( 311 ) 
It is therefore evident that we arrive from points m to points z 
by diminishing or enlarging the chords gm in a definite ratio. 
So the orbit of z is a circle touching the auxiliary circle (m) in g, 
whose tangent in g forms in that way the angle r with the line ga. 
If the quantities «, ft y, and rf are real, then a and g are real, 
whilst v = 0, therefore also r = 0. The points a and g therefore 
lie on the real axis and the orbit of 2 touches the real axis in the 
point g. If on the other hand ft = 0, then the centre of the orbit 
lies on the line ga. 
The way in which z changes with n we can read from the 
equations (10). 
If we suppose the point 2 to be furnished by the circle, which passes 
through q and 2 and whose centre lies on ga, then the first equation 
(10) tells us that the reciprocal value of the radius of that circle 
increases arithmetically uniformly that i. o. w. the radii of the circles 
through g whose centres lie on ga and which pass through z t , z 2 etc., 
form an harmonic series. If on the other hand we suppose that the 
point z is constructed as point of intersection of its orbit with the 
cirele through z touching the line ga in g, it then follows easily out 
of the second equation that also the reciprocal value of the radius 
of this circle increases arithmetically uniformly, that i.o.w. the radii 
of the circles touching ga in g and passing through the points 
etc. form an harmonic series. 
It is clear that for the case «, ft y, and 6 real, thus » = 0 and 
r — 0, only the first determination of the course of z can serve. 
