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construct as many functions *„(*) * one likes”, which in fact may 
be called solved in {a) and (6). 
\ 2 We shall now pass on to the investigation of a new standard 
form. We first imagine the .function y = 9 Of) •» be represented by 
the two parameterequations 
*W= g (<) 1 * = *!(*>. 
As the function /(*) is so to say the connecting link between the 
function <p and its iteral <p„ a relation, e.g. a differential equation, 
which is satisfied by all so also by 9. (*) = *• ma / wnt ®" 
as an equation, in which the function / is the independent variable, 
thus as a relation of the form 
(, d(p n d'(p n ^ — 0 
"7 
If here we exchange the independent variable with the dependent 
variable then we must be able to arrive at an equation 
l df d'f I 
*>■/(**> -i ’ ** .» 
retaining the same, so a constant value for all therefore also ' 
for (r.(.v) = ®; i.o.w. out of (!) and (A) a relation must be deduced 
of the form 
f j *,/(*),/' (*),/" (*),.•• I = -ff yJ(y\f ' W’{' to)- •••! = 
In .F no forms will be allowed to appear, in which a strange 
dy dx dy 
variable is present, therefore not ^ 5 ^ ’ dt 6 C 
By differentiating {A) according to t we find 
Q=f’{y)y' =■/'(*)*'• • • • * ■ ^ 
A first (trivial) solution we get by putting: = x > ory-—x-\-p f 
f (y)=f (x) = c, /(ar) = Gr + c', where {A) is then satisfied, when 
Differentiation of (3) produces 
{.'=/" ( S ) </’+/’(!/) / = P (*) *’* + /’ W 
or, on account of (3) 
1 /”<y) /'.‘W| . 
e l/”w /'*(«)! W *7 
