Statistical Relations of Radiant Energy. 655 



long period, the corresponding number for their extent or 

 capacity in the above wave-length comparison scale is in- 

 finite (go ). Thus if we are to take these ordinary molecular 

 coordinates as of* unit extent, then the extent o£ the coordinate 

 (j>n is oi\ n , u being a number to be made ultimately zero, but 

 which will depend essentially on the choice of a scale for the 

 statistics to be ultimately employed. 



It may be thought that such a conclusion as this is un- 

 tenable, inasmuch as it necessitates that the chance of a 

 given complexion in the radiation coordinates is vanishingly 

 small ; this is so, but it is to be remembered that the number 

 of coordinates of oscillatory type is infinitely large. In fact 

 the two systems of coordinates, dynamical and oscillatory, 

 are of a fundamentally different character. In gas theory 

 we have to deal with a very large but certainly finite number 

 of coordinates, and the chance of certain states in them is 

 certainly calculable ; but in such problems as the above we 

 have always an infinity of types of coordinate to deal with, 

 so that the chance of any condition among them is absolutely 

 incalculable, unless there exists, as above, some sort of con- 

 vergence of the chance of the condition in the various types 

 to a zero value, which enables us to write off all the types 

 beyond a certain limit as having practically no chance against 

 the rest in any distribution. The striking success of ihe 

 applications of the statistical methods in gas theory when 

 the molecules are treated as rigid, and therefore a number of 

 internal oscillatory degrees of freedom for each molecule 

 entirely ignored, is almost conclusive evidence in favour of 

 the above statement, which is here, however, derived from 

 the analytical restrictions necessarily implied in the analysis 

 employed. 



In some respects the analytical conclusions drawn from 

 the considerations of the present section are analogous to 

 those required by the considerations put forward in the 

 previous section, at least it appears possible to deduce the 

 one conclusion from the other*. Let us consider for this 

 purpose the wave-length spectrum plotted out on a straight 

 line, and divide the line into a large number of equal small 

 segments of length Bk. Each point on this line, and there 

 are ultimately an infinity of such in any length SX, represents 

 a possible oscillation in the thermal radiation, and may thus 

 be regarded as a possible receptacle of energy in the general 

 distribution : if the ultimate dimensions of the point are 

 zero it stands absolutely no chance of getting all the energy 



* Granted the possibility of definiteness in the final result. 



