( 77 ) 



paper but will be treated more in details in a following com- 

 munication. 



With the force field in S^ the vortex field in S^ dual to it has 

 been treated at the same lime. It is an integral of vortex fields as 

 thej run round the force lines of an elementary magnet and as 

 they run round the induction lines of an elementary circuit. 



The force field in Sn • 



The field E^. Voz again gives a double point, which furnishes a 



cos (p 

 scalar potential , where (p is the angle between radiusvector and 



^ ^,7i — 1 



axis of the double point; its gradient gi\es what we might call the 

 field of an elementary magnet in S^ 



lite field E^. Voy consists of equal plani vectors normal to a 

 small " — -sphere Spjj. To find the plani vector potential in a point 

 P, we call the perpendicular to the aS„ — i in which Sp,i is lying 

 OL, and the plane LOP the "meridian plane" of P ; w^e call 

 ip the angle LOP and OQ the perpendicular to OL drawn in 

 the meridian plane. We then see that all planivectors of Voy have 

 in common with that meridian plane the direction OL, so they can be 

 decomposed each into two components, one lying in the meridian 

 plane and the other cutting that meridian plane at right angles. The 

 latter components, when divided by the n — 2"'^ power of their 

 distance to P, and placed in P, neutralize each other two by two; 

 and the former consist of pairs of ecpial and opposite planivectors 

 directed parallel to the meridian plane and at infinitely small distance 

 from each other according to the direction OQ. These cause in P 



sin (p 

 a iilanivector potential lying in the meridian plane = c -. The 



field L\ is of this potential the v = V^, inid outside the origin is 

 identical to the field of an elementary magnet along OL. 



The force field in S,, can be regarded as if caused 1"^. by magnets, 

 2'"'. by vortex systems consisting of the plane vortices erected normal 

 to a snuill "—-sphere. We can also take as the cause the spheres 

 themselves with their iiidicatrices and say that the field is formed 

 by magnets and vortex spheres of n. — 2 dimensions (as in S^ the 

 cause is found in the closed electric current instead of in the vortices 

 round about it). 



Here also fields of a single plane voi-tex element are impossible. 

 Yet we can speak of the fictitious "field of a single vortex" although 



