456 ON PHYSICAL LINES OF FORCE. 



PROP. I. If in two fluid systems geometrically similar the velocities and 

 densities at corresponding points are proportional, then the differences of pres- 

 sure at corresponding points due to the motion will vary in the duplicate ratio 

 of the velocities and the simple ratio of the densities. 



Let I be the ratio of the linear dimensions, m that of the velocities, 

 n that of the densities, and p that of the pressures due to the motion. Then 

 the ratio of the masses of corresponding portions will be Pn, and the ratio of 

 the velocities acquired in traversing similar parts of the systems will be m ; 

 so that Fmn is the ratio of the momenta acquired by similar portions in 

 traversing similar parts of their paths. 



The ratio of the surfaces is Z 1 , that of the forces acting on them is l*p, 



and that of the times during which they act is ; so that the ratio of the 



m 



impulse of the forces is , and we have now 



m 



Pmn-fe, 



TO 



or m'n = p ; 



that is, the ratio of the pressures due to the motion (p) is compounded of 

 the ratio of the densities (n) and the duplicate ratio of the velocities (m 1 ), and 

 does not depend on the linear dimensions of the moving systems. 



In a circular vortex, revolving with uniform angular velocity, if the 

 pressure at the axis is p t , that at the circumference will be p 1 =p + ^pv', where 

 p is the density and v the velocity at the circumference. The mean pressure 

 parallel to the axis will be 



If a number of such vortices were placed together side by side with their 

 axes parallel, they would form a medium in which there would be a pressure 

 ]> t parallel to the axes, and a pressure p 1 in any perpendicular direction. If the 

 vortices are circular, and have uniform angular velocity and density throughout, 

 then 



If the vortices are not circular, and if the angular velocity and the density 

 are not uniform, but vary according to the same law for all the vortices, 



