THE STRUCTURE OF THE SERIES OF LINE- AND BAND-SPECTRA. 593 
from the band-series to the line-series. At present we are, of course, ignorant of the 
conditions which lead to the more general equation (18), but at least this remarkable 
and highly suggestive feature has been brought out by our investigation, that the 
addition of a single constant to equation (17), the acoustic theory of which is well 
established, leads to the RypBERG-THIELE formula, which, as we have demonstrated in 
the preceding sections, represents so completely all the phenomena of spectral 
regularities. . 
It is well known that if the three dimensions of the elastic body are altered in the 
same proportion, the wave-frequencies of its transverse vibrations change also in the 
same but inverse proportion, higher frequencies corresponding to smaller dimensions. 
Thus if we compare two rods of the same material, the one of length /, breadth b, and 
width w, and the other of length a./, breadth a.b, and width a.w, the wave-frequencies 
of their transversal vibrations are in the ratio 1:a, while the volumes are in the 
proportion 1:a*. Hence the wave-frequencies are inversely proportional to the cube 
| roots of the volumes. Now, it appears that in the case of vibrating atoms a similar, 
| although not quite so simple, relation obtains. In each group of chemically related 
elements, such as the alkalis for instance, the wave-frequencies of the tails, v», can be 
approximately represented by an equation 
oe ee (19) 
where m and n are constants and v denotes the atomic volume of the element. As an 
example, let us take the wave-frequencies v» of the following five elements, referring to 
the first components of the subsidiary series : 
v 1/v% Veo | Computed. 
ie? 118 0-43925 28589 | 28514 
Na: 23:7 0°34814 24486 24660 
K: 45:0 0:28115 21994 21827 
Rb: 56:0 0:26138 20965 20990 
Cs: 79:0 0:23306 19748 19792 
The values of the last column are those computed from the equation 
DS teal 
lv 
The atomic volumes have been taken from the data given in the article “‘ Chemistry ” 
in the Enc. Brit. To expect more than a merely approximate agreement seems 
scarcely warranted, considering the uncertainties in the values of v here adopted. In 
spite of the doubtless large discrepancies between the values of the last two columns, | 
am inclined to think that the asserted relation between the wave-frequencies v» and the 
linear dimensions of the atoms expresses a real physical law. The view is supported 
by the following interesting fact. In the vertical column of MENDELEJER’S system 
