236 Prof. J. H. Jeans on Temperature-Radiation 



these properties when it starts from any configurations except 

 an infinitesimal few, then these properties must be properties of 

 the "Normal State." 



For, if not, suppose that the system tends to assume some 

 property P which is not common to the whole of the gene- 

 ralized space, or even to the whole except for infinitesimal 

 regions, but is confined to some region S of the generalized 

 space. The " representative points " which at the beginning 

 of the motion in the generalized space occupied the whole of 

 this space (or the whole of it except for infinitesimal regions), 

 must, by the end of the motion, all lie within the region S — 

 a result which would be in opposition to equation (7), 



Dt ~ U * 



11. Of the properties of the " normal state/' that one 

 which is of primary importance for the present investigation 

 is the Equipartition of Energy *. 



Suppose that the energy E of the system can be expressed 

 as a function of the Lagrangian coordinates and velocities in 

 the form 



2E = a 2 pi 2 + u 2 p 2 2 + . . . + a n p n 2 +/(0i, 6 2 > - * • On), 



where p h p 2 , » . . p n are any quantities (coordinates, velocities, 

 or momenta) and l9 2 ? • • • #» are other quantities which may 

 or may not enter into « l5 a 2 , . . . &„. The law of equipartition 

 states that if n is very great, the energy represented by any 

 very great number s of the n first terms is, in the normal state, 

 proportional to s. We assume it to be JsRT. Then, if part 

 of the dynamical system consists of matter of any kind, T is 

 the temperature of this matter f. 



12. If the system is supposed, for the moment, to consist 

 solely of a non-dissipative vibrating medium, free from dis- 

 turbance by external agencies, the whole of the energy can 

 be expressed in the form 



2E =«ipi 2 + « 2 ^2 2 +'..• + *nPn\ 



in which p 1? p 2 , . . . p n represent the normal coordinates and 

 their rates of change. Thus each separate free vibration 

 contributes two terms to the energy. In the normal state, 

 the energy of any great number s of free vibrations must 

 be sPT. 



13. From a consideration of physical dimensions, it is clear 

 that in any medium whatever, the number of free vibrations 



* The proof that this is a property of the Normal State is purely 

 algebraic in its nature : see ' The Dynamical Theory of Gases,' p. 67. 

 t Jeans, 'Dynamical Theory of Gases,' §§ 77, 124. 



