TEE NEW STATISTICAL MECHANICS 389 



Unrecognizable as it may seem, this is actually a statement about the 

 spectrum of black radiation! This is because a photon of momentum p 

 and energy pc is associated with light-waves of wave-length given by the 

 "Eule of Correlation"; 



X = \ (76) 



P 



If I therefore multiply both members of (75) by pc, I have an expression for 

 the amount of energy associated with the waves ranging in wave-length 

 from h/p to h/{p -f- dp). 



There are instruments able to sort out the waves of different wave-lengths 

 with their associated photons; they are called spectroscopes. There are 

 instruments able to indicate the total energy borne by the photons thus 

 sorted out; they are called by such names as bolometer and thermopile. 

 There are people able to use these instruments; and so (75) can be tested. 

 It is customary to rewrite (75) so that either wave-length or frequency 

 becomes the independent variable, in place of p; but nothing would be 

 gained for the purpose of this article by doing so. The fact of experience is, 

 that (75) is a correct description of black radiation provided that three 

 modifications be made: 



a) For qm we are to write J^/V, presuming that this comes to the same as 

 though we had operated in sLx-dimensional space and put h as the volume 

 of the elementary cell therein (page 383); 



b) For B we are to put \/kT; 



c) We must double the right-hand member of (75), the factor 2 being 

 ascribed to the fact that light is polarizable. 



Making these modifications, and putting F = 1 so that the forthcoming 

 equation shall refer to the radiant energy contained in unit volume, we have 



for the number of photons in unit volume endowed with momenta between 

 p and p -f dp, energies between cp and c{p + dp). This is the distribution- 

 formula for black radiation of temperature T, commonly known as "Planck's 

 law." 



To have derived this law is the first, the great and the historic achievement 

 of the new statistics. Other ways have indeed been found for deriving it, 

 beginning with Planck's own; but the way of the new statistics is smoothest 

 and quickest. Quite different is this story from that of the theory of mate- 

 rial gases! There, the distribution law was correctly given by the old 

 statistics long before it was tested. Here, the distribution-law was found 



