22, 23] The Statistical Method 21 



If we now substitute this value for df'/dt in equation (17) we obtain 

 (cf. equation 15) 



+ lo g/') (ff ~ff) V - 2 cos e dudvdwdu'dv'dw'dco, 



an equation which is of course the same as (15) except that accented and 

 unaccented symbols are interchanged. If we add together the two values 

 for dH/dt which have been obtained, we have 



This equation expresses dH/dt as the sum of a number of contributions, 

 one from every possible class of collision. The typical class is taken to be 

 class a, in which 



u, v, w, u', v', w', 



become changed into 



u, v, w, u', v', w'. 



If we use the same equation, but take as the typical collision one of class 

 /3, in which 



u, v, w, u', v', w', 



become changed into 



u, v, w, u', v', w', 



we obtain, as a still different form for dH/dt, 



= i v fffl I ((( (2 + log //')( ff - ff) Va* cos e dudvdwdu'dv'dw'dco. . .(19). 



fj-t & I \ GJ J / \JJ J J ' \ ' 



JJJJJJJJ 



Equation (9) enables us to replace the product of the first six differentials 

 on the right-hand of this equation by dudvdwdu'dv'dw' and if we add this 

 modified value of dH/dt to that given by equation (18) we obtain 



d ** = lv (I [((((( (logff - log/7) (if ~ //') Va * cos dudvdwdu'dv'dw'dco 

 dt JjJJJJJJ 



(20). 



Now (logff logff) is positive or negative according as ff is greater 

 or is less than ff' and is therefore always of the sign opposite to that of 

 ff ff. Hence the product 



(log//'- log//') (//' -//'), 



if not zero, is necessarily negative. Since Fcos 0, the relative velocity along 

 the line of centres before impact, is necessarily positive "for every type of 

 collision, it follows that the integrand of equation (20) is always either 

 negative or zero. Hence equation (20) shews that dH/dt is either negative 

 or zero. 



