrary constants which enter into the complete solution, 
is an enquiry of considerable difficulty. 
Problem XXIX. 
Given the equation 
2, 2 \ n 
+ ,, ‘*(* 0 ') = F(x) 
This signifies, that after taking the second function relative 
to x, and then the second relative to y ; the result is consi- 
dered merely as a function of x, and its n th function taken rela- 
tive to that variable: lastly, the quantity to which this becomes 
equal, after performing these operations, is given. The man- 
ner of treating these equations is very simple; put 
f 2 > 2 (x, x) — x (f)t then our equation becomes 
2, 2 \ n 
4/’ 2j (x,y') = x n O) = F (0?) [y = ax] 
determine % from the equation x n x~F (a?) by Prob. XIII. 
Part I. and let its solution be F (a?), then we have 
1 
■ty 2 > 2 (x,y ) = F (a?) £ y = ax'] 
I 
H h 
MBCCCXVI. 
