Page:Philosophical magazine 21 series 4.djvu/187

and its velocity to diminish in the same proportion. In order that a medium having these inequalities of pressure in different directions should be in equilibrium, certain conditions must be fulfilled, which we must investigate.

Prop. II. — If the direction-cosines of the axes of the vortices with respect to the axes of x, y, and z be l, m, and n, to find the normal and tangential stresses on the coordinate planes.

The actual stress may be resolved into a simple hydrostatic pressure $p_{1}$ acting in all directions, and a simple tension $p_{1}-p_{2}$, or ${\begin{matrix}{\frac {1}{4\pi }}\end{matrix}}\mu v^{2}$,acting along the axis of stress.

Hence if $p_{xx}$, $p_{yy}$, and $p_{zz}$ be the normal stresses parallel to the three axes, considered positive when they tend to increase those axes; and if $p_{yz}$, $p_{zx}$, and $p_{xy}$ be the tangential stresses in the three coordinate planes, considered positive when they tend to increase simultaneously the symbols subscribed, then by the resolution of stresses^[1],

${\begin{array}{l}p_{xx}={\dfrac {1}{4\pi }}\mu v^{2}l^{2}-p_{1}\\\\p_{yy}={\dfrac {1}{4\pi }}\mu v^{2}m^{2}-p_{1}\ \\\\p_{zz}={\begin{matrix}{\dfrac {1}{4\pi }}\end{matrix}}\mu v^{2}n^{2}-p_{1}\\\\p_{yz}={\dfrac {1}{4\pi }}\mu v^{2}mn\ \\\\p_{zx}={\begin{matrix}{\dfrac {1}{4\pi }}\end{matrix}}\mu v^{2}nl\ \\\\p_{xy}={\begin{matrix}{\dfrac {1}{4\pi }}\end{matrix}}\mu v^{2}lm\end{array}}\!$

If we write

$\alpha =vl,\ \beta =vm,\ \mathrm {and} \ \gamma =vn$

then

$\left.{\begin{array}{ll}p_{xx}={\begin{matrix}{\dfrac {1}{4\pi }}\end{matrix}}\mu \alpha ^{2}-p_{1}\qquad &p_{yz}={\begin{matrix}{\dfrac {1}{4\pi }}\end{matrix}}\mu \beta \gamma \\\\p_{yy}={\begin{matrix}{\dfrac {1}{4\pi }}\end{matrix}}\mu \beta ^{2}-p_{1}&p_{zx}={\begin{matrix}{\dfrac {1}{4\pi }}\end{matrix}}\mu \gamma \alpha \\\\p_{zz}={\begin{matrix}{\dfrac {1}{4\pi }}\end{matrix}}\mu \gamma ^{2}-p_{1}&p_{xy}={\begin{matrix}{\dfrac {1}{4\pi }}\end{matrix}}\mu \alpha \beta \ \end{array}}\right\}\!$

(2)

Prop. III.—To find the resultant force on an element of the medium, arising from the variation of internal stress.

1. Rankine's 'Applied Mechanics,' art. 106.