178 Theory of a Non-Conservative Gas [CH. vm 



If, during the motion of the molecules, two or more of the surfaces S ly S z ... 

 intersect, an obvious slight modification must be made in the conventions just 

 stated. Instead of integrating expression (425) over the surfaces separately 

 to obtain the dissipation function, we must integrate only over the single 

 closed surface formed by such as intersect. Similarly instead of integrating 

 expression (424) throughout the volumes enclosed by the surfaces separately 

 to obtain the energy, we integrate only throughout the single connected 

 volume formed by such as intersect. In this way we fail to obtain the energy 

 and dissipation functions as the sums of contributions from individual mole- 

 cules, and obtain these functions only in the aggregate, a limitation which is 

 of no importance as regards the main results to be obtained. If, on the other 

 hand, we are willing to assume that there is a definite fixed line of demarca- 

 tion between matter and ether, and that two portions of matter cannot 

 occupy the same space, then these difficulties are obviated and we can 

 associate definite terms of the energy and dissipation functions with each 

 molecule. 



The General Dynamical Theory. 



210. We can now, in the main, follow the method of Chapter V. As 

 that method was based on Liouville's Theorem ( 72), we must begin the 

 present discussion by investigating a theorem to replace this when the con- 

 servation of energy does not apply. 



As in 72, we imagine a general dynamical system, specified by the n 

 coordinates q-i,q^...q n and the corresponding momenta p lt p^...p n , and we 

 suppose all possible configurations of this system represented in a space of 

 2n dimensions corresponding to the 2w variables 



Pi,p 2 ..>p n , qi, q*..> q n ........................ (428). 



If E is the energy of the system expressed as a function of the q's and qs, 

 we have 



and the equations of motion of the system are 



fc- .............. : ..................... (430), 



^__ 



9* ' dq. dq s ' 



where E is now expressed as a function of p lt p z . . . p n , q 1} q 2 ... q n , and F is the 

 dissipation function, expressed as a function of 



<?i, <?2 <?n, q\,q* jn- 



These, therefore, are the equations of motion of the fluid in the generalised 

 space, and the elimination of the time from these equations will lead to the 

 equations of the paths or stream lines along which the fluid moves. 



