Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/190

158 which, when the circuits are closed, becomes their mutual potential, then (27) may be written Rh

520.] Integrating, with respect to $$s$$ and $$s^\prime$$, between the given limits, we find

where the subscripts of $$L$$ indicate the distance, $$r$$, of which the quantity $$L$$ is a function, and the subscripts of $$F$$ and $$F^\prime$$ indicate the points at which their values are to be taken.

The expressions for $$T$$ and $$Z$$ may be written down from this. Multiplying the three components by $$dx$$, $$dy$$, and $$dz$$ respectively, we obtain

where $$D$$ is the symbol of a complete differential.

Since $$F\,dx + G\, dy + H\,dz$$ is not in general a complete differential of a function of $$x$$, $$y$$, $$z$$, $$X\,dx + Y\,dy + Z\,dz$$ is not a complete differential for currents either of which is not closed.

521.] If, however, both currents are closed, the terms in $$L$$, $$F$$, $$G$$, $$H$$, $$F^\prime$$, $$G^\prime$$, $$H^\prime$$ disappear, and Rhwhere $$M$$ is the mutual potential of two closed circuits carrying unit currents. The quantity $$M$$ expresses the work done by the electromagnetic forces on either conducting circuit when it is moved parallel to itself from an infinite distance to its actual position. Any alteration of its position, by which $$M$$ is increased, will be assisted by the electromagnetic forces.

It may be shewn, as in Arts. 490, 596, that when the motion of the circuit is not parallel to itself the forces acting on it are still determined by the variation of $$M$$, the potential of the one circuit on the other.

522.] The only experimental fact which we have made use of in this investigation is the fact established by Ampère that the action of a closed current on any portion of another current is perpendicular to the direction of the latter. Every other part of