Theory of Matter and on Radiation. 209 



Thus the only vibrations, which can give rise to observable 

 spectrum-lines, are 



for positive q : those o£ classes 0, —1, and occasionally — 2. 

 for negative q : those of classes 0, + 1, and occasionally + 2. 



That is, for a single ring of electrons, if we divide all the 

 frequencies into groups, each group corresponding to 

 vibrations of the same type, e. g. axial vibrations, orbital 

 vibrations and so on, but of various classes, n in number, 

 each group of frequencies can give rise only to two, or at 

 most three, observable spectrum-lines. 



§ 18. For example, Nagaoka* has discussed a system of 

 particles illustrating spectra ; he compares the free oscilla- 

 tions of a rotating ring with the vibrations giving rise to 

 spectrum-lines, and finds certain analogies between the 

 grouping of the lines in bands and series and that of the 

 oscillations of the ring, when their frequencies are compared. 

 The frequencies for the axial oscillations are given by a 

 quadratic, different for each class : these two groups he con- 

 siders analogous to the vibrations producing bands. The 

 frequencies for the orbital oscillations are given by a quartic 

 with a pair of real and a pair of imaginary roots ; the two 

 groups belonging to the real roots he considers analogous to 

 the vibrations giving series. Each line of a band or series 

 corresponds to a value of k (Ji in Nagaoka's notation), that 

 is, to an oscillation of that class. 



But we have just seen that any one group gives rise to at 

 most three spectrum-lines. On the other hand, the Banner 

 series of hydrogen has 29 lines, and few series are known with 

 less than 10. Even if we suppose all Nagaoka's 4 groups 

 to combine to give a single series we can only get 12 lines 

 at most; this is obviously inadequate. For bands the 

 difficulty is still greater. 



When we consider the whole series of waves emitted in 

 one group, we find that the intensities differ little for the first 

 2 or 3 members, given by k = Q, +1, +2 as the case may 

 be, but after that diminish with very great rapidity, faster 

 than the terms of a geometric progression whose ratio is 

 one to one million. Nothing like this is found for series or 

 for bands. 



1 think this difficulty is conclusive against Nagaoka's 

 view ; but apart from that the frequencies of the oscillations 

 used by Nagaoka are the frequencies relative to the ring 

 (n in his notation) ; the frequencies of the waves emitted by 

 the ring and received by a stationary observer are different 

 (7i+ h<o in his notation). The two sets only agree when the 



* Phil. Mag. ser. 6, vol. vii. p. 415. 



