8 CONSERVATION OF 



will represent algebraically the decrease of the number of 

 systems within the limits due to systems passing the limits p^ 

 and PI'. 



The decrease in the number of systems within the limits 

 due to systems passing the limits q and <?/' may be found in 

 the same way. This will give 



for the decrease due to passing the four limits p, p^", <?/, q^'. 

 But since the equations of motion (3) give 



^ + ^ = 0, (16) 



dpi dq l 



the expression reduces to 



(dD dD \ 

 d^ pi + d^ ?i ) * * dyi *-* (17) 



If we prefix 2 to denote summation relative to the suffixes 

 1 ... n, we get the total decrease in the number of systems 

 within the limits in the time dt. That is, 



T~ i* 



-dDd Pl ... dp n d Sl ... dq n , (18) 



d~ ^ ) dpl ' ' ' d d ' " d dt ~ 



or 



where the suffix applied to the differential coefficient indicates 

 that the JP'S and <?'s are to be regarded as constant in the differ- 

 entiation. The condition of statistical equilibrium is therefore 



If at any instant this condition is fulfilled for all values of the 

 p's and <?'s, (dD/dt} ptg vanishes, and therefore the condition 

 will continue to hold, and the distribution in phase will be 

 permanent, so long as the external coordinates remain constant. 

 But the statistical equilibrium would in general be disturbed 

 by a change in the values of the external coordinates, which 



