Page:Philosophical magazine 23 series 4.djvu/105

XIV. On Physical Lines of Force. By J. C. Maxwell, F.R.S.,

Professor of Natural Philosophy in King's College, London^[1].

Part IV. — The Theory of Molecular Vortices applied to the Action of Magnetism on Polarized Light.

THE connexion between the distribution of lines of magnetic force and that of electric currents may be completely expressed by saying that the work done on a unit of imaginary magnetic matter, when carried round any closed curve, is proportional to the quantity of electricity which passes through the closed curve. The mathematical form of this law may be expressed as in equations (9)^[2], which I here repeat, where $\alpha ,\beta ,\gamma$ are the rectangular components of magnetic intensity, and p, q, r are the rectangular components of steady electric currents,

$\left.{\begin{array}{l}p={\frac {1}{4\pi }}\left({\frac {d\gamma }{dy}}-{\frac {d\beta }{dz}}\right),\\\\q={\frac {1}{4\pi }}\left({\frac {d\alpha }{dz}}-{\frac {d\gamma }{dx}}\right),\\\\r={\frac {1}{4\pi }}\left({\frac {d\beta }{dx}}-{\frac {d\alpha }{dy}}\right).\end{array}}\right\}$

(9)

The same mathematical connexion is found between other sets of phenomena in physical science.

(1) If $\alpha ,\beta ,\gamma$ represent displacements, velocities, or forces, then p, q, r will be rotatory displacements, velocities of rotation, or moments of couples producing rotation, in the elementary portions of the mass.

(2) If $\alpha ,\beta ,\gamma$ represent rotatory displacements in a uniform and continuous substance, then p, q, r represent the relative linear displacement of a particle with respect to those in its immediate neighbourhood. See a paper by Prof. W. Thomson "On a Mechanical Representation of Electric, Magnetic, and Galvanic Forces," Camb. and Dublin Math. Journ. Jan. 1847.

(3) If $\alpha ,\beta ,\gamma$ represent the rotatory velocities of vortices whose centres are fixed, then p, q, r represent the velocities with which loose particles placed between them would be carried along. See the second part of this paper (Phil. Mag. April 1861).

It appears from all these instances that the connexion between magnetism and electricity has the same mathematical form as that between certain pairs of phenomena, of which one has a linear and the other a rotatory character. Professor Challis^[3]

1. Communicated by the Author.
2. Phil. Mag. March 1861.
3. Phil. Mag. December 1860, January and February 1861.