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

${\begin{array}{l}p_{xx}=p_{yy}=p_{zz}=p,\\\\p=\mu \left({\frac {d\xi }{dx}}+{\frac {d\eta }{dy}}+{\frac {d\zeta }{dz}}\right)\end{array}}$

(80)

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

$p_{xx}-p_{yy}=m\left({\frac {d\xi }{dx}}-{\frac {d\eta }{dy}}\right),\ etc.$

(81)

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

$p_{xx}=\left(\mu -{\frac {1}{3}}m\right)\left({\frac {d\xi }{dx}}+{\frac {d\eta }{dy}}+{\frac {d\zeta }{dz}}\right)+m{\frac {d\xi }{dx}};$

(82)

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

$p_{yz}={\frac {m}{2}}\left({\frac {d\eta }{dy}}+{\frac {d\zeta }{dz}}\right),\ etc.$

(83)

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

$\xi =exz,\ \eta =exy,\ \zeta =\left(x^{2}+y^{2}\right)+gz^{2}+d.$

(84)

Then

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

(85)

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

${\frac {d}{dx}}p_{zx}+{\frac {d}{dy}}p_{yz}+{\frac {d}{dz}}p_{zz}=0,$

(86)

which is satisfied in this case if

$m(e+2f+2g)+2\left(\mu -{\frac {1}{3}}m\right)(e+g)=0.$

(87)

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

$T=\left(p_{xx}-p_{zz}\right)\sin \theta \cos \theta +p_{xz}\left(\cos ^{2}\theta -\sin ^{2}\theta \right)$

(88)

$=2m(e+f-g)a\sin \theta \cos ^{2}\theta -{\frac {ma}{2}}(e+2f)\sin \theta .$

(89)

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

$g=e+f,\,$

(90)

$T=-{\frac {ma}{2}}(e+2f)\sin \theta .$

(91)