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Mathematics. — “Investigation of the functions ivhick can be built 
up by means of infinitesimal iteration By Mr. M. J. van 
Uven. (Communicated by Prof. W. Rapteyn). 
(Communicated in the meeting of October BO, 1909). 
In the paper presented by me in the preceding meeting I have roused 
the impression by an ambiguous mode of expression as if the method 
to obtain standard forms followed by me were the most serviceable. 
This is in nowise the case. I have exclusively intended to give a 
summary of what can be reached in this direction by differentiation. 
The last observation lying at hand and moreover already made 
by Schroeder according to which out of each function y = <f (f) 
constructive by means of infinitesimal iteration a whole series of 
suchlike functions can be deduced all of the form y = h- X [y |A(#)j] 
may have been made too much incidentally to weaken that impression 
properly. For, this last principle furnishes, as is immediately evident, 
an inexhaustible wealth of standardforms. With the aid of this 
principle we can i. a. just as well deduce the standardform (D). 
We have but to show that 
y 
4- ft 
follows directly out of the formula of Abel or Schroeder. By applying 
the standardform of Schroeder: 
we find out of 
g(y)=mg(x) 
y~g~\ {*»#■(*)! 
yx + d' 
where m, a, b, c and d must satisfy 
ad be bd ac 
ma—d a—rnd (1 -\-m)P —(l-j-" 1 )? 
80 that m is a root of 
ro! _“l±^±M m + ,=o. 
nd—py 
We then find without difficulty the form ( d ) for tp n (x,. In the 
parabolic case, ad — py = 0 , this calculation does not hold; we have 
then but to call in the aid of the equation of Abel. Let us put 
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