( 210 ) 
It is our task therefore to solve the functional equation (.4). This 
equation is called the functional equation of Abel * *). It is to be 
regarded as the logarithmic form of the equation 
==«*>), 
which bears the hame of the equation of Schroeder *). • 
Whilst Leau regards the solution of the equation of Abel (Schroeder; 
from the point of view of the theory of functions and Formenti and 
Spiess confine themselves to a few special cases, Schroeder deduces 
some groups of functions tp (a?) for which the function f{x) is easy 
to determine. It is now our intention to give to the results of 
Schroeder a more general form and to furnish in this way a con¬ 
tribution to the solution of the two functional equations mentioned. 
A first group of functions y — q>(x) for which ffn is easy to deter¬ 
mine is obtained by choosing for f{x) a definite function. 
Example I: 
f{x) = xv r yP s= xP -{7 1, y — if (,«) = \ /xp -j- 1, <p n (x) = \/xp -j- n. 
Example II; 
f(x)=sinx; siny= 8 in.v- f-1, y=q(x)-sin-'(sinx+l), <p„(x) = sin~' (*inx+n). 
It is as easy, however, to find the function f(x) and with it the 
expression for q n when in the equation 
g(y)= 9 (x) + P . (B) 
we choose for g(x) a definite function. For, in this case /(a*) 
p 
and 
<Pn{x) = g-x{g(x) + n$} . (*) 
Example III: 
gisc) — log x, log y == log x ft = log x -f log y, (y == eP) , 
y = if (x) — yx ; <p n ( a ) = y n x. 
If we were able to write every function y = q> (#) in the form 
(B), the problem of the solution of the functional equation of Abel 
would be very simple. It is clear that we stand a greater chance 
of bringing a function y == tp (x) in a certain standard form when 
this form contains more arbitrary constants. It is our aim therefore 
to increase the number of constants in the standard form. 
Though a given function resists a • transformation in the standard 
form found by us, no real difficulties adhere to the problem, “to 
b Oeuvres completes H, p. 36. 
*) See note 1 p. 208. 
