( 857 ) 
being nowhere zero or infinite and without singular points cannot 
exist). 
Between the above theorem 2 and the property deduced in a 
former communication *) that every continuous one-one transformation 
with invariant indicatrix of a sphere in itself shows at least one 
invariant point is a close connection. At first sight one might even 
suppose that they can be directly deduced out of each other. However 
this is not the case; on the contrary: they complete each other. 
Let us namely suppose on one hand the theorem about the vector 
distribution to be proved. If then is given a continuous one-one 
transformation of the sphere in itself, we can join each point 
P with its image P’ by an are of principal circle PP’, and consider 
that arc of circle in size and direction as a vector in P. But 
the univalence and the continuity of such a vector distribution is 
now assured only, if for no point P the image lies in the anti- 
podie point; and as this may not be assumed for an arbitrary 
continuous one-one transformation, a direct appearance of the 
theorem of the invariant point is excluded. 
Let us on the other hand regard as proved the theorem of the 
invariant point, and let a continuous vector distribution be constructed 
on the sphere. If we then make the points of the sphere undergo 
infinitesimal displacements proportionate to the vectors, and if we may 
suppose these displacements to generate at the limit a one-one trans- 
formation (of itself continuous and leaving the indicatrix invariant), 
we can conclude from it that the vector distribution must of neces- 
sity be somewhere zero or infinite. We are, however, sure of the 
one-one correspondence of that transformation only if by inde- 
finite decrease of the vectors we can make the vector variation 
anywhere smaller than the corresponding point variation, thus 
if the infinitesimal difference quotients of the given vector 
distribution do not exceed a certain maximum. And as in general 
this condition is not satisfied, the theorem of the vector distribution in 
its general form does not appear directly as a consequence of the 
theorem of the invariant point. 
A continuous vector distribution on the elliptic plane through a 
one-two correspondence determining a continuous vector distribution 
on the sphere, the two following theorems also hold: 
1) For monogenous complex functions this is a well-known theorem; for, a 
constant has in the point of the Neumann sphere representing the infinite a 
singular point. 
2) These Proceedings, page 797 of this volume. 
58 
Proceedings Royal Acad. Amsterdam. Vol. X{. 
