( 509 ) 
All 00 s initial points 2 0 furnish thus together all circles of the 
pencil of which g' and g" are the base points. 
Let us suppose point z determined on its orbit as point of inter¬ 
section of this orbit with an element of the conjugatedjpencil of 
circles, intersecting the real axis a. o. in a point s, then evidently 
g< z .g" 2 =sl g S :g"s holds, so that the equation (5) expresses that the 
quotient g's : g"s increases geometrically uniformly. (This property 
enables us to construct easily the points z belonging to given values 
of it). Farthermore holds g' = z—«, and g" = 2+ ® • 
C (a- d) 2 + 43y<0. 
The points g and g" lie symmetrically with respect to the real 
axis, p and q being conjugate complex. As mod. d = l we have 
— o. The ratio q’ ■ q" is now constant, so that the point describes 
a circle of Apollonius of A g ' g" z„, i.e. a circle of the pencil with 
d and g" as point circles. We can again regard the point 2 as if 
originated by intersection of the orbit with a circle of the conjugated 
pencil of circles. As the angle g'zg" increases uniformly with n we 
can easily construct with the aid of the conjugated pencil of circles 
the points 2 belonging to definite values of n. It is clear that the 
orbit of 2 when n increases indefinitely is described innumerable 
times, so that the function <p„(x) has as a function of n a real period. 
If v is commensurable with a, then this period is a mensurable 
number. 
If particularly « + d=0 holds, then * = *. This case is a. o. 
realized in the function y — -; here g =i, g — 
V. u = - , where (« — d)* 4^y = 0. 
Y* + d 
Eere we are in the parabolic case. 
so that 
