(m y 
each other an angle <. g n, are directed to the same side of the 
geodetic arc P X H X . 
Let K x be the last point of intersection of the arc P X H X of r with 
the geodetic arc P X H X . Then the arc K X E X of r and the geodetic arc 
K X H X form a simple closed curve, and we prove in the manner 
indicated on page 853 of our first communication, that either the 
pursuing branch of r from H x lies in the inner domain, and the 
recurring branch from K x in the outer domain, or the pursuing 
branch from H x in the outer domain, and the recurring branch 
from K x in the inner domain. 
Let us first assume that the pursuing branch lies in the inner 
domain , then certainly two points P t and Q 2 can be chosen on it, 
both at a distance e from the points zero and separated on r by at 
least one point at a distance from the points zero, whilst the distance 
between P 2 and — With the aid of those two points we 
construct in the same way as above now a simple closed curve, 
consisting of an arc K t H % of r and a geodetic arc H* in whose 
inner domain lies the pursuing branch of r from H t . 
Going on in this way we construct a fundamental series of closed 
curves u Xi m„ - lying inside each other. If there is a domain 
or set of. domains G, common to all the inner domains of these 
curves (which, as we shall presently show, is really the case) then 
the boundary of G can only be formed by points belonging to none 
of the curves u x ,u it u tt ... but being limit points of fundamental 
series of points lying on those curves. 
We assume q>p, and B to be a point of u q having a distance 
> 3 8 p and > 3 ^ from the points zero. Let C "be the first point 
when recurring from B , and D the first point when pursuing from 
B, which reaches a distance from B, then we shall assume for 
a moment that there exists on u f/ , but not on the arc CD, a point 
S lying at a distance from B, and. we shall show that 
this assumption leads to an absurdity. 
- Let SV •* “ recurring ~ j^-arc and 5 W a pursuing j -arc ' 
on u„ then the arcs CD and VW can have no point in comlon. 
