202 Mr. G. A. Schott on the Electron 



maximum kinetic energy 



for + k ' 



n {n + iy\<] H- q')'* snr (kir/n) 



that is, 7rhnp 2 q 2 <T 2 /n for &, and 7r 2 mp 2 g f2 cr 2 /n for — A\ The value 



of an is clearly less than — = — — — r ^ for 4- h. 

 J n sin {kir/n) — 



§ 12. The ideal problem of § 11 differs in two respects 

 from an actual case of ionization of a ring. The expulsion 

 of the electron cannot really be instantaneous, but must be 

 more or less gradual ; the disturbance produced will in reality 

 be less violent than in the ideal case, the values of a will be 

 less, and probably the vibrations of higher classes will be 

 weaker, not merely absolutely, but also relatively to those 

 of lower classes. Again, the effect of damping on account of 

 radiation has been neglected; inconsequence the values of a 

 diminish with the time according to an exponential law, those 

 of lower classes very rapidly, those of higher classes much 

 more slowly. We must now consider the effect of damping 

 on the radiation. 



]t has long been thought that the radiation from a gas, in 

 a vacuum-tube or in a flame, is connected with the ionization 

 of its atoms. Stark*, from his experiments on the Doppler 

 effect of canal rays, concludes that the line-spectra are emitted 

 bv positive ions produced from atoms by the expulsion of a 

 negative electron, and that band-spectra are emitted in the 

 recombination of the positive ion and negative electron. We 

 may compare a radiating gas to a system of a very large 

 number of rings like that of § 11. Suppose that on the 

 average a ring remains ionized for T seconds. Further, 

 suppose that the energy of an ionized ring due to a vibration 

 of class h and amplitude cr is E=rAcr 2 , and the corresponding 

 radiation R = Bcr' 2 . We have 



whence 1 <?E R B . , 

 = — _= — _-, a constant. 



E ctt E A 



Write B/A = 7 ; then 



E = E e-v<, R=E e-V, 



where t is reckoned from the instant of ionization. 

 * Physikalische Zeitschrift, vi. p. 892 ; ' Nature/ lxxiii.pp. 78, 389, 533. 



