( 853 ) 
a pare from G; so G and F are separated on r by at least three 
points A. 
Let FFE be a g-arc, then on FZ lies a point H not coinciding with 
A 
E in such a way, that no other point of that arc has a smaller distance 
from G. The are of principal circle GH is then in H perpendicular 
to 7, and as tbe vector directions in G and H form with each other 
an angle < */, 2, they are directed to the same side of that are of 
circle. The are of circle and the are GH of 7 have farthermore 
only their endpoints in common, and they form together a closed 
single curve k, of which the length is smaller than (p+3)y7V2, and 
which divides the sphere into two domains. 
If we pursue 7 from G, it first runs to H along the boundary of 
those two domains, and then at H enters one of those domains g,, 
and will leave it no more, for it no more meets, its own are GH 
according to the supposition, and if it were to meet the are of 
principal circle GH, that would be at a distance < 8 from H, 
thus with a pursuing tangent direction, which would lead it into 
g, but could not make it leave that domain; so this meeting will 
never be able to take place either *). 
And analogously, if we recur r from H, it first runs to G along 
1) On the same grounds it is clear that for 7, independent of the choice of Ap, 
G and H, certainly a minimum distance 3 can be pointed out, under which, 
after the (-are beginning in H, 7 will never be able to approach, between the 
two fl arcs beginning in G and in H, the arc of principal circle GH. 
