( 292 ) 
/ 
Fig. 6. 
I aa " When the first possibility of this figure appears, it follows 
from the correspondence of the arcs of curve ab and dc for the 
transformation, that on the arc of curve be must lie somewhere a 
juncture invariant for the transformation. 
When the second possibility appears, we notice that in be 
the arcs of curve dc and bk certainly cohere. For otherwise ab and 
cl would cohere there neither, and from both would follow, that in 
be between g r and & a segment s of the circumference of T r would 
be free of both $' r and the impossibility of which we have proved. 
Let us furthermore in be call L dc , L lc , and L ab the ends of the 
arcs of curve dc, le, and ab, then, habq having for the transformation 
as its image qdcl, and L ic jrnd L ab cohering with each other, the 
situation of L dc , L\ c and L ab with respect to each other is quite the 
same, as that of r r , g r and C /; the former thus allow of quite the 
same investigation as the latter, in which they are contained. If by 
this investigation we arrive again at the case la& we can investigate 
sti further partial sets which are in the same case, and can go on 
m this way. After a denumerable number of repetitions of this process 
we must then either have reduced this case to an other, or have 
tounct a point invariant for the transformation. 
V>. x r and & partly cover each other, or at least cohere on r;then 
the same holds for r } . and g r . 
Then, as r,. is the image of 5/, and $ r the image of tv, we can, 
