THE STRUCTURE OF THE SERIES OF LINE- AND BAND-SPECTRA. 583 
equation. Specially striking is the contrast between the values of a, , which for band- 
series is always very much smaller than for line-series. 
Professor THIELE has pointed out that in a band-series, proceeding from the head 
towards the tail, the distances between consecutive lines should first increase and 
afterwards diminish, and therefore reach a maximum at a certain point between the 
head and the tail. If we differentiate the RypBERG-THIELE equation 
=a,(m +p)? +, 
Veo —V 
with regard to vy and m, we find 
dv _ 2a,(m + p) 
dm  [a,(m+p)?+6,)? 
GV x © 2a, | _ 4a,(m + pw)? | 
dm? [a,(m+p)? +, }? a,(m+p)?+b,]° 
Hence the particular point m,+., for which the distance is a maximum, is found from 
the equation 
b 
y- 2G. +o)? 0: 1} 
3 ay 
= M, + ps)? = 
ACs (m, + 1) 
But from fig. 3 we know that 2 tan 6, hence 
1 
(m,+p)?=t tan B. 
We see at once that for a series, whose transversal is parallel to Ov, (e.g. the first 
Hydrogen Series) the maximum distance is at m,+"=0, 2.e. at the beginning (head of 
the series), since 6 = 0 ; whereas for a series whose transversal is parallel to Ov, [band- 
series satisfying DesLANDRES’ special equation], the maximum distance is at m,+"= © 
(tail of the series), since = but that in general it must be at a point between 
the head and the tail of the series. Thus we find without difficulty that in the a+ 
series of the Hydrocarbon-band (see p. 579) the greatest distance between consecutive 
lines occurs between the 85th and 86th line. 
Quite recently an important paper has been published by Dr JuneBLurH in the 
Astrophysical Journal, vol. xx. No. 4, in which the author discusses his exquisite 
measurements of the lines of the Cyanogen-band. I take this opportunity, therefore, of 
applying the Rypserc-THIELE formula to this very extensive band, which, as Professor 
Kayser has shown in his Handbuch, p. 479, seemed particularly inaccessible to a 
satisfactory representation by empirical formule. Before the appearance of Dr 
JounesLure’s paper I had already attempted to compute the wave-lengths of this band 
by means of the formula proposed in this paper. It was found that the RyppERc-THIELE 
formula gave excellent results up to about the 80th line, that the agreement could 
still be called very satisfactory up to the 140th line, but that further on the 
discrepancies increased enormously, and showed the formula to be not applicable 
throughout the whole extent of Professor Kayser’s observations. The question 
therefore arises, whether the formula, which has done such excellent work in the 
TRANS. ROY. SOC. EDIN., VOL. XLI. PART III. (NO. 24). 87 
