540 Lord Rayleigh on the Law of Complete Radiation. 



alike favoured ; and although for some reason not yet ex- 

 plained the doctrine fails in general, it seems possible that it 

 may apply to the graver modes. Let us consider in illustra- 

 tion the case of a stretched string vibrating transversely. 

 According to the Boltzmann -Maxwell law the energy should 

 be equally divided among all the modes, whose frequencies 

 are as 1, 2, 3, ... . Hence if k be the reciprocal of X, repre- 

 senting the frequency, the energy between the limits k and 

 k ±dk is (when k is large enough) represented by dk simply. 

 When we pass from one dimension to three dimensions, 

 and consider for example the vibrations of a cubical mass of 

 air, we have (' Theory of Sound/ § 267) as the equation for k 2 , 



k 2 =p 2 + q 2 + r 2 , 



where p, q, r are integers representing the number of sub- 

 divisions in the three directions. If we regard p. q, r as 

 the coordinates of points forming a cubic array, k is the 

 distance of any point from the origin. Accordingly the 

 number of points for which k lies between k and k-\-dk, pro- 

 portional to the volume of the corresponding spherical shell, 

 may be represented by ti 1 dk, and this expresses the distribution 

 of energy according to the Boltzmann-Maxwell law, so far as 

 regards the wave-length or frequency. If we apply this 

 result to radiation, we shall have, since the energy in each 

 mode is proportional to #, 



k 2 dk, (3) 



or, if we prefer it, 



0X~±dX (4) 



It may be regarded as some confirmation of the suitability of 

 (4) that it is of the prescribed form (1 ). 



The suggestion is that (4) rather than, as according to (2), 



X~ 5 dX (5) 



may be the proper form when Xd is great. If we introduce 

 the exponential factor, the complete expression will be 



d X- 4 e- c ^ 6 dX. . ... . . . (6) 



If, as is probably to be preferred, we make k the independent 

 variable, (6) becomes 



^ k* e-*' 9 dk (7) 



Whether (6) represents the facts of observation as well as 

 (2) I am not in a position to say. It is to be hoped that the 

 question may soon receive an answer at the hands of the 

 distinguished experimenters who have been occupied -with 

 this subject. 



