Page:Philosophical magazine 21 series 4.djvu/312

288 Prof. Maxwell on the Theory of Molecular Vortices

have by the conservation of force,

$\delta E+\delta W=0;$

(43)

that is, the loss of energy of the vortices must be made up by work done in moving magnets, so that

$-4\pi C\Sigma \left(2{\frac {d\phi _{1}}{dx}}m_{2}dV\right)\delta x+\Sigma \left({\frac {d\phi _{1}}{dx}}m_{2}dV\right)\delta x=0,$

or

$C={\frac {1}{8\pi }};$

(44)

so that the energy of the vortices in unit of volume is

${\frac {1}{8\pi }}\mu \left(\alpha ^{2}+\beta ^{2}+\gamma ^{2}\right);$

(45)

and that of a vortex whose volume is V is

${\frac {1}{8\pi }}\mu \left(\alpha ^{2}+\beta ^{2}+\gamma ^{2}\right)V.$

(46)

In order to produce or destroy this energy, work must be expended on, or received from, the vortex, either by the tangential action of the layer of particles in contact with it, or by change of form in the vortex. We shall first investigate the tangential action between the vortices and the layer of particles in contact with them.

Prop. VII. — To find the energy spent upon a vortex in unit of time by the layer of particles which surrounds it.

Let P, Q, R be the forces acting on unity of the particles in the three coordinate directions, these quantities being functions of x, y, and z. Since each particle touches two vortices at the extremities of a diameter, the reaction of the particle on the vortices will be equally divided, and will be

$-{\frac {1}{2}}P,\ -{\frac {1}{2}}Q,\ -{\frac {1}{2}}R$

on each vortex for unity of the particles; but since the superficial density of the particles is ${\tfrac {1}{2\pi }}$ (see equation (34)), the forces on unit of surface of a vortex will be

$-{\frac {1}{4\pi }}P,\ -{\frac {1}{4\pi }}Q,\ -{\frac {1}{4\pi }}R$

Now let $dS$ be an element of the surface of a vortex. Let the direction-cosines of the normal be l, m, n. Let the coordinates of the element be x, y, z. Let the component velocities of the