654 Mr. G. H. Livens on the 



conditions are satisfied if we can determine a positive number 

 /Lt x > such that 



T n 1 -^^ n 1+ t J - 



A x being a finite constant. Now, however, we see that, 

 since < fi x <co , the possible range of values of <j> n 2 and 

 n 2 (p n 2 for all finite values of n (i. e. practically in all attainable 

 regions of the spectrum) is given by the conditions 



0^^ 2 <~, 0^</> n 2 <-, 

 r n T n 



and this will be found absolutely sufficient for our purposes. 

 We need not, therefore, proceed with any further tests. 



Physically, various interpretations may be placed on this 

 latter restriction in the values of the coefficients. The simplest 

 and most obvious is that the capacities or extents of the 

 coordinates representing the different types of oscillation 

 are different, being in fact for any coordinate inversely pro- 

 portional to the integer corresponding to it, or directly 

 proportional therefore to the wave-length of the vibration 

 associated with it. This would also imply that in any dis- 

 tribution of energy, the chance of a certain specified amount 

 of energy getting into, say, the coordinate <f> n is proportional 

 to \„, and this is probably the most general form which can 

 be given to the restriction, which will be equally applicable 

 to any more complicated system, being entirely independent 

 of the particular conditions of the problem under immediate 

 discussion*. 



Here, then, we have the fundamental modification in the 

 general theorem of the partition of energy applied to radia- 

 tion, or in fact to any problem of the nature of those examined 

 by Rayleigh and Jeans. But before attempting a reformu- 

 lation of the theorem on this basis we must first choose a 

 standard capacity for a known type of coordinate with which 

 all the others can be compared. The obvious standard is 

 provided in the type of coordinate discussed in the dyna- 

 mical theory of gases: since, however, these coordinates may 

 be regarded as corresponding to oscillations of infinitely 



* This condition means that the mere chance of a specified collocation 

 among any set of coordinates of a particular type must be properly 

 ■weighted before being counted in the whole system. Thus if X n is the 

 number of ways in which an event can occur in connexion with the 

 coordinates of type <fi n , then the proper chance of its occurring in the 



general system is not simply X» but X a n . 



