( 769 ) 
they become respectively the images p and yw, and we suppose that 
( possesses no point invariant for the transformation. 
We can then surround p by a polygon approximating p at a 
distance «') and belonging entirely to y, in such a way that also the 
image of 9 for the transformation lies entirely inside yp, that in 
each point on or inside $s the length of the transformation vector 
(Le. the vector joining the point with its image for the trans- 
formation) does not fall below a certain minimum 4, and that each 
Bn 1 
point of ® allows itself to be joined with p by a path’) Gob 
Ja 
Qn P we then choose the points P,, P,,.... P,, which have this 
order in the sense of a positive circuit, and possess the property 
that in each are P,P,41 (to these also belongs the are PP) the 
] 3 
distance of the endpoints lies between ee and a b, and the distance 
3 
of two arbitrary points does not exceed we *). Let us now draw 
| ee we 
from each point P, to p a path Er 6 lying inside 9, then 
the are of simple curve ZirPr Pr lèp4i cannot cut its image for the 
transformation Opi 101H1. 
The ares RyPpPrpihepi we shall call skeleton ares; the ares 
OLALALLOK+1 image skeleton ares. 
If we represent by & the total angular variation described by the 
transformation vector for a positive circuit of 9, and by ag the total 
angle described by a vector of which the origin runs along the 
skeleton are PP, and the endpoint as a continuous function of 
the origin along the image skeleton arc 7.7041, then we have 
TAP 
By + we shall represent the point of the image skeleton arc 
07 Which is, if this are does not cut 9), identical to aj, and in the 
opposite case to its last point of intersection with 9. If then 8, designs 
the total angular variation of a vector of which the origin runs 
along the skeleton are PP and the endpoint as a continuous 
function of the origin along the image skeleton arc ryrg41, we have 
likewise 
D= Be 
We now distinguish three cases: 
1. On the circumference of p the segments Ry Regi and exept 
1) Scuoenriigs, Bericht über die Mengenlehre II, p. 104. 
2) ie. “Weg” in the sense of SCHOENFLIES. 
3) Scuoenrures, Bericht über die Mengenlehre II, p. 183. 
