370 Dr. G. J. Stoney on the 



time-integrals of the several terms of this expression are equal, 

 and that the average value of T is 1/Nth of the total energy 

 in the gas, N being the number of molecules ; in other words, 

 that the molecule exhibits one-sixth of its share of the total 

 energy in each of the following ways, viz. : — 1°, in its 

 journeyings east and west ; 2°, north and south ; 3°, up and 

 down ; 4°, 5°, 6°, in spinning on each of its three principal 

 axes. To simplify the conception as much as possible, we may 

 suppose our model of a molecule to be a smooth ellipsoid of 

 uniform density. 



Let us next represent the molecules by ellipsoids of revolu- 

 tion. The full expression for the energy of such a body is — 



T = i[M( W 2 + V 2 + ^) + Aa) 1 2 + B(a) 2 2 + c t )3 2 j], . (1) 



but for the purposes of the theorem it may * be reduced to — 



T / = i[M(^ + ? ; 2 + w; 2 ) + B(a) 2 2 + 6)3 2 )],. . . (2) 



since rotation round the axis of symmetry can neither be set 

 going by the kinetic energy with which the molecules collide, 

 nor if maintained in any other way can it in the least influence 

 the values of u, v, and ic. 



The model when dealt with in this way is instructive, be- 

 cause it illustrates, as Mr. Bryan has pointed out (Proceedings 

 of Cambridge Phil. Soc. of Nov. 2b', 1894), how a motion may 

 exist in a molecule which does not come under the theorem, 

 and which therefore may be going on with any amount of 

 energy. 



To describe the situation in other and very convenient 

 language, the motions of translation of the ellipsoids of revo- 

 lution between their encounters are A events, and the energy 

 of these events, viz. : — 



Average value of S^M(w 2 + v 2 + iv 2 ) 



* It is very necessary to bear in mind that, so far as the theorem is 

 concerned, it is optional with us whether we make this reduction from 

 6 to 5 terms or not. But if we retain the term | Aw,' 2 , we must remember 

 that the theorem only deals with motion subsequent to an initial condition 

 in which an equal partition of energy had been made among the terms. It 

 states that if this condition existed initially, it will continue subsequently ; 

 and this is evident so far as the rotation round the axis of sjmimetry is 

 concerned, since, if this rotation were once set up, it would continue un- 

 changed. For some purposes we must retain the six terms, e.g. in order 

 to see how the transition from the case of a solid of revolution to the case 

 of a solid which differs but little from a solid of revolution, takes place 

 without an abrupt change in the effect predicted by the theorem, so as to 

 comply with the general principle of continuity in dynamics. 



