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 another, 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

A

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 AEC 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.

B

Fig. 35.

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

589.]

ADDITION OF CIRCUITS.

213

not sensibly altered. This will evidently be the case if the dimensions of the portion of surface are small enough compared with its distance from the primary circuit.

If any closed curve be drawn on this portion of the surface, the value of p will be proportional to its area.

For the areas of any two circuits may be divided into small elements all of the same dimensions, and having the same value of p. The areas of the two circuits are as the numbers of these elements which they contain, and the values of p for the two circuits are also in the same proportion.

Hence, the value of p for the circuit which bounds any element ds of a surface is of the form

Ids,

where I is a quantity depending on the position of dS and on the direction of its normal. We have therefore a new expression for p,

where the double integral is extended over any surface bounded by the circuit.

589.] Let ABCD be a circuit, of which AC is an elementary portion, so small that it may be considered straight.

Let APB and CQB be small equal areas in the

same plane, then the value of p will be the same P for the small circuits APB and CQB, or

Hence

p(APB) = p(CQB).

p(APBQCD) = p(ABQCD)+ p(APB),

C

A

or the value of p is not altered by the substitution of the crooked line APQC for the straight line AC, provided the area of the circuit is not sensibly altered. This, in fact, is the principle established by Ampère's second experiment (Art. 506), in which a crooked portion of a circuit is shewn to be equivalent to a straight portion. provided no part of the crooked portion is at a sensible distance from the straight portion.

If therefore we substitute for the element ds three small elements, dx, dy, and dz, drawn in succession, so as to form a continuous path from the beginning to the end of the element ds, and if Fdx, Gdy, and II dz denote the elements of the line-integral corresponding to do, dy, and dz respectively, then

Jds Fdx+ Gdy + H dz.

(4)

590.] We are now able to determine the mode in which the quantity J depends on the direction of the element ds. For,

This is the expression for the resolved part, in the direction of ds, of a vector, the components of which, resolved in the directions of the axes of x, y, and z, are F, G, and H respectively.

If this vector be denoted by 2, and the vector from the origin to a point of the circuit by p, the element of the circuit will be dp, and the quaternion expression for J will be

The vector A and its constituents F, G, H depend on the position of ds in the field, and not on the direction in which it is drawn. They are therefore functions of x, y, z, the coordinates of ds, and not of l, m, n, its direction-cosines.

The vector 2 represents in direction and magnitude the timeintegral of the electromotive force which a particle placed at the point (x, y, z) would experience if the primary current were suddenly stopped. We shall therefore call it the Electrokinetic Momentum at the point (x, y, z). It is identical with the quantity which we investigated in Art. 405 under the name of the vectorpotential of magnetic induction.

The electrokinetic momentum of any finite line or circuit is the line-integral, extended along the line or circuit, of the resolved part of the electrokinetic momentum at each point of the same.

D

C

B

A

Fig. 37.

591.] Let us next determine the value of p for the elementary rectangle ABCD, of which the sides are dy and dz, the positive direction being from the direction of the axis of y to that of z. У

Let the coordinates of 0, the centre of gravity of the element, be xo, yo, 20, and let Go, Ho be the values of G and of H at this point.

The coordinates of 4, the middle point of the first side of the

approximately

and the part of the value of p which arises from the side A is

1 dG

Gody- dy dz.

2 dz

(9)

1 dH

dy dz.

2 dy

1 dG

dy dz.

2 dz

Adding these four quantities, we find the value of p for the

If we now assume three new quantities, a, b, c, such that

(10)

(A)

dx dy

and consider these as the constituents of a new vector B, then, by Theorem IV, Art. 24, we may express the line-integral of 2 round any circuit in the form of the surface-integral of B over a surface bounded by the circuit, thus

where is the angle between 2 and ds, and 7 that between B and the normal to dS, whose direction-cosines are l, m, n, and TA, TB denote the numerical values of A and B.

Comparing this result with equation (3), it is evident that the quantity I in that equation is equal to B cos 7, or the resolved part of B normal to dS.

592.] We have already seen (Arts. 490, 541) that, according to Faraday's theory, the phenomena of electromagnetic force and

induction in a circuit depend on the variation of the number of lines of magnetic induction which pass through the circuit. Now the number of these lines is expressed mathematically by the surface-integral of the magnetic induction through any surface bounded by the circuit. Hence, we must regard the vector B and its components a, b, c as representing what we are already acquainted with as the magnetic induction and its components.

In the present investigation we propose to deduce the properties of this vector from the dynamical principles stated in the last chapter, with as few appeals to experiment as possible.

In identifying this vector, which has appeared as the result of a mathematical investigation, with the magnetic induction, the properties of which we learned from experiments on magnets, we do not depart from this method, for we introduce no new fact into the theory, we only give a name to a mathematical quantity, and the propriety of so doing is to be judged by the agreement of the relations of the mathematical quantity with those of the physical quantity indicated by the name.

The vector B, since it occurs in a surface-integral, belongs evidently to the category of fluxes described in Art. 13. The vector 2, on the other hand, belongs to the category of forces, since it appears in a line-integral.

593.] We must here recall to mind the conventions about positive and negative quantities and directions, some of which were stated in Art. 23. We adopt the right-handed system of axes, so that if a right-handed screw is placed in the direction of the axis of x, and a nut on this screw is turned in the positive direction of rotation, that is, from the direction of y to that of z, it will move along the screw in the positive direction of x.

We also consider vitreous electricity and austral magnetism as positive. The positive direction of an electric current, or of a line of electric induction, is the direction in which positive electricity moves or tends to move, and the positive direction of a line of magnetic induction is the direction in which a compass needle points with the end which turns to the north. See Fig. 24, Art. 498, and Fig. 25, Art. 501.

The student is recommended to select whatever method appears to him most effectual in order to fix these conventions securely in his memory, for it is far more difficult to remember a rule which determines in which of two previously indifferent ways a statement is to be made, than a rule which selects one way out of many.