necessary for Equipartition of Energy. 589 



If we exclude (as we legitimately may) the case of systems 

 which remain permanently at rest in an equilibrium con- 

 figuration, it follows that we must have — —^ different from 



Qt 



zero. Hence equation (13) assumes the form 

 and the most general solution is 



P = <M^l> +S> • • • • ^2n-l) . . . (14) 



in which cj) is the most general function of the (2n — 1) variables. 

 § 5. If the systems are not subject to external disturbance 

 there is little more to be said. Of the 2n — 1 quantities 

 t/tj, ^r 2 , • • • ^2n~i. one (say ^) may, without loss of gene- 

 rality, be taken to be identical with E : the remaining (2n — 2) 

 yjrs are necessarily functions of quantities other than E. 

 Thus it appears that although 



P = *(B), 



(the solution leading to equipartition of energy), is a parti- 

 cular solution of the general equations, it is by no means the 

 only solution. In other words, equipartition, although pos- 

 sible, is not necessary. This is as it should be, for Maxwell's 

 condition of continuity of path is not satisfied. 



§ 6. Maxwell and Rayleigh now suppose that the system 

 is subject to certain external agencies, and postulate that 

 these agencies shall be such that by them each system is 

 made to pass through all phases which are consistent with 

 the conservation of energy. From the point of view of this 

 paper, they postulate that the elements of fluid are moved out 

 of their stream-lines, and this in such a way that every 

 element is made to pass over the whole of the particular 

 surface 



E = constant 



to which it initially belongs. If this postulate is granted 

 their proof is unassailable, but they do not prove that the 

 postulate is true in the case of any single system, and it 

 seems to the present writer that for a large class of natural 

 systems the postulate cannot possibly be true. 



Consider, for instance, the case of a particle moving upon 

 a horizontal plane, in which the disturbing influence is 

 supplied by a system of rigid barriers. As a preliminary, 

 suppose these barriers replaced by a continuous field of force, 

 such that the potential becomes infinite over certain lines 

 a 3 h y c, . . . in the plane. If this potential is included in the 



