OF ENERGY IN A SYSTEM OF MATERIAL POINTS. 723 



Then by the equation of energy 



Of these quantities, A n is a function of the co-ordinates only, because E is 

 given and V is a function of the co-ordinates, A n-1 is a function of the co- 

 ordinates and a n , A n _, of the co-ordinates and of a n and a n _ l} and so on. 



Also by equation (2) 



6 -= = 



To integrate the expression 



we begin by integrating with respect to a a , thus 



|(7(6 1 )- 1 c?a 3 =|(7( ) a 1 ^)- 4 (^ i -a 2 5 )-^a 2 .................. (37), 



the limits of integration being +A Z . The result is 



-u; ........................... (38) . 



For the next integration we have 



3 (A?-aWda 3 = T ^A? ................ (39). 



Hence after r integrations, r being any number less than n, the result is 



Putting r = n 1 and remembering that p n A n * = 2E 2V, we find 



<lvf? .................... (41). 



This is the number of systems whose configuration is specified by the 

 variables 6,... 6., while the momenta may have any values consistent with the 

 equation of energy. 



912 



