210 Mr. G. A. Schott on the Electron 



frequencies n are very large compared with the angular 

 velocity co, which is not the case. If* it be true that the 

 values of n crowd together when li becomes large, as Nagaoka 

 supposes, it follows that the frequencies of the lines produced 

 approach to coincidence with an arithmetic progression, with 

 difference equal to co. This corresponds neither to a band 

 nor to a series, but to a set of lines of constant frequency 

 difference. 



§ 19. Nagaoka's model may be modified in two ways. 

 (1) The controlling field, in which the ring moves, may be 

 altered ; for instance, the ring may be made stable as in 

 J. J. Thomson's model *. A comparison of Nagaoka's 

 equations (9), (12) with Thomson's equations (4), (3), and 

 indeed with MaxwelPs equations (14 ) (22) f, shows that in 

 all these eases the frequency equations are fundamentally the 

 same ; we have the same six groups of vibrations, each of 

 the same number n of classes, but the values of the frequencies 

 are altered ; this is true however we modify the controlling- 

 field. Hence in all these cases the system can give rise to 

 observable spectrum-lines for at most three classes in each 

 group, that is 18 lines in all. 



(2) In the cases just considered the velocity has been 

 supposed negligible compared with that of light, and there- 

 fore higher powers of /3 have been omitted. Let us remove 

 this restriction. 



The frequency equations reduce as before to a separate 

 equation for axial and orbital vibrations, but each equation is 

 transcendental and therefore has an infinite number of roots 

 for each of the n classes. For stability all these must be 

 real, if damping be neglected ; if damping be taken into 

 account they must be complex with positive imaginary part. 

 Can each of these give rise to 3 spectrum-lines ? If so, we 

 have a sufficient number to account for series and bands. 



§ 20. I have found the complete frequency equations for a 

 ring rotating with large velocity in a given controlling field, 

 but the results are very complicated ; as an example I give 

 the frequency equation for axial vibrations ; in the notation of 

 the present paper Nagaoka's equation (9) may be written 



where the central positive charge is equal to v of the negative 

 charges, and 



"" f Ti 4Lsin 8 (wi/»)! 



i^- 1 sin 2 (fari/ri) 



* J. J. Thomson, Phil. Mag. ser. 6, vol. vii, p. 237. 

 f Collected Papers, vol. i. pp. 315, 31(5. 



