of Molecular Vortices. 213 



trifugal force will cause a pressure to be exerted in all directions 

 against the inside of the vessel. To determine the mean inten- 

 sity of that pressure irrespectively of periodical variations, con- 

 ceive the contents of the vessel to be divided into two parts by 

 an imaginary plane, and consider what will be the mean inten- 

 sity of the force with which the circulating streams tend to drive 

 asunder the portions of matter at the two sides of that plane. 

 The 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 6 for three planes at right 

 angles to each other must be = 1 ; therefore the mean value 



of cos 2 is = 5-*.; and, finally, the mean intensity of the centri- 

 fugal pressure is given in absolute units per unit of area by the 

 equation 



"<T CD 



§ 4. Energy of Steady Circulation compared with Centrifugal 

 Pressure. — The actual energy t of the steady circulation in a 

 unit of volume is expressed in absolute units of work as follows : — 



pw 2 

 ~2~ 



; (2) 



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

 result : — 



pw 2 2 pw 1 

 P = 3 "g" 2 5 ■■■■■<?) 



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 a 

 unit of volume. This is one of 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 



* 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 is -. 

 o 

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



