WILSON. — RECTILINEAR OSCILLATOR THEORY. 127 



certain differential equation, nor in pointing out that the integration 

 of the equation leads to a physical point d'arret, but in the general 

 inferences or alternative inferential possibilities of importance to 

 physical science which the discovery of the point d'arret may suggest. 



It is clear that we shall not have to abandon the statement of physi- 

 cal problems in terms of differential equations possessing mathemati- 

 cally continuous solutions in order to introduce into physics discontinu- 

 ities even more startling than those introduced by the adherents of cur- 

 rent theories of quanta. The statement of simple cases of rectilinear 

 motion in equations of the second degree and second order has led to 

 cusps in the space-time locus and, by virtue of the irreversibility of 

 time, to physical discontinuities. (Such a conclusion could hardly be 

 drawn from Borel's work, for his equation led to no point d'arret.) 



If we regard the existence of the point d'arret as inconceivable, 

 we must look about for a method of doing away with it. One possi- 

 bility is to accept literally the method of calculation employed by J. J. 

 Thomson as cited above, namely, to regard the motion of the electric 

 charge as determined by an ordinary equation of mechanics, which 

 omits altogether the reaction of the radiation, and to compute the 

 radiation subsequently. This would mean that the radiant energy 

 did not come out of the field or motion (potential and kinetic energy) 

 of the system but was a phenomenon inherently arising when charges 

 moved. 14 This would be a shock to those who have implicit confidence 

 in the conservation of energy as now generally understood, but would, 

 not perhaps be so serious to those who take the position of Ritz 15 : 

 La localisation de l'energie doit done etre comptee au nombre des con- 

 ceptions logiquement inutiles (et peut-etre parfois nuisibles) de la 

 theorie . . . Dans le cas le plus general du rayonnement eleetromagne- 

 tique, la conservation de l'energie n'est plus une loi, mais une con- 

 vention. 



Another method of escape would be to deny the accuracy of the law 

 that the radiation is proportional at each instant to the square of the 

 acceleration. The law is indeed only an approximation to the general 

 form given by Abraham and Heaviside (with the apparent assumption 

 of the quasi-stationary state) and proved by Lewis and me (on the 

 basis of the principle of relativity) to be rigorously true for the point 

 electron moving with any velocity and acceleration, 16 — for with the 



14 This may remotely suggest Bateman's double-barreled gun in his Corpus- 

 cular Radiation, Phil. Mag. (6) 26, pp. 579-585, Oct. 1913. 



15 Gesammelte Werke, pp. 343, 345. 



16 Wilson and Lewis, loc. cit., (5). 



