WILSON. — RECTILINEAR OSCILLATOR THEORY. 



109 



The modification of the equation is thus based directly on the 

 principle of energy. The form in which F is thrown depends some- 

 what on the point of view. 



If the oscillator is considered as a Hertz dipol and the ordinary 

 solution for the electromagnetic field set up by this radiator at large 

 distances from its center is obtained, we find by the application of 

 Poynting's theorem that a certain amount of energy is radiated off 

 per period and, in particular, that the amount is proportional to the 

 square of the acceleration. Then the equation, written for a com- 

 plete integrated period T is 



/ 



J to 



<o+ T 



d 



dt 



dx 



dt 



+ lrx 1 



+ 2 M 



d?x 

 df- 



dt= 0. 



An integration by parts transforms the third term. Whence 



Jto 



fo+ T 



d 

 dt 



+ 2m 



£r+« 



-2 M 



dx d?x 

 dtdJ 



dt 



'dxd*x 

 dt df 2 



<o+ T 



= 0. 



(3) 



Now if f be an epoch for which either the velocity or the accelera- 

 tion vanishes, the last term drops out. Under these conditions 



/ 



J to 



u + t j d_ r fdx 



r* \ dt \_ \dt 



The conclusion is then drawn that 

 d r /d*Y 



dt I U 



+ F.r 



-2 M 



dx d 3 x 

 dtd? 



dt= 0. 



+ k 2 .v- 



2m 



dx r/ :! .r 



diw 



= o. 



(4) 



Exceptions can be taken to this conclusion whether the argument is as 

 above or as given in slightly different form by Planck. 2 It by no means 

 follows that, because the integral of a function over a complete period 

 from t -\-nT to t G J r(n-\-l)T vanishes, the integrand must therefore 

 be identically null. 



What we may infer is that 



d_ 

 dt 



dxv 

 dt 



+ tfz> 



-*££-» 



where a(t -\-(n J rl)T) — aito+nT) = 0. There is nothing at all to 



2 Planck, Vorlesungcn iibcr die Theorie der Warmestrahlung, first edition, 

 110 (1906). Abraham, Theorie der Elektrizitat, second edition, 70 (1908). 

 Rudenberg, Annalen der Physik, 25, 346-466 (190S). (In his second edition. 

 Planck introduces the quantum hypothesis early and omits this derivation of 

 the equation of the oscillator.) 



