592 Mr. J. H. Jeans on the Conditi 



ions 



the new space. Hence equation (16) may be replaced by 

 the condition that a shall be constant along a stream-line. 



Let p, p' be the densities at points occupied by the repre- 

 sentative points of the two component molecules, at the for- 

 mation of a double molecule, and let p, p' be the densities at 

 the points representative of the same two molecules at the 

 dissolution of the double molecule. Then pp' and pp' are the 

 two values of cr at the two ends of a single stream- 

 line in the 4n-dimensional space, and, therefore, by equation 

 (18), _ 



PP'=PP', (19) 



the same result as is obtained by Boltzmaim's well-known 

 H-theorem. 



Since the motion is dynamically reversible we may take 

 p, p to be the densities at formation, then p, p' will be the 

 densities at dissolution, and the same result holds. 



From this it follows that in equation (17) the decrease in 

 p caused by the formation of double molecules of any specified 

 kind is exactly counterbalanced by the increase caused by 

 the dissolution of double molecules of the same kind. Hence 

 in equation (17) dpjdt may be taken to be the change in p 

 caused solely by the continuous motion of the fluid, and may 

 be treated as in § 4. 



§ 11. To sum up, we have found that the equations of 

 steady motion, on the hypothesis of binary encounters, may 

 be expressed as follows : — 



(a) Throughout the 2n-dimensional space, p must be con- 

 stant along every stream-line. 



(j3) Throughout the 4w-dimensional space a must be con- 

 stant along every stream-line. 



(7) At every point on the boundary of the 4n-dimensional 

 space we must have 



<T.= Pf>'. 



To these may be added a fourth condition — 



(S) At every point on the boundary of the 2?i-dimensional 

 space (i. e. at infinity) the flow across the boundary must be 

 nil, or what is the same thing, we must have 



p=0. 



These conditions are necessary and sufficient for steady 

 motion. 



Ternary and Higlier Encounters. 



§ 12. By a simple extension of the method already ex- 

 plained the possibility of encounters of ternary and higher 



