(D) in order 
in that way ne 
( 217 ) 
How it is impossible to bring the term with into the form 
e 5 1 st ./(»)./(?)-••!— 0 — ••■•)! 1 
in that way we can neither break up (22) into a differential equation 
for y = <p(x) and one for /(a?). ' , , ' ,, 
The way chosen hy us leads therefore no further than to the 
standard form (D) with three arbitrary constants. 
An attempt to start from the equations ( C ) o 
treat them in the same manner as {A) and to find 
standard forms, offers no more chance of success. 
If namely we differentiate 
g(y) = + £ 
twice, then after elimination of the constants we find 
= —+ ^ = . (23) 
g'iy) y' ?'(*) x 
In this equation we must endeavour to separate by means of 
differentiation the expressions x'./. rf etc. from the forms g , 'J e c. 
We first find 
g'%) ,, „, m y- + C - £ = 
g'iy) y g’iy) 9 g'iS) I 9 9 
_ *'< - *» + - - V- 
= J(«) A*) *' * 
As now however y' as well as f (resp. x' as well as x") appear 
as factors of forms' with g\y), g\vl- ( ie9 P- 9^9 <*)-•> and we 
have but one equation namely (23) at our disposal, we cannot separate 
the two expressions from each other, so that in this direction also 
further investigation will prove in vain. 
We can therefore do nothing else but repeat the above final con¬ 
clusion: with the method followed here we get no further than the 
standard form ( D ). , 
From all standard forms it is evident, that when y = w) can be 
built up by infinitesimal iteration, the same holds for y — h^\\^\h{x)\\ 
where h{x) represents an arbitrary function of x. For h { \g~i (x)\ 
is the inverse function of gWft This result has even been found 
already by Schroeder (l.c. p. 301). 
$ 5. Here may follow some examples. 
Applications of { C ) and (c) : ^ 
Example IV. <*«) = * ; y = »(«) = «* + Pi »>»<*) = «"* + ^ 9- 
