428 M. Wladimir Michelson on the Distribution 



the atom produces a wave of the same period in the surround- 

 ing asther, and if we consider that the waves produced by the 

 radial motion indicated above will not only be the most fre- 

 quent, but also the most intense, we must admit, as a first 

 approximation, that it is only these motions which determine 

 the composition of the radiations emitted by our solid. 



But the period of vibration t corresponding to this radial 

 motion is evidently connected with the instantaneous velocity 

 v of the atom by the relationship 



'-* ;<« 



This relationship may be employed for the transformation of 

 the known formula of Maxwell, which expresses the most 

 probable distribution of energy amongst a large number of 

 any material points whatever continually exchanging their 

 velocities by means of forces having a potential*. If we denote 

 by N the total number of atoms, by m the mass of each of 



3 

 them, by e the base of Napierian logarithms, by k= ;r=2 a 



constant inversely proportional to the mean vis viva —^- °* an 



atom, Maxwell's law gives the number of atoms whose velo- 

 cities lie between the limits v and v + dv ; this number is 



4N 

 v v =^L{km)h- km ^dv (5) 



V 7T 



4p 

 Replacing v by — , according to (4), we obtain 



256N 8/7 , 3 -i^ 2 AJ 

 v T — —= p\hni)%e ^ T-*dr. ... (6) 



V7T 



This formula gives the number of atoms of which the prin- 

 cipal periods of vibration lie between the limits t and T + dr. 

 It is the probable law of distribution of the periods of vibra- 

 tion between the atoms of our solid. 



In consequence of the law of superposition of vibrations, we 

 may admit that the intensity of a simple radiation of period t 

 ought to be : — 



(1) Proportional to the number r T of atoms of the source of 

 radiation vibrating in the same period. 



* See upon this subject L. Boltzmann, " Ueber die Beziehnng zwiscben 

 dem zweiten Hauptsatze der mecbaniscben Warmetbeorie und der 

 Wabrscbeiulicbkeitoechnung &c." (Sitzunffsber. der Wiener Ahad. der 

 Wiss. Bd. lxxvi. pp. 878-485, Oct. 1887) ; and also Watson, 'A Treatise 

 on tbe Kinetic Tbeory of Gases ' (Oxford, 1870), p. 12, &c. 



