Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/244

 Rh contributes something to the value of p, and that the part contributed by each portion of the circuit depends on the form and position of that portion only, and not on the position of other parts of the circuit.

This assumption is legitimate, because we are not now considering a current the parts of which may, and indeed do, act on one an other, but a mere circuit, that is, a closed curve along which a current may flow, and this is a purely geometrical figure, the parts of which cannot be conceived to have any physical action on each other.

We may therefore assume that the part contributed by the element ds of the circuit is Jds, where J is a quantity depending on the position and direction of the element ds. Hence, the value of p may be expressed as a line-integral where the integration is to be extended once round the circuit.

587.] We have next to determine the form of the quantity J. In the first place, if ds is reversed in direction, J is reversed in sign. Hence, if two circuits ABCE and AECD have the arc AEG common, but reckoned in opposite directions in the two circuits, the sum of the values of p for the two circuits ABCE and AECD will be equal to the value of p for the circuit ABCD, which is made up of the two circuits.

For the parts of the line-integral depending on the arc AEC are equal but of opposite sign in the two partial circuits, so that they destroy each other when the sum is taken, leaving only those parts of the line-integral which depend on the external boundary of ABCD.

In the same way we may shew that if a surface bounded by a closed curve be divided into any number of parts, and if the boundary of each of these parts be considered as a circuit, the positive direction round every circuit being the same as that round the external closed curve, then the value of p for the closed curve is equal to the sum of the values of p for all the circuits. See Art. 483.

588.] Let us now consider a portion of a surface, the dimensions of which are so small with respect to the principal radii of curvature of the surface that the variation of the direction of the normal within this portion may be neglected. We shall also suppose that if any very small circuit be carried parallel to itself from one part of this surface to another, the value of p for the small circuit is