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V. Equipartition of Energy and Radiation Theory. 

 By W. F. Gf. Swann, D.Sc, A.R.C.S.* 



Introduction. 



IT is usually maintained that the theorem of equipartition 

 of energy is a necessary consequence of the application 

 of the so-called "ordinary dynamics" to a system, and 

 further, that an application of the principle to the problem 

 of radiation necessitates the conclusion that Rayleigh's 

 formula ~E\d\ = 87rRTX~ 4 ^A, ought to be true. Though 

 much has been written in this connexion, it has always 

 seemed to the present author that there are several points 

 which require a somewhat closer scrutiny than is usually 

 given, and which concern themselves chiefly with the ques- 

 tion as to how far the results to which the theorem leads 

 have really any connexion with the physical phenomena 

 which they are generally considered to represent. The 

 present paper forms a discussion of some of the points in- 

 volved in this connexion. Since much of what is here 

 written involves ultimately a careful distinction between 

 the results to which the theorem apparently leads and those 

 to which it really does lead, it will perhaps not be super- 

 fluous to commence by giving a brief survey of the essentials 

 involved in the theorem. 



Remarks on the Proof of the ± heorem. 



In the first place, it is to be noted that the theorem itself 

 is as follows. Suppose that the system is specified by a large 

 number n of coordinates and a large number, n, of momenta, 

 and that a certain quantity which is constant (the energy), 

 is a homogeneous quadratic function of the coordinates and 

 momenta of such a kind that in it there are no terms in- 

 volving products of coordinates and momenta. Then, if we 

 separate out from the energy all those squared terms in- 

 volving coordinates and momenta which do not appear in the 

 energy otherwise than in the said squared terms, and if from 

 these squared terms we pick out two sets P and Q each 

 containing a large number of terms and take the average 

 value in each case, we shall find that the average for the 

 P group is the same as that for the Q group. Each group 

 may if we choose involve some terms from the potential 

 energy and some from the kinetic. The terms " coordinate - " 



* Communicated by the Author. 



