/; 



( 129 ) 

 dr 



sinh^^~^ r 



z= 10 u {')•), 



and it has in the origin an isolated divergency of kn (if k^ /'"~' ex- 

 presses the spherical surface of the Euclidean space Spn)- 



IV. For two scalar distributions <p and ip the theorem of Gheen 

 holds (conip. Opitz., I.e.): 



I <p— . dOn-l — I <ƒ W' • (^'^n = I tp— . dOn—l — jtp V^» • dt» 



(=ƒ 



S{\7 (f, V tf') . dXn 



If at infinity <p and tp both become whilst at the same time 



Urn (/)if>e("-i)'- = 0, 



then for an "—'sphere with infinite radius the surface integrals dis- 

 appear and we have left 



i<p . V'tp . dtu = 1 1 



integrated over the whole space. 



If here we take an arbitrary potential function for <p and iVn (r) 

 for if?, where r represents the distance to an arbitrarily chosen 

 point P — these functions satisfying together the conditions of the 

 formula — we have : 



^ri <Pp = I W'„ (r) . ^7" (fi . dtn. 



If thus we postulate for the vector distributions under consideration 

 the field property (which remains defined just as for Sp^) we have, 



if we put iOn{ir)^F^{r), for an arbitrary o^: 



IX^ST/ j)lL^F,{r)dr: (/) 



from which we deduce (compare A § VI) that there is no vector 

 field which has in finite nowhere rotation nor divergency ; so that 

 a vector field is uniformly determined by its rotation and its divergency. 



V. So a vectorfield is an arbitrary integral of: 



1. Fields E^, of which the second derivative consists of two 

 equal and opposite scalar values close to each other. 



2. Fields E^, of which the first derivative consists of planivectors 

 distributed regularly in the points of a small "--sphere and perpen- 

 dicular to that "—^spijgpe. 



9 

 Proceedings Royal Acad. Amsterdam. Vol. IX. 



