711 BOLWMAXN'* THEOREM ox THE AVERAGE DISTRIBUTION 



1 1 U abewn in treatises on dynamics* that 



' (6). 



j-j- ' ' _'/'' ..(7). 



f] if* sis* \ / 



The indices r and * in this equation may be the same or different. 



Also if t' and t are the values of the time at the beginning and at the end 



of the motion, 



/.!_, . . 



dp, dt' , . , dp,' dt 



Ik ' noe ,//:= -<i< ]r < 9 > and -d-E=-dq~; < 10 )- 



In the course of our investigation we shall have to compare the product 

 ..f the differentials of the co-ordinates and momenta at the beginning of the 

 motion with the corresponding product at the end of the motion. We shall 

 write for brevity ds = dq l ...dq n for the product of the differentials of the co- 

 ordinates, and da- = dp l ...dp n for the product of the differentials of the momenta, 

 and we shall use the product ds'dsdE as a middle term in comparing ds do-' <li' 

 with tlsda-iJt. 



Now ds da' dt' = ds' ds dE2 ( d fi-. . . ^ '} .-(ID 



\2i dq n dE) 



_, (<1>\ dp' dt'\ 



where 2 ^- --p. ) 



\dq l dq n dLJ 



denotes the functional determinant 



dp; dp t ' 

 "dq n ' dE 



'//>.' dpS (12). 



dq t ' " dq n ' dE 



dq n ' dE 



* Thomson and Tail's Natural Philosophy, 330. 



