Page:Philosophical magazine 23 series 4.djvu/33

applied to Statical Electricity, 17

The normal stress on the surface at any point is

${\begin{array}{c}N=p_{xx}\sin ^{2}\theta +p_{yy}\cos ^{2}\theta +2p_{zz}\sin \theta \cos \theta \\=2\left(\mu -{\frac {1}{3}}m\right)(e+g)a\cos \theta +2ma\cos \theta \left((e+f)\sin ^{2}\theta +g\cos ^{2}\theta \right);\end{array}}$

(92)

or by (87) and (90),

$N=-ma(e+2f)\cos \theta .\,$

(93)

The tangential displacement of any point is

$t=\xi \cos \theta -\zeta \sin \theta =-\left(a^{2}f+d\right)\sin \theta .$

(94)

The normal displacement is

$v=\xi \sin \theta +\zeta \cos \theta =\left(a^{2}(e+f)+d\right)\cos \theta .$

(95)

If we make

$a^{2}(e+f)+d=0,$

(96)

there will be no normal displacement, and the displacement will be entirely tangential, and we shall have

$t=a^{2}e\sin \theta .$

(97)

The whole work done by the superficial forces is

$U={\frac {1}{2}}\Sigma (Tt)dS,$

the summation being extended over the surface of the sphere.

The energy of elasticity in the substance of the sphere is

$U={\frac {1}{2}}\Sigma \left({\frac {d\xi }{dx}}p_{xx}+{\frac {d\eta }{dy}}p_{yy}+{\frac {d\zeta }{dz}}p_{zz}+\left({\frac {d\eta }{dz}}+{\frac {d\zeta }{dy}}\right)p_{yz}+\left({\frac {d\zeta }{dx}}+{\frac {d\xi }{dz}}\right)p_{zx}+\left({\frac {d\xi }{dy}}+{\frac {d\eta }{dx}}\right)p_{xy}\right)dV,$

the summation being extended to the whole contents of the sphere.

We find, as we ought, that these quantities have the same value, namely

$U=-{\frac {2}{3}}\pi a^{5}me(e+2f).$

(98)

We may now suppose that the tangential action on the surface arises from a layer of particles in contact with it, the particles being acted on by their own mutual pressure, and acting on the surfaces of the two cells with which they are in contact.

We assume the axis of $z$ to be in the direction of maximum variation of the pressure among the particles, and we have to determine the relation between an electromotive force R acting on the particles in that direction, and the electric displacement $h$ which accompanies it.

Prop. XIII. — To find the relation between electromotive force and electric displacement when a uniform electromotive force R acts parallel to the axis of $z$.

Take any element $\delta S$ of the surface, covered with a stratum