( 212 ) 
To arrive at an equation of the form (2) the second member of 
(4) must be reducible by means of an equation obtained by differen¬ 
tiation of (1) to a form W{y,y\y'\...) _ ypfa — W x , 
which in its turn must allow of being brought to the form 
9 [ 0 lyJiy^f (y)./" (y). •• 1 -- (.*),/" (*)..•}!=p(* y -#x)= 
= y'f (y) *y — x f (*) so that W y — W x — y'f (y) 4> y - x'f (a) 4> x . 
From this ensues that f (y) <P y may contain y, but not/(y),/(y),... 
or i. o. w. that f (y) *P y = tp (y) and then likewise f (x) 4> r = ip(j?) 
must hold; therefore <P y must be of the form ty{y)'f(y\ and 0 x of 
the form ip (x) :f (x). Hence the transformation of the second member 
of (4) looks as follows 
. - & -?)■ = * (y) - y - * ItS-tSI- 
If therefore the given function y = rp (a?), represented by (1), satisfies 
the differential equation of the second order 
p— p- + -y' —<)>(*).*' = o, . . ■ • (5) 
then f (y) (resp. /(*)) is determined out of 
/" (y) /" («) »l> (y) f (*) 
r (y) /’*(«) /’(y) /'(*)’ 
/" (») _ ( 6 ) 
/*<*) /» .. 
and we are able to build up the function (p by infinitesimal iteration. 
The equation (5) passes when we put t = x into 
J + <f(y)-y'-«!>(*) = °.. (7) 
from which ensues by integration 
y(y) = «$(*) + £• 
where g(x) is put equal to dx. 
With this we have obtained a new standard form, one 
arbitrary constants. By putting either a = 1, or /? = 0 we 
of this the two standard forms of Schroeder (1. c. p. 303). 
We have now but to bring a given function y.= *P ($) 
form (C) or by differentiation into one of the forms (7) or 
hjx) Hjy) 
. . y ““s’to) - "AW *B{») 
By integrating (6) we find in connection with ( A ), 
with two 
> get out 
into the 
. • (8) 
