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 acting in all directions, and a simple tension ,
or ,acting along the axis of stress.
Hence if , , and be the normal stresses parallel to
the three axes, considered positive when they tend to increase
those axes; and if , , and 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],
If we write
then
(2)
Prop. III.—To find the resultant force on an element of the medium, arising from the variation of internal stress.