closed curve the extremity of the perpendicular will trace a second Let the length of the second closed curve be o, then

closed curve.

the solid angle subtended by the first closed curve is

This follows from the well-known theorem that the area of a closed curve on a sphere of unit radius, together with the circumference of the polar curve, is numerically equal to the circumference of a great circle of the sphere.

This construction is sometimes convenient for calculating the solid angle subtended by a rectilinear figure. For our own purpose, which is to form clear ideas of physical phenomena, the following method is to be preferred, as it employs no constructions which do not flow from the physical data of the problem.

419.] A closed curves is given in space, and we have to find the solid angle subtended by s at a given point P.

If we consider the solid angle as the potential of a magnetic shell of unit strength whose edge coincides with the closed curve, we must define it as the work done by a unit magnetic pole against the magnetic force while it moves from an infinite distance to the point P. Hence, if σ is the path of the pole as it approaches the point P, the potential must be the result of a line-integration along this path. It must also be the result of a line-integration along the closed curve s. The proper form of the expression for the solid angle must therefore be that of a double integration with respect to the two curves s and σ.

zero.

When P is at an infinite distance, the solid angle is evidently As the point P approaches, the closed curve, as seen from the moving point, appears to open out, and the whole solid angle may be conceived to be generated by the apparent motion of the different elements of the closed curve as the moving point approaches.

As the point P moves from P to P' over the element do, the element QQ of the closed curve, which we denote by ds, will change its position relatively to P, and the line on the unit sphere corresponding to QQ will sweep over an area on the spherical surface, which we may write

To find ПI let us suppose P fixed while the closed curve is moved parallel to itself through a distance do equal to PP′ but in the opposite direction. The relative motion of the point P will be the same as in the real case.

420.]

GENERATION OF A SOLID ANGLE.

39

During this motion the element QQ will generate an area in the form of a parallelogram whose sides are parallel and equal to QQ and PP'. If we construct a pyramid on this parallelogram as base with its vertex at P, the solid angle of this pyramid will be the increment do which we are in search of.

To determine the value of this solid

angle, let 0 and be the angles which ds and do make with PQ respectively, and let & be the angle between the planes of these two angles, then the area of the projection of the parallelogram ds. do on a plane perpendicular to PQ or r will be

ds do sin 0 sin e' sin p, and since this is equal to r2 do, we find

1

P

Fig. 3.

do Пds do = sin sin 6' sin o ds do.

p.2

(2)

420.] We may express the angles 0, 6', and

and its differential coefficients with respect to s and σ, for

in terms of r,

A third expression for II in terms of rectangular coordinates may be deduced from the consideration that the volume of the pyramid whose solid angle is do and whose axis is r is

} r3 dw = { r3 Пds do.

But the volume of this pyramid may also be expressed in terms of the projections of r, ds, and do on the axis of x, y and z, as a determinant formed by these nine projections, of which we must take the third part. We thus find as the value of II,

This expression gives the value of II free from the ambiguity of sign introduced by equation (5).

421.] The value of w, the solid angle subtended by the closed curve at the point P, may now be written

1=

Пds do + wor

(7)

where the integration with respect to s is to be extended completely round the closed curve, and that with respect to o from A a fixed point on the curve to the point P. The constant wo is the value of the solid angle at the point A. It is zero if A is at an infinite distance from the closed curve.

The value of at any point P is independent of the form of the curve between A and P provided that it does not pass through the magnetic shell itself. If the shell be supposed infinitely thin, and if P and P' are two points close together, but P on the positive and P' on the negative surface of the shell, then the curves AP and AP' must lie on opposite sides of the edge of the shell, so that PAP' is a line which with the infinitely short line PP forms a closed circuit embracing the edge. The value of w at P exceeds that at P' by 4, that is, by the surface of a sphere of radius unity.

Hence, if a closed curve be drawn so as to pass once through the shell, or in other words, if it be linked once with the edge of the shell, the value of the integral II ds ds do extended round

both curves will be 4π.

This integral therefore, considered as depending only on the closed curves and the arbitrary curve AP, is an instance of a function of multiple values, since, if we pass from A to P along different paths the integral will have different values according to the number of times which the curve AP is twined round the curve s.

If one form of the curve between A and P can be transformed into another by continuous motion without intersecting the curve 8, the integral will have the same value for both curves, but if during the transformation it intersects the closed curve n times the values of the integral will differ by 4 n.

If s and σ are any two closed curves in space, then, if they are not linked together, the integral extended once round both is

zero.

If they are intertwined n times in the same direction, the value of the integral is 4 n. It is possible, however, for two curves

422.]

VECTOR-POTENTIAL OF A CLOSED CURVE.

41

to be intertwined alternately in opposite directions, so that they are inseparably linked together though the value of the integral See Fig. 4.

is zero.

It was the discovery by Gauss of this very integral, expressing the work done on a magnetic pole while describing a closed curve in presence of a closed electric current, and indicating the geometrical connexion between the two closed curves, that led him to lament the small progress made in the Geometry of Position since the time of Leibnitz, Euler and Vandermonde. We have now, how

Fig. 4.

ever, some progress to report, chiefly due to Riemann, Helmholtz and Listing.

422.] Let us now investigate the result of integrating with. respect to s round the closed curve.

One of the terms of II in equation (7) is

the integrals being taken once round the closed curve s, this term

Collecting all the terms of II, we may now write

This quantity is evidently the rate of decrement of w, the magnetic potential, in passing along the curve o, or in other words, it is the magnetic force in the direction of do.

By assuming do successively in the direction of the axes of x, y and z, we obtain for the values of the components of the magnetic force

The quantities F, G, H are the components of the vector-potential of the magnetic shell whose strength is unity, and whose edge is the curve s. They are not, like the scalar potential w, functions having a series of values, but are perfectly determinate for every point in space.

The vector-potential at a point P due to a magnetic shell bounded by a closed curve may be found by the following geometrical construction:

Let a point travel round the closed curve with a velocity numerically equal to its distance from P, and let a second point R start from A and travel with a velocity the direction of which is always parallel to that of Q, but whose magnitude is unity. When has travelled once round the closed curve join AR, then the line AR represents in direction and in numerical magnitude the vector-potential due to the closed curve at P.

Potential Energy of a Magnetic Shell placed in a Magnetic Field. 423.] We have already shewn, in Art. 410, that the potential energy of a shell of strength placed in a magnetic field whose potential is V, is

dv dV +m

da dy

dz)ds,

(12)

where l, m, n are the direction-cosines of the normal to the shell drawn from the positive side, and the surface-integral is extended over the shell.

Now this surface-integral may be transformed into a line-integral by means of the vector-potential of the magnetic field, and we may write

M=

dx ds

dy dz

+ Gay

ds

+H1⁄4%)ds,

(13)

where the integration is extended once round the closed curve s which forms the edge of the magnetic shell, the direction of ds being opposite to that of the hands of a watch when viewed from the positive side of the shell.

If we now suppose that the magnetic field is that due to a