( 505 ) 
It is clear that the orthogonal projection of the twisted curve of 
x on the plane V x coincides with the orbit x ;/„■ 
We shall for the present occupy ourselves only with the study 
of such an orbit x —>y n - 
To find the orbit x ^ y n we have but to solve the functional 
equation of Abel. We have namely to find that function /(.r) of x 
increasing with a when for x is substituted y n =z<p„{x)-, this function 
increases for the process of iteration with real contributions, i.e. the 
quantity g = f(x) = U + %V describes in its complex plane the right 
line V= c parallel to the real axis. If once we know the form 
of the function 5 = f [x\ then we also know the orbit of the 
quantity x=f— i (5)- , . t TT 
The value of V and the initial value ( n = 0) of the real part L 
of g represent together two arbitrary constants, of which we do not 
dispose until we choose the initial value of x. 
We shall indicate the current point (y„) of the orbit x y n by 
whilst we shall point out x by z 0 ; we then have 
/(*)=/(*«) + " 
or 
U + iV= U a + iV a + n, 
so that 
U = U 0 + n , v= v„. 
The choice of the initial point z 0 now determines the values U„ 
and F„. 
shall not always follow 
unnecessarily lengthy in 
When working out some examples w 
the systematic way sketched above, as it 
simple cases. 
In reference to the broken linear function y = 
*±£ 
.vyi-d 
notice 
that this has been thoroughly investigated already by Poincare * l ) and 
Klein’), the latter having also included complex values of «, [1 y, and d m 
the study. Klein too allows the tunction y = *° ai * Se ^ ra< ^ u 
ally out of x and regards the orbit described thereby. For the non¬ 
parabolic cases he builds up the function by infinitesimal iteration 
in' the sense indicated by us. For the parabolic case, on the other 
hand, he takes as parameter of the function in its orbit not the 
iteration-index n, but a complex multiple of it. In consequence of 
!) Poincare. Acta Mathematica I (1882), p. 1. 
i) Klein_Fiucke. Vorl. ii. d. Theorie der elk Modulfunktionen 
(Teubner, 1890), 
34* 
