( 854 ) 
We shall suppose that the vector becomes nowhere zero or infinite; 
in any point the direction is then univalently determined, and that 
direction varies continuously from point to point. By placing the 
sphere into a Kuclidean space and by projecting there an arbitrary 
spherical shell upon its base plane, we then deduce from the theorem 
of Prano that in an arbitrary point of the sphere we can at 
least start one single curve, which is tangent curve to the vector 
distribution. Let 7 be such a tangent curve; we shall then say, that 
we pursue r if we describe it in the direction of the vector and that 
we “recur” it, if we describe it in the direction opposite to the vector. 
We now introduce a spherical distance 8 with the property, that 
within an arbitrary circle described on the sphere with radius @ the 
angle‘) of any two vectors is < */, 7. *). 
Let us now start the curve 7 ia A, and let us pursue it up to a 
point P in such a way, that all points of the described are A,P 
have a distance < 8 from A,, then the radius vector drawn from A, 
to an arbitrary point of the arc A,P will enclose with the direction 
of the vector in A, an angle <'*/,. For, if one of the two ares 
of principal circles, which in A, make an angle */, zr with the vector, 
were transgressed by 7 between A, and P, then according to the 
supposition the vector is directed in that point of intersection to the 
inner side of the angle formed by those two circular ares; so if we 
pursue 7 from A, to P, it can enter the just-mentioned angle, but 
it cannot leave it; then however it must always remain inside that 
angle. 
It is likewise evident, that, if 7 is an arbitrary point on the are 
A,P, the radius vector drawn from 7’ to an arbitrary point of the 
are 7’P encloses with the vector direction in 7’ an angle < '/, x. 
Let Q now be an arbitrary point of 7 between A, and P, we 
then know that the vector direction has in Q a component in the 
direction of the radius vector A,Q; thus, if Q moves along 7 from 
1) For the definition of the angle between two vectors not starting from the 
same point in an arbitrary non-Euclidean space, comp. these Proceedings Vol. 
IX, 1906, page 121, 122. To determine that angle here on the sphere we transfer 
the two vectors to a point of the principal circle, which joins them, maintaining 
their angle with that circle. 
2) That such a spherical distance 6 can always be indicated, is evident as follows: 
If a point D approaches indefinitely to a point C, in which the vector is not 
zero, then on account of the continuity of the vector distribution also the angle 
between the vectors in D and C converges to zero, and farthermore that con- 
vergence takes place for different points of convergence C uniformly, the vector 
distribution being uniformly continuous on account of the sphere being a closed 
set of points, 
