( 209 ) 
*(.r) passes after an iteration n- 1 times repeated into a given 
function »(«), so that «.„(*) = ?« is satisfied. With the answer to 
this question among others E. Schroeder'), Formbnti 1 2 3 4 ), Leau'), Spusss*) 
have occupied themselves. 
We too will take the problem of iteration from* this side and we 
will endeavour to find an expression for the (n—l) st iteration 9>»W 
of g>(x) which retains its importance for all real values of the 
iterationindex n. When we have found that expression we have but 
to put n = - to obtain the function if = <*> . , furnishing y after an 
iteration n — 1 times repeated so that 
(,r) = tpx m (*) = 9i (*) = V (*)• 
By increasing n with infinitesimal amounts we can as it were build 
up the function <p by infinitesimal iteration. 
It is necessary to just distinguish here between two kinds of 
infinitesimal iteration. If namely we consider (see Sfiess) *) the 
operation x + <fx, where then by iterating this operation 
(m-1) times we obtain Lira (T-M)”* = *• *• We on the contrary 
wish to occupy ourselveTwith the case in which not an operation 
but a function proper is iterated. 
It is evidently the point to express in the form of a conti¬ 
nuous function of x and n. However we can put the problem also 
tike this: which form must function fix) have to be increased with 
unity by the substitution y = p (*) instead of x, i. 0. w.: for which 
function / (*) holds for a given y = !P(®) 
/( y) =/{<f (*)! =/W+ l > . 
therefore also 
/(<*>«) =/(") + n f 
If the function f{x) is determined, then <p„ follows out of 
w + »i..< a) 
in which /_, represents the inverse function of /. We have then 
reached our aim: we have written *p n as function of x and n. 
!) Math. Ann. Bd. Ill (1871), p. 296. 
2 ) Rendiconti d. R. Istituto Lombardo (2) vol. Vill (1875), P* i7b - 
3 ) Ann. d. 1. Fac. des sc. de Toulouse t. XI (1897), E. 
4 ) Mitteil. der Nalurforsch. Gesellsch. in Bern, 1901, P-106. 
See also the article of Pincherle in the Encycl. der Math. Wiss. Bd. II 1, Heft 
6, p. 791. 
