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,

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

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

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

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

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,

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