’ 
552 DR J. HALM ON 
in a geometrical form, by means of which we are enabled to collect all the line- and 
band-series into one single diagram revealing a community of properties between the 
two classes of spectral regularities and their individual members. This new geometrical 
connection between the series appears to be of theoretical importance, inasmuch as it 
shows a striking similarity between the vibrations of a radiating system of atoms 
and the nodal vibrations of elastic bodies. From this point of view an im- 
portant relation has been discovered between the wave-frequencies of the “tails” of 
line-series and the atomic volumes of the emitting elements. In the course of this 
paper we shall have frequent opportunities of observing regularities in the constants 
of equation (2), and of drawing from them conclusions which cannot but be of some 
importance, however small, in connection with the theory of the phenomena of spectral 
regularities—a region into which the speculative mind has so far vainly attempted to 
penetrate. The outcome of the investigation must, I think, be to convey the impression 
that equation (2) is to be considered as more than a merely empirical formula, and that, 
if it does not represent the physical law itself, it 1s at least a remarkably close 
approximation to it, sufficiently reliable, perhaps, to guide the theorist in his search 
for the ultimate cause of the spectral regularities here considered. 
Before entering upon the first part of our investigation, viz. that dealing with the 
question how far equation (2) is capable of representing the observed wave-frequencies, 
it will be useful to derive other forms of this equation, which we shall employ later on. — 
First of all, we see at once that (2) may be expressed by the following series : 
l by b? 
~a,(m+p)2 a,*(m + p)4 x a,3(m + p)® zh 
V =Vo 
and we notice that in this form it represents a more general case of RyDBERG’s formula, 
into which it converges, for b,=0. In order to express the fact of its belonging to this 
type, and at the same time to distinguish it from Rypsere’s more special equation, I 
propose for it the name “ Rypperc-THIELE” equation, recognising thereby Professor 
THIELE’S merit in having first introduced its present form into spectroscopic science. 
Equation (2) assumes a simpler and in some cases a more convenient form by intro- 
ducing v), the wave-frequency corresponding to m+m=0. In most cases there is no 
line referring to this special value of », which obviously must lie close to the “ head” of 
the series. But for the sake of convenience we may be permitted to speak of vp in the 
following formulze as the wave-frequency of the “beginning” of the series. Introdue- 
ing v), we find from (2) : 
A=) 
=,(Ve —Vy). (M+ p)?=a,(m + p)?. (3) 
Vo —V 
Similar equations are at once obtained if wave-lengths are substituted for wave- 
frequencies : 
1 
A— Xx 
ea Bae ie ae 2 
soy ed ae ee a,(m + 1) + B, 
(4) 
Xr -- Neo 
PWS veces. x2 (m +)? =a,.(m + ph)? 
