( 121 ) 



coshr 

 — ;-- — sin y, 

 smh^r 



directed perpendicular to the meridian plane. 



XI. For the fictitious field of an element of current (having mean- 

 while everywhere current, i. e. rotation) introduced in this way we 

 shall find a linevector potential V, everywhere "parallel" (see above 

 under § X) to the element of current and the scalar value of which 

 is a function of r only. 



Let us call that scalar value U, and let us regard a small elemen- 

 tar}^ rectangle in the meridian plane bounded by radii vectores from 

 the origin and by circles round the origin, then the line integral of 

 V round that rectangle is : 



Ö , . d 



— ^ I Usin (p sinh r dw] dr — —-iUcos (p dr] dw. 

 or d(fi r ^ r 



This must be equal to the current of force through the small 

 rectangle : 



c^sli r 



— 7-— sin <p . sinh r d(p. dr, 

 sinh r 



from which we derive the following differential equation of Ü 

 with respect to ?' : 



d 



* U — ^ \U sinli r\ z=z coth r, 



or 



the solution of which is: 



U =z cosech r — \r sech^ ^ r -{- c . sech^ ^ r. 



Let us take c = 0, we shall then find as vector potential V of 

 an element of current unity E: 



cosech r — ^ r sech'' ^ r ^ F^ (?-), 

 directed parallel to E. 



Let us now apply in an arbitrary point of space a vector G, then 

 the vector V has the property that, when integrated in G along an 

 elementary circuit whose plane is perpendicular to G, it indicates 

 the force in the direction of G, caused by the element of current 

 E, or likewise the vector potential in the direction of E caused by 

 an elementary magnet with intensity unity in the direction of G. 

 . So, if we call of two vectors unity is and 7"^ the potential x (-£"» i'), 

 the symmetric function /% {r, cos y, where r represents the distance 

 of the points of application of the two vectors and (p their angle 

 after parallel transference to a selfsame point of tlieir connecting line, 

 we know that this function x gives, by integration of e.g. E over 



