410 



THE ELECTROMAGNETIC FIELD. [PT. III. CH. XI. 



We shall in future denote the magnetic force by H, its com- 

 ponents by L, M, N, the magnetic induction by 33, its components 

 by , S 3ft, 91, and the magnetic potential by H, reserving the nota- 

 tion F, X, F, Z, g, , g), 3> V> f r the corresponding electric 

 quantities. For the electric inductivity we shall use the letter e, 

 leaving yu, for the magnetic inductivity. These distinctions have 

 not before been necessary, since we have not at the same time 

 considered both electrical and magnetic quantities, as we must do 

 from now on. If we form the line integral of magnetic force from 

 a point A to a point B, we have 



(I) 



fB 



I Ldx+Mdy + Nd**=l A - 



which must be independent of the path AB, for otherwise, by 

 changing the path infinitely little, we should, starting with the 

 given value 1 A , cause H^ to change by an infinitely small amount, 

 and could thus cause 1 B to take at the same point a series of con- 

 tinuously varying values. The integral is accordingly the same 

 for all paths that can be changed into one another by continuous 

 deformation. If, however, the current separates two paths ACB, 

 ADB, the integral is not the same for both. In other words, while 

 the integral around any closed path not linked with the circuit is 

 zero, the integral around a path linked with the circuit is not. 

 But the integral around any two closed paths each linked once 

 with the circuit is the same, for they may be continuously 

 deformed into each other. Or in other words, 

 we may connect two such paths 1 and 2, Fig. 

 84, by a path PQ. The integral around the 

 circuit ABPQDCQPA, which is not linked 

 with the current, is zero, but this is equal to 

 the sum of the integrals PABP around 1 in 

 the positive direction, together with the in- 

 tegral QDCQ around 2 in the negative direction, 

 while the integrals over the coincident paths 

 FIG. 84. PQ, QP in opposite directions destroy each 



other. Accordingly 



We shall say that two geometrical circuits are linked positively, 



