WILSON. — RECTILINEAR OSCILLATOR THEORY. 125 



We have now concluded a tolerably complete qualitative and 

 quantitative discussion of the equation 



dxfcPx . „ \ 2e*fd?x\* , 1C . 



which arises by the application of the principle of energy with the 

 usual law of instantaneous radiation, and which would arise from the 

 integrated form of the energy equation by the same reasoning as is 

 applied, after a transformation by integration by parts, namely, that 

 if an integral vanishes, its integrand also vanishes, to obtain the 

 equation 



fdPx, ,\ 2#d*x nR . 



m U*- + A 7 = 3?*- (16) 



It is probable that if the non-linear equation (15) had been easily 

 integrable and had given a motion essentially simple harmonic and 

 damped, the transformation to (16) would never have been made. 

 And the fact that (15) is difficult of integration, and when actually 

 integrated gives an aperiodic discontinuous motion does not seem a 

 very satisfactory justification for making the transformation. 



6. Conclusions. One of the first inferences of our work is that, 

 whatever we may think of the merits of either of (15) or (16), the 

 Abraham-Planck-Riidenberg derivation of (16) is without much mathe- 

 matical or physical validity. The derivation of Lorentz and Ritz 11 

 is not open to the same objections; for it proceeds directly from the 

 retarded potentials to (16) through a series of approximations. The 

 law of radiation which this equation states is that the rate of radiation 

 is 



dxfdtx 2 \ 2fdxd*x 



m dt\df + **;- 3 C 3 ^ d?' 



which for oscillatory motions is nearly proportional to the square of 

 the velocity and is physically nearly equivalent to a friction propor- 

 tional to the velocity. The average rate of radiation is, of course, 

 that which corresponds to that found by Hertz. 



Our work throws some light on the method of presenting the differ- 

 ential equation for the oscillator of radiotelegraphy. We may assume 

 that the oscillations are damped simple harmonic; we know that the 

 simplest mathematical representation of such motion is by an equa- 



11 Lorentz, Theory of Electrons, loc. cit. ; Walther Ritz, Gesammelte Werke, 

 401 ff. 



