this the orbit of z found by Klein differs a little from ours. Although 
after stating and annulling this difference we might suffice with a 
reference to the results of Klein, we will dwell a little longer on 
the function y — , the more so as, differing from Klein, who 
treats first simple cases and then applies the principle of transformation 
of the circle correspondence, we shall immediately investigate the 
most general case. 
Examples: 
I. y — x-\-p, y n z=x.-{- np or z = z 0 np. 
The point z describes the right line connecting the points z = z a and 
z = “h in such a way that the distance from 2 to 2 0 is pro¬ 
portional to n. 
II. y = ax, y n — a n x or z = a n z 0 . 
Let us put 2 — 2 0 = a = oe* , then 
Qe id =14J”e* nT Q 0 ei\ 
from which ensues 
9 = 8=z8 0 + nr . . . . , . ( 1 ) 
q =5= a ' — cefi . 
Point z describes a logarithmic spiral round the origin. The polar 
angle 0 increases uniformly with n, i.o.w. the polar angle 8 increases 
arithmetically uniformly ; it is clear that the radius vector q increases 
geometrically uniformly. 
If a is real, then r = 0. The second equation (1) tells us that the 
polar angle remains constant, so that point 2 moves along the line 
connecting 0 and 2 9 and that with a geometrically uniform increase 
of p. 
If moda — \, then <y= 1. The first equation (1) then indicates, 
that the radius vector remains constant, so that point 2 describes the 
circle round O as centre, passing through point z 9 . The polar angle 
6 increases arithmetically uniformly. 
If r is commensurable with it, i. e. if a is a root out of unity, 
then y = ax leads back to x after a whole number of iterations. 
(* + fi) 
in. ,=« + A /(.)= L 
- = g, therefore f(z) = JZijl — 
— 1 toga 
