Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/72

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

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

$\omega =\iint {\Pi ds\,d\sigma }+\omega _{0},$

(7)

where the integration with respect to s is to be extended completely round the closed curve, and that with respect to σ from A a fixed point on the curve to the point P. The constant ω₀ 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 ω 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 P'P forms a closed circuit embracing the edge. The value of ω at P exceeds that at P' by 4π, 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 $\iint {\Pi dsd\sigma }$ extended round both curves will be 4π.

This integral therefore, considered as depending only on the closed curve s 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 s, 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 4πn.

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 4πn. It is possible, however, for two curves