Law of Partition of Energy. 587 



is a quadratic function of the velocities which define a phase 

 for the same particle, and of those velocities only. That is, 

 given x and y, / is a function of (y? + u 2 ) . It follows that 

 for each of Rayleiglr's particles ?r = v 2 . 



6. But if particles of one system have mass m, and 

 those of another system m', and their respective velocities 

 are u u', no conclusion can be drawn as to the relation 

 between mu 2 and m'u 12 . For we have two alternatives : 

 (1) The particles do not undergo collisions or encounters 

 with each other, whereby a particle would gain or lose 

 energy, and so change its path ; or (2) such encounters do 

 take place. If we choose alternative (1), every distribution 

 of energy between the classes m and m! is permanent. If 

 we choose alternative (2), the method fails to prove that 

 f — f, and is inapplicable. 



7. Rayleiglr's method is easily generalized as follows: — 

 Instead of a particle moving in two dimensions, the system 

 may be a particle or elastic sphere in three dimensions. 

 Or it may be defined by n generalized coordinates q x . . . q lt 

 with the corresponding momenta p { . . . p n . And we may 

 denote by f(q p)dq x . . . dp n the number of system & which 

 are in the phase (q p). It then follows on precisely the 

 same conditions as in the case of the particles in two dimen- 

 sions, that / is constant throughout a path. Also E is 

 constant. And if E be the only other constant, / is a 

 function of E, that is of V + T, say/ = 0(E). 



In this general case Rayleigh argues that T can always be 

 expressed in terms of squares of the velocities,, as 



T = A,</ x 2 + A 2 </ 2 2 + &c. 



I pass over this discussion, because it does not concern 

 translation velocities, with which alone I am dealing. Hence 

 follows, that if q x q 2 &c. define a phase of any system, 



A^! 2 = A 2 </ 2 2 = &c. 



The result is proved — if it is proved — only for velocities 

 which belong to the same system, while each system fulfils 

 Maxwell's condition. 



8. If therefore we are to prove by Rayleiglr's method 

 that, for different masses m and M, mw 2 = MlP, we nrast 

 make u and U velocities of the same system, one of a class 

 of systems to which Maxwell's principle is assumed to apply. 

 I take the simplest case. The system consists of two elastic 

 spheres whose masses are m and .M, and whose velocities 

 are u v w and U V W respectively. To simplify matters 

 further, we will assume the field of force to be uniform. 



2T2 



