426 M. Wladimir Michelson on the Distribution 



special memoir upon the radiations of solids at different tem- 

 peratures, induce me to publish at once, at least in abstract, 

 some theoretical considerations upon this subject. I hope to 

 give a more complete discussion of the question when the new 

 data of Prof. Langley's spectro-bolometric researches shall 

 enable me to confront my theory with experiment in a more 

 detailed manner. 



1. Hypothesis and General Law. — The absolute continuity 

 of the spectra emitted by solids can only be explained by the 

 complete irregularity of the vibrations of their atoms. The 

 discussion of the distribution of radiant energy amongst the 

 simple vibrations of different, period is, then, to be undertaken 

 by the calculus of probabilities. 



Let us consider a homogeneous isotropic solid of which all 

 the atoms are in identical circumstances, so that, for example, 

 they are not grouped into separate molecules. Each atom 

 has a definite position of equilibrium towards which it is con- 

 tinually driven back by the surrounding atoms, and about 

 which it describes infinitely small oscillations. I express this 

 fact by supposing that each atom moves freely in the interior 

 of a spherical elastic shell of infinitely small radius p, which 

 has the position of equilibrium as its centre. The atom- 

 rebounds from the interior surface of this sphere according to 

 the law of impact for perfectly elastic bodies, preserves its 

 absolute velocity during several free paths, and then changes 

 its velocity in consequence of the unsymmetrical action of the 

 surrounding atoms. 



Let us endeavour to find upon this hypothesis what w r ill be 

 the most probable trajectories of the atom in the interior of 

 the sphere of displacement. Let us admit that, for the initial 

 position of the atom, all possible distances from the position 

 of equilibrium are equally probable ; then the probability that 

 this distance shall lie between the limits r and r + dr will be 

 expressed by jr 



P 



Let us take the radius ON (fig. 1) of the sphere passing 

 through the initial position, M, of the atom for polar axis. 

 Let us denote the angle N M P, which the direction of motion 

 of the atom makes with this axis, by cj>. Let us admit, as in 

 the theory of gases, that all directions are equally probable. 

 Then the probability that <£ will lie between the limits </> and 

 + <tywillbe isintfxty. 



Let us call the angle of incidence MPO, 8. This angle is 

 connected with <j> by the relationship 



r sin <f> = p sin S. 



