284 Free Path Phenomena [CH. xv 



If we substitute for -~ from equation (660) and further simplify, we find 



A / 2 2\ A 1 , > 



Aw (u* + v 2 + w 2 ) = 4>qu U -- 1 -5-! ' + =-! ' -f 

 /te * \dx dy 



dv du Q \ fdw du \ dq 



-^ + - 2 2w - + ^- + 5 9a ...... 



dt dz / ox 



THE DYNAMICS OF COLLISIONS. 



340. We now proceed to calculate the values of AQ which are required, 

 directly from the dynamics of collisions, assuming the law of force between 

 molecules to be that of the inverse fifth power of the distance. 



With a view to a subsequent investigation of diffusion, we shall suppose 

 there to be gases of two kinds, the masses of the molecules being m 1 and m^ 

 respectively. 



Let us consider a collision between two molecules of different kinds. 

 Their mutual potential energy at distance r is 



r K 



I m^m^~ dr, 



* er\ * 



where Km l replaces our former p for the first kind of molecule, and Km 2 

 replaces p for the second. Integrating, we get 



rr 



= -m l m 2 (665). 



. -' 4>r* 



If at any instant x 1 , y lf z^ are the coordinates of the first molecule, and 

 X 2> 2/2> z 2 those of the second, the equations of motion of the first molecule are 



K . -TT . 



[QG\ ~ ^ ~f" TTlj^i. j GuC.j 



while those of the second are 



m 2 x 2 = ^ [- m 2 X, etc. 



ox 2 



From these we obtain 



v N ) = wl __ m _ ...(666). 



t*/2/ l*"i-- 2^__ \ / 



Let us write x 2 x^ = x, etc., so that x, y, z are the coordinates of the 

 second molecule relatively to the first. Then 



r 2 = x 2 + y 2 + z 2 , 

 so that since <!> is a function of r, 



d<&_ 8<I>__9<& 



3#j dx 2 dx ' 



