Page:Philosophical magazine 21 series 4.djvu/364

340 Prof. Maxwell on the Theory of Molecular Vortices

any set of three axes. We shall first consider the effect of three simple extensions or compressions.

Prop. IX. — To find the variations of $\alpha ,\beta ,\gamma$ in the parallelopiped x, y, z when $x$ becomes $x+\delta x;$; $y$, $y+\delta y$; and $z$, $z+\delta z$, the volume of the figure remaining the same.

By Prop. II. we find for the work done by the vortices against pressure,

$\delta W=p_{1}\delta (xyz)-{\frac {\mu }{4\pi }}\left(\alpha ^{2}yz\delta x+\beta ^{2}zx\delta y+\gamma ^{2}xy\delta z\right);$

(59)

and by Prop. VI. we find for the variation of energy,

$\delta E={\frac {\mu }{4\pi }}(\alpha \delta \alpha +\beta \delta \beta +\gamma \delta \gamma )xyz.$

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The sum $\delta W+\delta E$ must be zero by the conservation of energy, and $\delta (xyz)=0$, since $xyz$ is constant; so that

$\alpha \left(\delta \alpha -\alpha {\frac {\delta x}{x}}\right)+\beta \left(\delta \beta -\beta {\frac {\delta y}{y}}\right)+\gamma \left(\delta \gamma -\gamma {\frac {\delta z}{z}}\right)=0.$

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In order that this should be true independently of any relations between $\alpha$, $\beta$, and $\gamma$, we must have

$\delta \alpha =\alpha {\frac {\delta x}{x}},\ \delta \beta =\beta {\frac {\delta y}{y}},\ \delta \gamma =\gamma {\frac {\delta z}{z}}.$

(62)

Prop. X. — To find the variations of $\alpha ,\beta ,\gamma$ due to a rotation $\theta _{1}$ about the axis of $x$ from $y$ to $z$, a rotation $\theta _{2}$ about the axis of $y$ from $z$ to $x$, and a rotation $\theta _{3}$ about the axis of $z$ from $x$ to $y$.

The axis of $\beta$ will move away from the axis of $x$ by an angle $\theta _{3}$, so that $\beta$ resolved in the direction of $x$ changes from 0 to $-\beta \theta _{3}$.

The axis of $\gamma$ approaches that of $x$ by an angle $\theta _{2}$; so that the resolved part of $\gamma$ in direction $x$ changes from 0 to $\gamma \theta _{2}$.

The resolved part of a. in the direction of $x$ changes by a quantity depending on the second power of the rotations, which may be neglected. The variations of $\alpha ,\beta ,\gamma$ from this cause are therefore

$\delta \alpha =\gamma \theta _{2}-\beta \theta _{3},\ \delta \beta =\alpha \theta _{3}-\gamma \theta _{1},\ \delta \gamma =\beta \theta _{1}-\alpha \theta _{2}.$

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The most general expressions for the distortion of an element produced by the displacement of its different parts depend on the nine quantities

${\frac {d}{dx}}\delta x,\ {\frac {d}{dy}}\delta x,\ {\frac {d}{dz}}\delta x,\ {\frac {d}{dx}}\delta y,\ {\frac {d}{dy}}\delta y,\ {\frac {d}{dz}}\delta y,\ {\frac {d}{dx}}\delta z,\ {\frac {d}{dy}}\delta z,\ {\frac {d}{dz}}\delta z;$

and these may always be expressed in terms of nine other quantities, namely, three simple extensions or compressions,

${\frac {\delta x'}{x'}},\ {\frac {\delta y'}{y'}},\ {\frac {\delta z'}{z'}}$