between Radiation and Free Electrons. 21 



The solution is easily found to be 



<K9)=c<f, (ii) 



In which C and n are constants. 



5. It is at once obvious that this form cannot include 

 Planck's formula. Thus we have seen that there is nothing 

 (so far as the evidence of the radiation formula goes) which 

 is untrue in the ordinarily assumed laws for free sether, or 

 in the ordinarily assumed thermodynamical laws, but it now 

 appears that there must be something untrue in the equations 

 which have been used as the basis of the analysis of § § 2, 3. 

 It is somewhere in these equations that the break between 

 the classical dynamics and the true dynamics must occur. 



6. Any attempt to localize still more definitely the exact 

 point of failure of the classical equations must of necessity 

 involve a detailed discussion of the equations of which we 

 have made use. 



In equations (1) it may first be noticed that the terms in 

 square brackets, representing emission, have not been used 

 at all — equations (1) were immediately replaced by equa- 

 tions (2) because it was found that the terms in equation (1) 

 in square brackets were negligible for the particular crucial 

 problem we were testing. Nothing, then, is to be gained by 

 discussing whether the terms in square brackets are accurate 

 or inaccurate. But two assumptions have been made about 

 the emission — first that the emission terms in equation (1) 

 may be neglected for the special problem under considera- 

 tion, and, second, (in § 3), that a simple-harmonic motion of 

 a free electron results in an emission of light of the same 

 frequency. Both these assumptions must of course be 

 regarded as being under suspicion. 



For the rest, we have assumed the truth of equations (2), 

 which are simply the equations of action of electric and 

 magnetic forces on an electron as a whole. Experiments on 

 electrons moving in electric and magnetic fields (e. g. deter- 

 minations of e/m) seem to indicate that these equations are 

 at least true for steady fields, and, so long as we assume that 

 the action on an electron at any instant depends solely on 

 the field at that instant, it is hard to see how the equations 

 can fail to be true for varying fields also. 



One other assumption has been implied : namely, that a 

 single free electron in a field of radiant energy is a possible 

 dynamical system. It may be that the simplest system 

 which can be considered is not a single free electron but a 

 tube of force with an electron at one end and a positive 

 charge at the other. 



