( 510 ) 
z -y-jzr ff + ~7^-"~jzr g + ((»+*»>)*• ... ( 9 ) 
Tbe difference between our method and that of Klein arises from 
the fact that Klein allows the quantity . n to increase really. 
If we take a and g as auxiliary origins and if we put 
z —a = z' = v'e ®, z — g = z " = ( ,V r , 
then the equation (9) takes the form of 
+ (f + ir)n 
from which ensues 
«* = 4 «* (»'-#'.) + tm, 1 
;* ! . . . . (10) 
V - sin sin + m . | 
If we put n — amsx, v = adnx o. w. we find out 
of (10) when eliminating n. 
sin (8>-8"-x) = sin {8\-8\-x) ^ ... (I1 ) 
It is clear, that the orbit as found by Klein follows from ours 
putting x = 0. The orbit of Klein can thus serve as iteration- 
orbit for real values of the quantity thus of 
To investigate the curve determined by the equation (11) we 
imagine the circle passing through g and a and of which the arc 
ga amounts to 2r, so that from each point of the supplementary arc 
the line ga is seen under the angle r. (See fig. p. 511). 
If we connect g with z g and 2 , the connecting lines will meet 
the circle in m„ and m. 
Now ^ gvfia = £ gm 9 a ~ x 
Farthermore ^ zam — 8'~ 8” — r, £ z 0 am 0 = 8\ — 8\ — r. 
If we let fall the normals z 0 n 0 and zn on am* and am, then 
= q’o ■«« (^o — — r) and zn = p' sin {8’ — 8" — r). 
The equation (11) now demands 
