Page:Philosophical magazine 23 series 4.djvu/111

and the energy

${\begin{array}{ll}E&={\frac {1}{2}}\Sigma dmr^{2}\omega ^{2}={\frac {1}{2}}A\omega ,\\\\&={\frac {1}{8\pi }}\mu \alpha ^{2}V\ \mathrm {by\ Prop.\ VI} .\end{array}}$[1]

(144)

whence

$A={\frac {1}{4\pi }}\mu \alpha V$

(144)

for the axis of $x$, with similar expressions for the other axes, V being the volume, and $r$ the radius of the vortex.

Prop. XIX. — To determine the conditions of undulatory motion in a medium containing vortices, the vibrations being perpendicular to the direction of propagation.

Let the waves be plane-waves propagated in the direction of $z$, and let the axis of $x$ and $y$ be taken in the directions of greatest and least elasticity in the plane $xy$. Let $x$ and $y$ represent the displacement paralll to these axes, which will be the same throughout the same wave-surface, and therefore we shall have $x$ and $y$ functions of $z$ and $t$ only.

Let X be the tangential stress on unit of area parallel to $xy$, tending to move the part next the origin in the direction of $x$.

Let Y be the corresponding tangential stress in the direction of $y$.

Let $k_{1}$ and $k_{2}$ be the coefficients of elasticity with respect to these two kinds of tangential stress; then, if the medium is at rest,

$X=k_{1}{\frac {dx}{dz}},\ Y=k_{2}{\frac {dy}{dz}}.$

Now let us suppose vortices in the medium whose velocities are represented as usual by the symbols $\alpha ,\beta ,\gamma$, and let us suppose that the value of $\alpha$ is increasing at the rate ${\tfrac {d\alpha }{dt}}$, on account of the action of the tangential stresses alone, there being no electromotive force in the field. The angular momentum in the stratum whose area is unity, and thickness $dz$, is therefore increasing at the rate ${\tfrac {1}{4\pi }}\mu r{\tfrac {d\alpha }{dt}}dz$; and if the part of the force Y which produces this effect is Y', then the moment of Y' is $-Y'dz$,so that $Y'=-{\tfrac {1}{4\pi }}\mu r{\tfrac {d\alpha }{dt}}$.

The complete value of Y when the vortices are in a state of

1. Phil. Mag. April 1861.