( 74 ) 



so it is the general aS„ integral of the v^' of an isolated P-> vector, 

 consisting of equal /'vectors normal to the surface of an "-/'sphere with 

 infinitesimal radius described round the point of the given isolated 

 vector in the R,i-p+u normal to that vector. 



From this follows: 



Theorem 8. The general p V is an arbitrary integral of elemen- 

 tary fields E, and E^, where : 

 p-i 



ƒ» Zdv P-^ 

 , where » Z consists of the y"* vectors in the 



surface of an infinitesinal P- ' sphere aS/;., (1) 



p+i 



JYdv P'^^ 

 , where ^ Y consists of the p+^ vectors normal 



to the surface of an infmiiesinal "-P-^sphere Sp^^. ... (2) 



For the rest the fields E^ and E^ must be of a perfectly identical 

 structure at finite distance from their origin; for two fields ^i and ^, 

 with the same origin must be able to be summed up to an isolated 

 ^vector in that point. 



We can call the spheres >§/>>// and Spz vvitli their indicatrices the 

 elementary vortex si/stems Voy and Fo-. A field is then uniformly 

 determined by its elementary vortex systems and can be regarded 

 as caused by those vortex systems. 



We shall now apply the theory to some examples. 



The force field m S^. 

 The field E^. The elementary sphere Sp. becomes here two points 

 lying quite close to each other, the vortex system Vo^, passes into 

 two equal and opposite scalar values placed in tiiose two points. It 



cos If) 



furnishes a scalar potential — - in which <ƒ denotes the angle of the 



radiusvector with the *S\ of Yo~, i. c. the line connecting the two 

 points. The elementary field is the (first) derivative of the potential 

 (the gradient); it is the field of an agens double point in two di- 

 mensions. 



The field E^. The elementary sphere Sp,, again consists of two 

 points lying in close vicinity, the elementary vortex system Vo,, has 

 in those two points two equal and opposite planivectors. The plani- 

 vector potential (determined by a scalar value) here again becomes 



COS iD 



; so the field itself is obtained by allowing all the vectors of 



