102 The Laiv of Distribution [CH. v 



It is not necessary to make any assumptions as to the form of f (TST}, but 



/go 

 f(&) dm 1, any constant multiplier being absorbed 

 ) 



into the constant A, so that as in equation (30), 



. /It 3 1 1 i s 



V 7T 3 



The number of collisions per unit time per unit volume for which V lies 

 between V and V+ dV, is from expression (45), 



3/vij3 



li.._T.^-rro i-ii- (244) 



Hence the mean value of V' 2 averaged over all collisions is 



4 



f- 



Jo 



'o 



where c 2 is now the mean value of c 2 averaged over all the molecules of the 

 gas (cf. equation (161)). 



The mean value of -5r 2 + c7' 2 averaged over all collisions is clearly 2w-, 

 where or 2 is the average value of -cr 2 taken over all the molecules of 

 the gas. 



Substituting these values for {&'* + -or' 2 } and {F 2 } in equation (242), we 

 obtain 



(246). 



Let us write 



H = |m& 2 'BT 2 , 



K = |mc 2 , 



so that K is the mean energy of translation, and H is the mean energy of 

 rotation, excluding rotation about the axis of symmetry. Then equation (246) 

 can be written 



115. There are v molecules per unit volume, and, also per unit volume 

 there are, by formula (46) 



collisions per unit time. Hence summing equation (247) over all the 

 collisions which occur in time dt, we find that the change in the sum of the 

 values of c 2 for all the v molecules, which is produced by collisions in 

 time dt, is 



