( 261 ) 



an isolated line vector. For elementary o^ ^^'e shall thus have to put the 

 field of a divergency double point. 



r dr 



The Schering elementary potential I — ^ t'„ (r) is here a plu- 



J sin " — 'r 



r 



rivalent function (comp. Klein, Vorlesungen über Nicht-Euklidische 

 Geometrie II, p. 208, 209) ; it must thus be regarded as senseless. 



II. The unilateral elliptic Sp^ is enclosed by a plane Spn—\, 

 regarded twice with opposite normal direction, as a bilateral singly 

 connected *S/9„-segment by a bilateral closed Spn—\- If we apply to 

 the Spn enclosed in this way the theorem of Green for a scalar 

 function (p having nowhere divergency, and for one having in two 

 arbitrary points P^ and P^ equal and opposite divergencies and 

 fartheron nowhere (such a function will prove to exist in the follo- 

 wing), we shall find : 



i. 0. w. ^ is a constant, the points P^ and P, being arbitrarily chosen. 



So no u^ is possible ha\ing nowhere divergency, so no X having 

 nowhere rotation and nowhere divergency; and from this ensues: 



A linevector distribution in an elliptical Spn is uniformly deter- 

 mined by its rotation and its divergency. 



III. So we consider : 



1. the field E^, with as second derivative two equal and opposite 

 scalar values quite close together. 



2 . the field E^ with as first derivative planivectors regularly distri- 

 buted in the points of a small "—-sphere and perpendicular to that 

 small "—^sphere. 



At finite distance from their origin the fields E^ and E^ are of 

 identical structure. 



IV. To find the potential of the field E^ we shall represent it 

 uni-bivalently'' on the spherical Spn; the representation will have as 

 divergency two doublepoints in opposite points, where equal poles 

 correspond "as^opposite points ; it will thus be the field (rf), deduced 

 under F § II, multiplied by 2 : 



