ON PHYSICAL LINES OF FORCE. 457 



where p is the mean density, and C is a numerical quantity depending on the 

 distribution of angular velocity and density in the vortex. In future we shall 



write *- instead of Co, so that 



4?r 



where ft is a quantity bearing a constant ratio to the density, and v is the 

 linear velocity at the circumference of each vortex. 



A medium of this kind, filled with molecular vortices having their axes 

 parallel, differs from an ordinary fluid in having different pressures in different 

 directions. If not prevented by properly arranged pressures, it would tend to 

 expand laterally. In so doing, it would allow the diameter of each vortex to 

 expand 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 I, m, and n, to find the normal and tangential 

 stresses on the co-ordinate planes. 



The actual stress may be resolved into a simple hydrostatic pressure p l acting 

 in all directions, and a simple tension p l p,, or ptf, acting along the axis 



of stress. 



Hence if p^, p m> and p a be the normal stresses parallel to the three axes, 

 considered positive when they tend to increase those axes ; and if p^, p lx , and 

 Pxy be the tangential stresses in the three co-ordinate planes, considered positive 

 when they tend to increase simultaneously the symbols subscribed, then by 

 the resolution of stresses*, 



1 



Pm-fr 



i 



^ 



i 



* Rankine's Applied Mechanics, Art. 106. 

 VOL. I. 58 



