586 Mr. S. H. Burbury on the 



left to itself, will sooner or later pass through every phase 

 consistent with the conservation of energy (Maxwell's paper, 

 p. 548). It must therefore sooner or later return to its first 

 state, the motion being in fact cyclic. Further the motion 

 is evidently reversible. 



5. The line of argument is sufficiently shown in Eayleigh's 

 treatment (Phil. Mag. January 1900, pp. 102-107} of the 

 system of particles moving in two dimensions in a field of 

 force. He defines as follows : — (1) If x, y denote the 

 coordinates, u, v the component velocities of a particle, then, 



when x y u v lie within the limits x . . , x -\- dx . . . 



v ■. . . v + dv, the particle is in the phase (# y u v). 

 (2) f{x y u v) dx dy du do is the number of particles which 

 at any instant are in the phase (x y u v). The path of any 

 system, and of any one of Rayleigr?s particles as a particular 

 case, is the series of successive states through which the 

 system passes in unguided motion with total energy constant. 

 The path in which the total energy is E may be called the 

 path E. Then f argues Rayleigb, the particles which at a 

 given instant (£ = 0) are in the phase (x y u v) are the 

 identical particles which will at time t be in the phase 

 {x' y' v! v 1 ), and no other particles will at time t be in the 

 last mentioned phase. Therefore 



f(x y u v)dx dy du dv = f{x' y f u v'}dx 'dy'du' 'dv' '. 



But by a known theorem, which owes much to Rayleigb,. 



dx dy du dv = dx' 'dy 'du 'dv\ 

 Therefore 



f{x y u v) =/V y'u f v), or shortly / = /'. 



That is, the number of particles which initially are in the 

 phase (x y u v) is equal to the number which after time t 

 will be in the phase (%' y' u' v'). If therefore there be at the 

 initial instant the same number of particles in every phase 

 of the path E ; the motion will as regards these particles be 

 stationary. 



In stationary motion, then, /is constant for all phases on 

 the same path. E- is also constant for all phases on the 

 same path, and we will assume for the present with Rayleigh 

 that it is the only other constant. We will now assume 

 that there are many particles on the same path for each of 

 which E is constant, and many paths for each of which/ and 

 E are constants, and no other thing is constant; but E, and it 

 may be / will vary between one path and another. It 

 follows that / is a function of E. Now E = V + T } where 

 V is a function of the coordinates, and T, the kinetic energy, 



