( 858 ) 
Trrorem 4. A vector direction varying continuously on a singly 
connected, onesided, closed surface must be indeterminate at least in 
one point. 
Trrorem 5. A continuous vector distribution anywhere univalent 
on a singly connected, onesided, closed surface must vanish or become 
infinite at least in one point. 
By the following elementary example theorem 4 is illustrated : 
If we wish to adjoin in the projective plane by linear relations 
between the respective coordinates to any point ? a straight line 
passing through that point, this_is only possible by taking for that 
line the line which joins P with a fixed point Q. Theorem + informs 
us that if we wish to select in any point P? one of the two half 
lines joining P and Q, this cannot be done continuously. 
We really see that if moves along a straight line and if at the 
same time the half line PQ varies continuously, after a circuit of P 
that half line has not remained the same. 
Finally we notice that theorem 5 has asa direct consequence the 
theorem of the invariant point for the elliptic plane. *) 
For, for a continuous one-one transformation of the elliptic plane 
in itself the two straight line segments, which join / and its image 
P’, determine, it is true, two oppositely directed vectors, but a 
selection out of them for one point determines a selection every where, 
varying continuously in the whele plane. This is immediately proved, 
if we let P move along a unilateral curve; /” describes then like- 
wise a unilateral curve, and the selected segment PP’, after having 
varied continuously during the circuit, has remained the same as 
before. 
As farthermore the vector distribution, constructed in this way, 
becomes nowhere infinite, it must vanish at least in one point; this 
point is invariant for the transformation. 
1) These Proceedings, page 798 of this vol. 
