ELECTROMAGNETISM. 



411 



-f- Rig hi 



210] 



when, given a direction of circulation about each circuit, the 



direction of circulation in one circuit agrees with 



the forward motion of a right-handed screw, 



whose rotation corresponds to the direction of 



circulation in the other circuit. Fig. 85 repre- 



sents two circuits linked positively above and 



negatively below. By an extension of the above 



reasoning we see that the integral around any 



circuit linked n times in the positive manner 



with the current is nJ, where J is the integral 



around any circuit linked once. Accordingly 



the potential at any point is an infinitely 



valued function, whose values differ from each other by integral 



multiples of J. We may however make the potential a uniform 



function, if we prevent passage from one point to another by 



paths not continuously deformable into each other, 



that is, if we reduce the doubly connected space 



about the current to a singly-connected one by 



means of a diaphragm covering the current 



circuit. Then no two paths can be separated 



by the current. If we consider the potential Fm - 86 - 



at two points infinitely near each other but lying on opposite 



sides of the diaphragm, Fig. 86, to get from one to the other we 



must perform a closed circuit about the current, so that their 



potential differs by the amount J, accordingly in crossing the 



diaphragm, the potential is discontinuous, the amount of the 



discontinuity being 



where A is on the positive side of the diaphragm. There is, how- 

 ever, no discontinuity nor lack of uniformity in the derivatives 

 of 0. 



If we now consider all space, except a small sphere of radius R 

 with center at the point P, and apply to it Green's theorem 



where for V we put the magnetic potential fl, and for U the 

 function 1/r, where r is the distance from P, the volume integrals 

 vanish, and the surface integrals are to be taken over the infinite 



