278 Free Path Phenomena [CH. xv 



331. From this general equation, we can obtain the equations of 

 Chapter VII. as special cases. If we put Q = 1, so that Q= 1, the equation 

 becomes 



dv 9 , _ 9 _, 9 , _. 



(649), 



the equation of continuity already obtained in equation (352). 

 Similarly, on putting Q = mu, we obtain 



m T (I/MO) = m (vu*) + 5- (vuv) + ^- (vuw) \+ vX + AQ. . .(650). 



CLv I vCC ulJ u2 



When the molecules collide only with other molecules of the same kind, 

 AQ of course vanishes, and equation (650) now becomes identical with 

 equation (358). 



332. If we multiply equation (649) by Q, and subtract from equation 

 (648), we obtain as a new form for the general equation, 



V ~dt~ 



where 2 denotes summation with respect to the three coordinates x, y 

 and z. 



Let us now write 



u = u + u, etc. 



so that u, V, w are components of molecular velocity. Then 



u = u , u = 0, uQ = u Q+ uQ = u Q+ uQ 

 Hence 



I <*> - 1 V =$1 ^ - L (-) - 1 



We further have 



du 9 (u + u) du ' 

 where Q is now expressed as a function of u , u, etc., so that 



__ 

 \du \du / du ' 



Making these substitutions, equation (651) becomes 



dQ ^ f 3Q d / ^ , v v dQ~\ 

 v~ = ^\ -vu ^-^-(vuQ) + -X^ 

 at ax dx m 9w J 



