88 The Law of Distribution [CH. v 



where \, /JL, v, as in Chapter II., are the direction-cosines of the line of 

 centres. The value of dS corresponding to collisions of the type specified 

 will clearly be 



.. d% t m du.'dv'dw'o*d& ...... (211). 



A 



If we substitute the value for =r from equation (210) and for dS from 



Oli 



equation (211) into expression (209), and equate the value so obtained to 

 expression (208), we obtain, after dividing through by common multipliers, 



Pipi=p2 ................................. (212). 



This equation may be regarded as giving the density p 2 at all points on 

 the surface S in the 4w + 8 dimensional space, in terms of the densities at 

 points in the 2w + 3 dimensional space. 



99. Since the systems represented in the 4n + 9 dimensional space are 

 not acted upon by any external forces, we have, as in 72 (equation (131)) 



where DjDt denotes differentiation with respect to the time as we follow the 

 fluid in its motion. We may however write 



Dt " dt dt ds ' 



where dp 2 /dt denotes the rate of increase at a fixed point, ds/dt is the velocity 

 along a stream line, and dp 2 /ds is the increase of p 2 per unit length along 

 the stream line. If we combine equation (213) with the condition for a 

 steady state (equation (205)), equation (214) reduces to 



so that p z must be constant along every stream line. 



Let p lt pi be the densities in the %n + 3 dimensional space, at points 

 occupied by the representative points of the two component molecules at the 

 formation of a double molecule, and let p lt p/ be the densities at the points 

 representative of the same two molecules at the dissolution of the double 

 molecule. Then by equation (212) p 1 p 1 ' and jjjpY are the two values of p 2 at 

 the two ends of a single stream line in the 4<n + 9 dimensional space, and, 

 therefore, by equation (215), 



Pi/>/ = Pi/5i' .............................. (216), 



the same result, it will be noticed, as is obtained by the .ff-theorem of 

 Chapter II. (cf. equation (21)). 



Since the motion is dynamically reversible we may equally well take p lt J5,' 

 to be the densities at formation, then p 1} p/ will be the densities at dissolu- 

 tion, and the same result holds. 



