﻿384 Mr. G. H. Livens on the Law of 



conceivable mechanical scheme which can lead to Planck's 

 law is one in which elements o£ energy of a definite size play 

 an important and essential part. 



In a previous paper on the statistical theory of radiation, 

 I have attempted to show that the application of certain 

 tentative ideas in the statistical theories of both Planck and 

 Jeans enable us to arrive at Planck's formula, but without 

 any very unnatural assumptions. The ideas there tenta- 

 tively introduced, which may appear rather crude and un- 

 certain, involve a modification, not of our ordinary mechani- 

 cal notions but merely of the one additional fundamental 

 principle * that all dynamical coordinates which enter into 

 the usual expression for the energy of any system are 

 necessarily equally probable as receptacles of energy ; this 

 modification f applies most particularly to those coordinates, 

 infinite in number, introduced by the use of Fourier's series 

 and usually expressed by the coefficients in these series. 



It may well be asked, why is it that these coordinates are 

 not equally probable, seeing that they all enter into the 

 energy function in precisely the same way ? But it may be 

 equally well asked whether the mode o£ appearance in the 

 energy function, or more generally in the dynamical theory, 

 is sufficient criterion for the probability of the coordinate as 

 a general receptacle of energy in any statistical problem. 

 Besides the assumption that all the coordinates are alike in 

 this one respect is nothing if not impertinent, seeing that it 

 implies a good deal more knowledge of the higher order 

 coefficients in the Fourier series than any mathematical 

 theory would allow. And after all the coefficients in the 

 Fourier series are at least differentiated from each other by 

 their places in the series, and I see no reason to suppose that 

 their differences end at this. 



If we are prepared to adopt such a modification of the 

 theoretical bases of the statistical method in mechanics, it 

 can be shown that the form of the theory which is to agree 

 with experience is at least not inconsistent with our usual 

 Newtonian mechanical notions. 



We assume quite generally that the state of any dynamical 

 system is determined at any instant by its state at the 

 previous instant, and that this state can be specified by the 

 value of certain definite generalized coordinates. The motion 



* It is very important to recognize that this principle underlies all 

 deductions of the equi-partition law, and is additional to the mechanical 

 principles involved. 



t A similar modification is implied in Planck's theory, but a dyna- 

 mical reason, involving- a discrete atomic structure for the energy, is 

 assigned for it. 



