OF MOLECULAR VORTICES. 559 



plane will cut the streams that flow across it, some normally, others obliquely ; 

 and the tangents to those streams will have all possible directions relatively to a 

 normal to the plane, subject to the condition, in the case of isotropic action, 

 that the mean value of cos 2 6 must be the same for all positions of the plane. 

 But the sum of the mean values of cos 2 9 for three planes at right angles to each 



other must be = 1 ; therefore the mean value of cos 2 # is =,;* and finally, the 



mean intensity of the centrifugal pressure is given in absolute units per unit ot 

 area, by the equation, 



P = V ( L ) 



§ 4. Energy of Steady Circulation compared with Centrifugal Pressure. — 

 The actual energyf of the steady circulation in an unit of volume, is expressed 

 in absolute units of work, as follows :- — 



*£ • • (2); 



which, being compared with equation (1), gives the following result: — 



?? _^_ 2 P^ toy 



that is to say, the intensity of the centrifugal pressure on the unit of area is two- 

 thirds of the energy of the steady circulation in an unit of volume. This is one ot 

 the propositions of the paper of 1849-50, p. 151, eq. v.; but it is now shown to be 

 true, not merely, as in the former paper, for molecular vortices arranged in a 

 particular way, but for molecular vortices arranged in any way whatsoever, 

 provided their action is isotropic, and their mean velocity uniform. 



A similar proposition has been proved by Waterston, Clausius, Clerk 

 Maxwell, and others, for the pressure produced by the impulse of small particles 

 flying about in all directions within a closed vessel, and rebounding from its 

 sides. 



§ 5. Vortices with Heterotropic Action. — It is conceivable that in solid bodies, 

 molecular vortices may be so arranged as to produce centrifugal pressures of 

 different intensities in different directions. In such cases, it is to be recollected 

 that the sum of the mean values of cos 2 6 for the obliquities of any set of lines to 

 any three planes at right angles to each other is = 1 ; whence it follows, that if 

 //, //', and p'" be the mean intensities of the centrifugal pressures in any three 

 orthogonal directions, we have 



p' + p" + p'" = pw* (4); 



* There is a well-known integration by which it is easily proved, that for a number of 

 directions equally distributed round a point, the mean value of cos 2 6 is - . 



o 



f Called by Thomson and Tait the " Kinetic Energy." 



