( 260 ) 



directed parallel to the acting vortex element and projecting itself on 

 that plane according to the tangent to a concentric circle; whilst tp 

 is the angle of the radiusvector with the Spn—2 perpendicular to 

 the vortex element. 



V. In the same way as in C § IX we deduce from this the 

 plant vector potential F of a vortex element directed everywhere 

 parallel to the vortex element and of which the scalar value is a 

 function of r only. That scalar value U of that vector potential is here 

 determined by the differential equation : 



\U COS (fi . dr . cs sin ^—^r cos ^~^(p \ d(p — 



0^ ( ) 



\U sin<p . sin r d(p . ce sin "— 3r cos »— 3y | dr z=z 



br I ) 



r= /„ [r) sin <p . sin r dip . dr . ce sin f—^r cos "~^g}. 



dU 



(w— 2) ü sinr — {n — 2>) U cosr ■=. Xn {r) sm r. 



dr 



dU 



_ {n-2) Utghr=-yin W- 

 dr 



^ = orn 2U • r^^' '^""'^ 2 ^ • Xn (r) dr , 



cos 2(.n— 2;i r J 



r 



a function vanishing in the opposite point, which we put ^ F^ (r). 

 We then find for an arbitrary flux : 



lX:^\f/J^^FAr)dT (//) 



And taking an arbitrary vector field to be caused by its two deri- 

 vatives (the magnets and the vortex systems) propagating themselves 

 through space as a potential according to a function of the distance 

 vanishing in the opposite point, we find : 



X = V j ƒ ^ F, (r) dr + ƒ ^ ^. W ^^1 



G. The Elliptic Sp»^ 



Also for the elliptic Sp^ the derivative of an arbitrary linevector 

 distribution is an integral of elementary vortex systems Voy and 

 VOz, which are respectively the first and the second derivative of 



