WILSON. — RECTILINEAR OSCILLATOR THEORY. Ill 



The calculation of the instantaneous rate of radiation of energy 

 from a moving electron, as contrasted with the integrated rate hitherto 

 used, has been carried out by a number of authors in different ways. 

 The rate is almost universally found proportional to the square of the 

 acceleration when the square of the velocity is negligible relative to 

 the square of the velocity of light. 5 The mathematical statement 

 of this result is contained in (6), and this equation therefore might 

 more properly be taken as representative of the motion of the oscil- 

 lator than the equation (5), — provided, of course, that we are willing 

 to use the electron model of the oscillator. 



A comparison of (5) and (6) will throw some light on the legitimacy 

 of the approximations used in deriving (5). The terms to be com- 

 pared are 



dx d 3 x . (d 2 x\ 2 



-^M and M [d¥ 



On the supposition that the oscillator executes in the main a simple 

 harmonic motion, the signs of the two terms are the same. But the 

 phases of the magnitudes are very different; for the first term is a 

 maximum as the particle passes through the position of equilibrium 

 and vanishes in the positions of extreme elongation, whereas the 

 second term is a maximum in the extreme positions and vanishes in the 

 position of equilibrium. The average magnitudes of the terms are 

 the same, but it is difficult to see how two terms of equal average 

 magnitude could be more different than these two in their effect upon 

 the equation of motion. 



We shall now discuss the two forms (5) and (6), the first rather 

 briefly, because it has been tolerably well treated, the second more 

 in detail, because it seems to have been neglected. 



2. Discussion of (5). As we have to deal with cubic equations 

 and their (approximate) solutions, it is necessary to determine the 

 order of magnitude of the coefficients and it is desirable to put the 

 equations into their simplest form. If m is the mass of the vibrating 

 particle, c the charge in electrostatic units, c the velocity of light, and 

 if we adopt the customary coefficient for the radiation term, we have 



5 For two different lines of deduction see J. J. Thomson, Electricity and 

 Matter, chap. 3 (1904), and Wilson and Lewis, The Space-time Manifold of 

 Relativity, These Proceedings, 48, 389-507 (1912), with especial reference to 

 p. 480. A. Macdonald in his Electric Waves, 75 (1902), finds a steady rate of 

 radiation from the oscillator. 



