Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/73

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

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, however, 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 Π in equation (7) is

${\frac {\xi -x}{r^{3}}}{\frac {d\eta }{d\sigma }}{\frac {dz}{ds}}={\frac {d\eta }{d\sigma }}{\frac {d}{d\xi }}\left({\frac {1}{r}}{\frac {dz}{ds}}\right).$

(8)

If we now write for brevity

$F\int {{\frac {1}{r}}{\frac {dx}{ds}}},\quad G\int {{\frac {1}{r}}{\frac {dy}{ds}}},\quad H\int {{\frac {1}{r}}{\frac {dz}{ds}}},$

(9)

the integrals being taken once round the closed curve s, this term of Π may be written

${\frac {d\eta }{d\sigma }}{\frac {d^{2}H}{d\xi ds}},$

and the corresponding term of $\int {\Pi ds}$ will be

${\frac {d\eta }{d\sigma }}{\frac {dH}{d\xi }}.$

Collecting all the terms of Π, we may now write

${\begin{aligned}-{\frac {d\omega }{d\sigma }}&=-\int {\Pi ds}\\&=\left({\frac {dH}{d\eta }}-{\frac {dG}{d\zeta }}\right){\frac {d\xi }{d\sigma }}+\left({\frac {dF}{d\zeta }}-{\frac {dH}{d\xi }}\right){\frac {d\eta }{d\sigma }}+\left({\frac {dG}{d\xi }}-{\frac {dF}{d\eta }}\right){\frac {d\zeta }{d\sigma }}.\end{aligned}}$[1]

(10)

This quantity is evidently the rate of decrement of ω, the magnetic potential, in passing along the curve σ, or in other words, it is the magnetic force in the direction of dσ.

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

1. Fixed typo in the second term in the second line, see Errata, page 1.