Page:Philosophical magazine 23 series 4.djvu/32

16 Prof. Maxwell on the Theory of Molecular Vortices Let $$\mu$$ be the coefficient of cubic elasticity, so that if

Let $$m$$ be the coefficient of rigidity, so that

Then we have the following equations of elasticity in an isotropic medium,

with similar equations in $$y$$ and $$z$$, and also

In the case of the sphere, let us assume the radius = $$a$$, and

Then

{{MathForm2|(85)|$$\left.\begin{array}{l} p_{xx}=2\left(\mu-\frac{1}{3}m\right)(e+g)z+mez=p_{yy},\\ \\p_{zz}=2\left(\mu-\frac{1}{3}m\right)(e+g)z+2mgz,\\ \\p_{yz}=\frac{m}{2}(e+2f)y,\\ \\p_{zx}=\frac{m}{2}(e+2f)z,\\ \\p_{xy}=0.\end{array}\right\} $$}}

The equation of internal equilibrium with respect to $$z$$ is

which is satisfied in this case if

The tangential stress on the surface of the sphere, whose radius is a at an angular distance $$\theta$$ from the axis in plane $$xz$$,

In order that T may be proportional to $$\sin\theta$$, the first term must vanish, and therefore