' 1 heory of Matter and on Radiation. 213 



electrons are as crowded as this, the value of k/3 is as great 

 as 26, and that of k as great as 30, so that the line is far too 

 weak to be observable. 



We conclude that the large roots of the frequency equations 

 of the ring cannot give observable spectrum-lines; so that 

 the most general type of single ring cannot account either 

 for spectrum-series or for bands. 



§ 22. The argument of this paper may be summarized 

 briefly as follows. After some preliminary work (§§ 1-10) 

 an ideal radiating system is considered, consisting of a large 

 number of mutually independent similar rings of electrons, 

 each in orbital motion in a suitable controlling field, of which 

 a fraction are ionized owing to the previous expulsion of an 

 electron (§§ 11-12). 



Each ion executes a number of vibrations; these can be 

 arranged in six groups according to the frequency (q) relative 

 to the rotating ring ; each group includes n classes of vibra- 

 tions, n beino- the number of electrons in the ionized ring 

 (ion), and the class (k) giving the number of segments in 

 which the ring vibrates (§ 7). 



Each vibrating ion emits a corresponding number of waves 

 (§ 8), of which the frequencies to a stationary observer 

 are (q + kco), co being the angular velocity of rotation of 

 the ring. 



The intensities of these waves after the first two or three 

 classes fall off with such great rapidity (§§ 9, 14), that 

 waves of classes +3 and upwards are far too weak to 

 giye rise to observable spectrum-lines ; waves of classes + 1 

 and give rise to lines whose intensities are of the same 

 order of magnitude as those of spectrum-lines (§§ 14, 16); 

 waves of classes +2 do so when the ionization is large 



(§ 14 )- 



"W hen the velocity of the electrons in steady motion is very 

 small compared with that of light, there are six frequencies (q) 

 for each class (/:); hence the ring gives at most 18 observable 

 lines (§§18, 19). When the velocity is comparable with that 

 ot light, the frequency equations are transcendental, and there 

 are an infinite number of frequencies (q) for each class (£); 

 but of these frequencies only six can give rise to observable 

 lines, so that the maximum number for the ring is only 18 

 as before (§§ 20, 21), that is, too small to account for series 

 or bands. 



