necessary for Kquipartition of Energy. 587 



of these coordinates at any subsequent instant, so long as the 

 system is not acted upon by any forces which are not included 

 in the energy-function. In this way we find a "path" in 

 the generalized space which is described by the system in 

 question. In this way we may map out the whole of our 

 generalized space into " stream-lines." It is obvious that 

 there will be one, and only one, stream-line through every 

 point of this space, and that stream-lines which are adjacent 

 at one point remain adjacent throughout their whole course. 

 The motion of the representative points may, therefore, be 

 replaced by a hydrodynamical motion, this motion being 

 continuous as regards both space and time. 



Let us denote differentiation with respect to a fixed point 

 in this space bv d/dt, that with respect to a moving element 

 by D/Dt. 



The velocity at any point is, under all circumstances, a 

 function of the coordinates only. The necessary and sufficient 

 condition for a steady state is therefore 



£-0 w 



From the hydrodynamical equation of continuity, 



dt~ Dt (2n) dgd*' x 



where £ is any one of the 2n coordinates of scheme (3), and 

 the summation extends to all. 



Xow the molecules which at any given instant occupy the 

 element of volume dq x dq 2 . • • will be precisely those which 

 at some subsequent instant will occupy some other element 

 dqi dq 2 f • • ., and. by a known theorem, 



clj l dq 2 ... —dqi dq. 2 f (6) 



In terms of our present notation equation (6) may be 

 expressed concisely in the form 



sr- p> 



Hence from equation (5) the condition for a steady state is 

 seen to be 



2 'dp'dS ={) ( - 



(2n)d£a* [J 



§ 4. Let the total energy of the system, supposed expressed 

 in the Hamiltonian form, be denoted by E. The energy in- 

 cludes the potential and kinetic energies of the system. It 



2Q2 



