Partition of Kinetic Energy. Ill 



are an infinite number) which allows T to be expressed in the 

 form (35). 



In the case where the " system " consists of a single 

 particle, (35) is justified by any system of rectangular coor- 

 dinates ; and although we are not bound to use the same 

 system for different positions of the particle, it would conduce 

 to simplicity to do so. If the system be a rigid body, we may 

 measure the velocities of the centre of inertia parallel to three 

 fixed rectangular axes, while the remaining momenta refer to 

 rotations about the principal axes of the body. If Maxwell's 

 assumption hold good, a permanent distribution is such that 

 in one, or in any number of positions, the mean energy of 

 each rotation and of each translation is the same. And under 

 the same restriction a similar assertion may be made respecting 

 the time-averages for a single rigid body. 



There is much difficulty in judging of the applicability of 

 Maxwell's assumption. As Maxwell himself showed, it is easy 

 to find cases of exception ; but in most of these the conditions 

 strike one as rather special. It must be observed, however, 

 that if we take it quite literally, the assumption is of a severely 

 restrictive character; for it asserts that the system, starting 

 from any phase, will traverse every other phase (consistent 

 with the energy condition) before returning to the initial phase. 

 As soon as the initial phase is recovered, a cycle is estab- 

 lished, and no new phases can be reached, however long 

 the motion may continue. 



We return now to the question of the distribution of 

 momenta among the systems which occupy a given configu- 

 ration, still supposing the coordinates so chosen as to reduce 

 T to a sum of squares (35) . It will be convenient to fix our 

 attention upon systems for which E lies within narrow limits, 

 E and E-fdE. Since E is given, there is a relation between 

 pu TP^i ' • • Vm an d we may suppose p n expressed in terms of 

 E and the remaining momenta. By (35) 



p n dp n = dT — dE, 



since the configuration is given, and thus (33) becomes 



f(®)dE.dq 1 ...dq n . I >-id Fl ...d Fn _ 1 . . . (37) 



i-pose the latter 

 r e to consider is 



d Pl dp 2 . . . dp n _ x 



^ { 2T-p^-p^-...-pi_ l y • ■ • W 



in which T, being equal to E— V, is given. For the moment 

 w e may suppose that 2T is unity. 



For the present purpose the latter factors alone concern us, 

 so that what we have to consider is 



