650 Mr. G. H. Livens on the 



another as at first sight appears, and that if certain plausible 

 assumptions are made the correct formula given by Planck 

 can be obtained from both theories, and now without any 

 implication of a fundamental difficulty such as appeared to 

 be involved in Planck's theory. 



2. The constitution of natural radiation. 



In both the methods of Jeans and Rayleigh and of Planck, 

 it is generally assumed that the electromagnetic field at any 

 point traversed by a stream of steady radiation can be re- 

 solved into its harmonic constituents by a Fourier series of 

 type 



Z a»cos(— 6 n ), 



n=l \ A- n / 



wherein Xa = -, X being a constant, introduced for the pur- 

 poses of the analysis, which is arbitrarily chosen in Planck's 

 case, but is directly related to the dimensions of the elastic 

 solid mass in the discussions of Jeans and Rayleigh. 



An assumption of this nature implies that the complete 

 spectrum of thermal radiation can be regarded as composed 

 of perfectly harmonic constituents of wave-lengths 



X A x x 



1' 2' 3' n' 



But whatever view we may take as to the conditions govern- 

 ing the production and distribution of thermal radiation, it 

 seems, to me at least, very difficult to avoid the assumption 

 of theoretically perfect continuity in its spectrum. When 

 we consider the great variety of actions which are effective 

 in altering to various extents the period of any mono- 

 chromatic vibrator in a piece of matter, it will be difficult 

 for us to avoid the assumption of equal opportunity for every 

 possible wave-length. Perfect continuity is apparently a 

 necessary hypothesis in a statistical theory of this subject. 



The conclusion is therefore to be drawn that such a series 

 of wave-lengths as that given above cannot under any cir- 

 cumstances be regarded as forming the complete spectrum 

 of thermal radiation. In technical language the point in- 

 volved appears to be that the complete linear set of points 

 on a line indefinitely extended in one direction has the 

 power of the continuum, and cannot in any way be regarded 

 as made up of an enumerable set of points. 



If the spectrum of the radiation is perfectly continuous 



