180 Theory of a Non-Conservative Gas [CH. vni 



We now have 



s = l 



s=nt=n 



by equation (429), expressing A^ as a function of the qs only. Since 

 E and .F are both positive for all values of the variables it follows that AF is 

 positive throughout the space, and hence from equation (435) it is clear that 

 as we follow the motion of any element, p continually increases. Also, as we 

 follow the motion E decreases, so that the motion of the points in our 

 imaginary space is one of concentration about minimum values of E and, in 

 particular, about E = 0. 



213. The rate of change in E as we follow any element in its motion is 

 of course 



. 



Dt , \dq s dt dp s dt 



s=n $E dF 



= Z ^ ^-r, by equations (430) and (431) 

 *=i ops uQs 



=n 9j p 



= - 2 p s ^- 

 s=i dp* 



= -2F ............................................. (437), 



since F is a homogeneous quadratic function of the momenta p lt ... p n . 

 From equations (435) and (437) it appears that as we follow the motion of 

 any element 



al 'tr .............. - ................. < 438 >' 



and D may now be regarded as the symbol of differentiation along a stream 

 line in the imaginary space. The equation just obtained has therefore the 

 integral 



JAP 



r-jjr ^ ................................. ( 439 >> 



where the integration is taken along any stream line, and p is the value of p 

 at the lower limit of integration. 



214. Equation (439) is an equation between p and the coordinates in 

 the imaginary space : it does not involve the time. Hence it follows that 

 provided the initial distribution of points in the space is such as to satisfy 



