(Sia) 
XXIV.—On the Structure of the Series of Line- and Band-Spectra. By J. Halm, 
Ph.D., Lecturer on Astronomy in the University of Edinburgh. 
(Read July 4, 1904. MS. received October 14, 1904. Issued separately July 3, 1905.) 
In a preliminary note read before the Society on July 4, 1904, 1 drew attention to 
the fact that a number of line-series, forming a group which includes the first series 
of Hydrogen, can be represented by an equation of the form 
1 
Vo —Vv 
=am?+b,, (1) 
where v denotes the wave-frequency of any line of the series, v. that of the so-called 
“tail” of the series (m=), and a, 6, constants depending on the nature of the 
emitting substance ; the frequencies of successive lines being obtained by substituting 
successive integers for m. We see at once that this equation is a generalisation of 
Batmer’s formula, into which it is transformed by equating }, to zero. In the same 
note I also pointed out the existence of another group represented by an equation of the 
same form, if (m+) is substituted form. As a special case (b,=0) this group contains 
the second Hydrogen series discovered by Professor PickERING in the spectrum of 
¢ Puppis. Subsequent investigations convinced me, however, that, although a consider- 
able number of line-series may be classified into these two groups, there are numerous 
instances where the more general formula 
oS 1 = a,(m + p.)? + b, (2) 
Vo = VV 
must be employed, in which u represents various fractional numbers. Also, in studying 
more thoroughly the literature on the subject, I found that the equation, in the last- 
mentioned form, is merely a modification of a mathematical expression already employed 
by Professor Tu1rLE in his investigations on the band-series of the carbon spectrum 
and on the line-series of Helium. But, convinced as I was from my own computations 
of the accuracy and general importance of this equation, I was surprised to find it 
rejected by Professor THIELE, on the ground that it did not sufficiently represent 
the observed wave-lengths. This statement appeared to be so far from acceptable, 
that I resolved to demonstrate, by an exhaustive examination of all the known series, 
not only the general applicability of equation (2), but also its great superiority over 
any other formula hitherto proposed. The demonstration of this fact will be the first 
object of the present communication. It will be shown that the equation represents not 
only all the line-series hitherto known, but also the band-series, with an accuracy which 
leaves nothing to be desired. But far more important even than the demonstration 
of equation (2) as an empirical formula of practical usefulness are certain con- 
clusions which may be drawn from the character of this equation and the numerical 
values of its constants. For instance, we shall see that equation (2) can be represented 
TRANS. ROY. SOC. EDIN., VOL. XLI. PART III. (NO. 24). 83 
