Statistical Relations of Radiant Energy. 651 



the energy in any enumerable infinity of perfectly mono- 

 chromatic constituents must be zero. It would appear, in 

 fact, that it is only when the aggregate of all possible wave- 

 lengths for the system has a non-zero measure that a finite 

 amount of energy can be associated with it in the steady 

 state. 



If we accept this conclusion we must of course deny the 

 possibility of resolving the radiation by a Fourier analysis. 

 But this is, after all, what we should be inclined to expect. 

 In fact, any function can only be so resolved if d has in the 



interval ( 0, — ) of U finite number of singularities (discon- 

 tinuities or turning values); and when we consider that a 

 ray or filament of light even sensibly homogeneous of in- 

 finitesimal range (pX) of wave-length, is not a train of uniform 

 waves, but the aggregate of a vast complex of trains of 

 limited lengths, coming from the various molecules or 

 electrons that take part in the radiation, it would appear 

 that no such condition as that mentioned can be fulfilled by 

 any component of the electromagnetic field at a point 

 traversed by the radiation, when examined over a finite 

 time. 



The argument does not, of course, materially affect the 

 arguments of Gouy and Rayleigh regarding the origin of 

 the periodicity of white 'light which is exhibited on dispersion. 

 The considerations offered by these authors are in reality 

 probably valid for each constituent pulse in the complex 

 ray, and therefore also, by superposition, for the whole ray. 



It may, however, be thought that considerations such as 

 are offered above entirely remove the whole subject beyond 

 the powers of our analysis. When, however, it is remembered 

 that we are after all merely concerned with averages, and 

 thus for the purposes of a statistical theory every possible 

 oscillation with a wave-length in a given small range h\, is 

 practically identical with all the others, then it is seen that 

 it will be sufficient to examine some simple system, as Planck 

 and Jeans have in fact done, in which only one oscillation 

 of each type occurs, one in each of the specified ranges, and 

 then multiply by an infinitely large number JN\ the amount 

 of energy deduced as corresponding to wave-lengths in the 

 specified ran^e for the simple system, a process which is 

 practically equivalent to superposing a vast number of prac- 

 tically identical systems, which are, however, sufficiently 

 different to fill out each small range 8\ so as to obtain 

 a completely continuous spectrum. 



2 U 2 



