( 75 ) 



a field E^ to rotate 90''. As on the other hand it has to be of an 

 identical structure - to A\ outside the origin we may call the Held 

 E^ resp. E, "dual to itself". 



In our space the field E^ can be realized as that of a plane, 

 infinitely long and narrow magnetic band with poles along the edges ; 

 the field E^ as that of two infinitely long parallel straight electric 

 currents, close together and directed oppositely. 



The planivector {vortejc) field in S^. 



The field E^. The elementary sphere /S/>, is a circlet, the elementary 



vortex system Vo~ a current along it. It furnishes a linevector 



siti (f 

 potential =i — — directed along the circles whicii ])roject themselves 



on the plane of Vo^ as circles concentric to Voz, and where </ is 

 the angle of the radiusvector with the normal plane of Vo-,. The 

 field is the first derivative (rotation) of this potential. 



Jlie field E,. The elementary sphere Sjyy is again a circlet, tiie 

 elementary vortex system Vo^j assumes in the points of that circlet 

 equal 'vectors normal to it. The ' F-potential consists of the ' I^'s 

 normal to the potential vectors of a field E^ ; the field E^ is thus 

 obtained by taking tlie normal i)lanes of ail planivectors of a field ^i. 

 As on the other hand E^ and E^ are of the same identical structure 

 outside the origin, we can say here again, that the field E^ resp. E, 

 is dual to itself 



So we can regard the vortex field in S^ as caused by elementary 

 circular currents of two kinds; two equal currents of a different kind 

 cause vortex fields of ecpial structure, but one field is perfectlv 

 normal to the otiier. 



So if of a field the two generating systems of currents are 

 identical, it consists of isosceles double-vortices. 



The force field in S,. 



The field E^. Vo, gives a double point, causing a scalar 



cos (p 

 potential -—- , where </ is the angle of the radiusvector with the 

 r 



axis of the double point; the derivative (gradient) gives the wellknown 

 field of an elementary magnet. 



The field E^. Voy consists of equal planivectors normal to a 

 small circular current. If we represent the [»lanivector potential by the 



linevector normal to it, we shall find for that linevector ^-^ directed 



