( 122 ) 



a closed curve e not onlj the negative energj of a magnetic scale 

 with intensity unity bounded by e in the üeld of an element of 

 current unity F, but also the component along F of the vector 

 potential caused by a current unity along e. 



From this ensues for the yector V of an element of current, 

 that v^hen the element of current is integrated to a closed current 

 it becomes the vector potential of that current determined uniformly 

 on account of its flux property. 



So really the vector potential of a 2X i- e. of a field of currents 

 is obtained as an integral of the vectors T^of the elements of current. 



XII. We can now write that in an arbitrary point: 



IX=^/ j^:il^F,{r)dx, {II) 



where we first transfer in a parallel manner the vector elements 

 of the integral to the point under consideration and then sum up. 



Let us now consider an arbitrary force field as if caused by its two 

 derivatives (the magnets and currents), we can then represent to our- 

 selves, that both derivatives, propagating themselves according to a 

 function of the distance vanishing at infinity, generate the potential 

 of the field. 



The field X is namely the total derivative of the potential : 



The extinguishment of the scalar potential is greater than that of the 

 vector potential ; for, the former becomes at great distances of orderc^^-'", 

 the latter of order re~''. Farther the latter proves not to decrease 

 continuously from go to 0, but at the outset it passes quickly 

 through to negative, it then reaches a negative maximum and 

 then according to an extinguishment re~^' it tends as a negative (i. c. 

 directed oppositely to the generating element of current) vector to zero. 



XIII. The particularity found in Euclidean spaces, that 



Fj (r) z=. F^ {r) =T - — , is founded upon this, that in Euclidean spaces 

 r 



the operation of twice total derivation is found to be alike for scalar 



distributions and vector distributions of any dimensions (I.e. p. 70). 



Not so in non-Euclidean spaces; e. g. in the hyperbolic Sp^ we 



find for the V of a scalar distribution u in an arbitrary point 



