THE STRUCTURE OF THE SERIES OF LINE- AND BAND-SPECTRA. 567 
the differences of the wave-frequencies of their tails, computed on the one hand by 
Kaysir’s formula, and on the other by the RypBerc-THIELE equation : 
K-R. Ry-Th. K-R. = Ry-Th. 
Ibis + 81 -11 Hg: +58 -13 
Na: + 57 +9 In: +20 + 3 
K: + 31 + 5 O: —14 = 5 
Mg: + 41 + 1. ae qe Jl 
Ca: +121 + 83 S: — 9 + 3 
Sr: + 35 +18 Se: +20 +28 
Zn: + 10 + 5 He: — 26 -— 38 
Cd: + 42 +457 - 7 -— l 
These figures show clearly that the law in question is more closely represented by 
the Rypserc-THieLE formula. [Errors of 10-15 units are possibly accounted for 
by the uncertainty of the data from which the constants have been derived, because it 
can be shown that an error of only one unit in one of the observed wave-frequencies may 
sometimes alter the computed position of the tail by more than ten times this amount. 
The discrepancies in the [Ry-Th.] column, except those for Ca and Se, may therefore 
be considered as admissible, if the still existing uncertainties of the observed wave- 
lengths are taken into account. 
With regard to », it has doubtless been noticed that in a considerable number of 
eases this constant is zero. Indeed, among the 44 series mentioned above we have no 
less than 19 instances of this kind. _We find that, with the exception of Mg, 
Ca and Sr, all the first subsidiary series belong to this particular group, which, as has 
been already pointed out, includes the first Hydrogen-series. There is also evidence 
of the existence of smaller groups, for instance »=0°5 [H, Mg, 8, Se]. Now 
in any such group, if we denote by » the wave-frequency of a line of one series 
and by n the wave-frequency of a line of another series, since « is the same in both 
series, we have the relation 
= 5 +5 (7) 
where a, and b; are constants while v, and n, are the wave-frequencies of the «th 
lines of the two series. This relation obviously enables us to express one series of 
lines by means of another belonging to the same m-group. A similar relation obtains 
for the wave-lengths : 
+B; (8) 
It may be of interest to prove the existence of such a relation between the series 
of different spectra in some special cases. Let us take, for instance, the group »=0, 
and let us assume as the series of comparison the well-known Hydrogen - series 
Tepresented by Batmer’s formula. In accordance with the foregoing equations, we 
call the wave-frequencies and wave-lengths of the hydrogen-lines n and 1 respectively, 
TRANS. ROY. SOC. EDIN., VOL. XLI. PART III. (NO. 24). 85 
