applied to Magnetic Phenomena. 167 
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. I1.—If the direction-cosines of the axes of the vortices 
with respect to the axes of x, y, and z be-/, 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, acting in all directions, and a simple tension p, —po, 
or rae acting along the axis of stress. 
Hence if py», Pyy, and pzz be the normal stresses parallel to 
the three axes, considered positive when they tend to increase 
those axes; and if p,,, Pex, and p,, 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 
Psa= Ty hh Pr 
1 Qin 
Pyy= nee 1 ay 
i 2,2 
pe ge UE 
Pyz= ipeemn 
1 2 
Pz 7 nl 
1 
Pay = Fe poorlm. 
If we write 
| a=vl, B=vm, and y=v, 
then 
! ] 
Pes= go MO — Py Pyz= G BBY | 
oe Cie 
Py = rate —Pi Paa= Ge we} (2) 
ae. a Sate 
Pas eehY TP Pay= go Mob. j 
Prop. I11.-—To find the resultant force on an element of the 
medium, arising from the variation of internal stress. 
* Rankine’s ‘ Applied Mechanics,’ art. 106. 
