456 Mr. G. A. Schott on Frequencies of Free Vibrations 
correspond to observable spectrum-lines, namely those for /=0, 
±1, and exceptionally k= ±2. 
Orbital vibrations. 
In the same way as above, remembering that N, L are even 
and M is odd, and that N , L + 2K, M all vanish, we obtain 
the set of undamped waves 
g = 4y i -L 1 -2K + 43I 1 
^ 2 + 4X 1 -L 1 - 2K-4 3I 1 3 + 4N 3 -L 2 -2B:-43X 2 
1 o_4y i -L 1 -2K-4 M 1 3 _4X 2 -L 2 -2K-4M 2 
2^ 2v 
1 _ i _43y 2 -L 2 -2K+4M s 
2v 
4K-L -2K+4M a 
1- 
Lastly we obtain a set of damped waves, and a set of waves 
of instability of the same frequency, given by 
V ' ~ + V ' 
2M, 2M S 
=H 
(17) 
I V ' ** V >'"J 
§26. Remembering that the suffixes 0, 1 correspond to 
relatively strong, and 2 to weak lines, we see that a single 
ring can give rise to the following lines : 
, v A , r , „n 4Ni— L,-2K + 4M li 
(a) A strong line or small frequency co —^ 
(b) An octet of 5 strong lines and 3 very weak lines, all of 
frequency co very nearly. 
(c) A quintet of 3 strong lines and 2 very weak lines, all 
of frequency 2g> very nearly. 
(d) A triplet of 3 very weak lines, of frequency 3g> very 
nearly. 
In general the very weak lines will not be observable. 
If all the frequencies are such that all the lines fall within 
the limits of the visible spectrum, we get three strong lines, 
simple or complex, and occasionally a fourth very weak line. 
If Nagaoka's assumption that v is very large compared with 
n be not true, the results are altered to a certain extent, but 
not very materially until the quantity K becomes comparable 
with v ; this requires that n be comparable with v, a con- 
tingency which can hardly occur unless v be small. 
(16) 
