590 DR J. HALM ON 
and vice versd. In general, the wave-lengths of the lines of any line- or band-series 
appear to be made up by two terms, the one satisfying the Ryppere formula (15) and. 
the other the more general DESLANDRES equation (16). This interpretation of the 
RyYDBERG-THIELE formula shows perhaps more concretely than any other the fundamental 
character of its structure, and also its importance as the universal expression of spectral 
regularities. 
C. GENERAL CONCLUSIONS. 
We are now prepared to enter upon the discussion of some results of a more general 
character. The geometrical property of the Rypserc-TureLe formula, as already 
indicated, enables us to represent on one single diagram every possible line- or band- 
series as a transversal on which the successive lines of the series are indicated by 
the points of intersection with the rays Ov. On fig. 4 accompanying this paper I 
have indicated the positions of these transversals for a limited number of cases. The 
construction of the diagram is made sufficiently clear by the explanations already given, 
and therefore requires little additional comment. After the rays O) and Ow had 
been constructed at right angles to each other, a line was drawn parallel to Ow, and, 
starting from its point of intersection with O,, the successive values of (m+ )*, on a 
conveniently chosen scale, were measured off. Through O and the points thus obtamed 
lines were then drawn which are marked at their ends in our figure by the correspond- 
ing values of (m+ ). Obviously, in order to obtain the true inclinations 6 of the 
transversals, the parallel should be drawn at unit distance from the ray Ow. It was 
found, however, that under this condition the diagram would occupy too much space 
to be conveniently reproduced here, and I therefore decided to draw the parallel 
at a distance of 10 units. Consequently the inclinations of the transversals in the 
figure, which we may call 6), are considerably smaller, the two angles being in the 
relation tan 8,= 75 tan 8. The rays O,, 0,, 03... , which correspond to integral 
values of (m+), have been represented by slightly stronger lines. Now, in the lower 
part of the diagram we find nearly all the line-series of the group «=0. The chemical 
elements to which the series belong are indicated at the two ends of the transversals, 
and, where not otherwise stated, the latter refer to the 1st subsidiary series. ‘If on any 
of these transversals we measure off the distances between consecutive points of inter- 
section, these distances will be found to be exactly proportional to the corresponding 
wave-frequencies of the series to which the transversal refers. In this arrangement, 
therefore, all the spectral lines lie precisely on the rays O,, O,,... . and all the 
tail-ends on the ray O.. Hence, if we imagine these rays to be represented by thin 
pencils of light, we may at once obtain the exact arrangernent of the lines in any of 
these series by interposing a plane screen in a direction parallel to the corresponding 
transversal, since the centres of the luminous dots on the screen must then mark the 
true relative positions of the lines in the spectrum. There would perhaps be m0 
difficulty in constructing an apparatus for lecture-purposes by which the correctness of 
