Page:Philosophical magazine 21 series 4.djvu/187

Rh 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 ,

$$ \begin{array}{l} p_{xx} =\dfrac{1}{4\pi}\mu v^2l^2-p_1 \\ \\ p_{yy} = \dfrac{1}{4\pi}\mu v^2m^2-p_1\ \\ \\ p_{zz} = \begin{matrix}\dfrac{1}{4\pi}\end{matrix}\mu v^2n^2-p_1 \\ \\ p_{yz} = \dfrac{1}{4\pi}\mu v^2mn\ \\ \\ p_{zx} = \begin{matrix}\dfrac{1}{4\pi}\end{matrix}\mu v^2nl\ \\ \\ p_{xy} = \begin{matrix}\dfrac{1}{4\pi}\end{matrix}\mu v^2lm \end{array} \!$$

If we write $$\alpha=vl,\ \beta=vm, \ \mathrm{and} \ \gamma=vn$$ then {{MathForm2|(2)|$$ \left. \begin{array}{ll} p_{xx} = \begin{matrix}\dfrac{1}{4\pi}\end{matrix}\mu \alpha^2-p_1 \qquad & p_{yz} = \begin{matrix}\dfrac{1}{4\pi}\end{matrix}\mu \beta \gamma \\ \\ p_{yy} = \begin{matrix}\dfrac{1}{4\pi}\end{matrix}\mu \beta^2-p_1 & p_{zx} = \begin{matrix}\dfrac{1}{4\pi}\end{matrix}\mu \gamma \alpha\\ \\ p_{zz} = \begin{matrix}\dfrac{1}{4\pi}\end{matrix}\mu \gamma^2-p_1 & p_{xy} = \begin{matrix}\dfrac{1}{4\pi}\end{matrix}\mu \alpha \beta\ \end{array} \right\} \!$$}}

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