¥ 
576 DR J. HALM ON 
the two phenomena, the line- and band-series, being comprised under one and the — 
same mathematical, and probably also physical, conception. 
From what has been found in the case of the line-spectra it appears now extremely 
probable that Drstanpres’ formula represents only a limited number of band-spectra 
and is but a special case of our more general equation, just as BaLmer’s formula is a 
particular instance of the same equation in the case of line-spectra. A considerable 
part of the remainder of the present communication will be devoted to the investigation 
of this point. Before entering upon this new side of the question, however, I should 
like to discuss briefly, for the convenience of those readers who are interested in the 
foregoing computations, a method by which the four constants of the RyDBERG-THIELE 
formula may be determined from one of the two equations (5). If »,, vy, v2, ¥» are the 
wave-frequencies of any four lines of the series, we have from the second of (5): 
i Se ae Se 
Ve—vz (2+)? — (e+ pw)? 
1 a. 
ware rer Bya) 
1 a, 
Se eS aD c 
ian¥y (ey t Dwr)? () 
1 a, 
pee tt ee et he aii 
Vo — Vz (w +24 2u)(w—2) z 
(a) 
(=e eee 
; (@@4+e4Quje—z)” 
(4) 
Subtracting (c) from (a) and (d) from (b), and dividing, we obtain after slight 
transformations : 
(ety t2n)\(z+w+t Qu) _ _(2—2%)(w— y) N@p= va) Wa Ve) 
(w+2+2Qu)(ytwt2p) (y- a)(w—2z) (v,—Vz)(V — Vy) 
= ¢, say, (e) 
or by substituting 
y-“=p 
2-2=q 
wW-x=Pr 
[p+2(m+a)[gtr+2(ut+e)]_ 
[g+2(u+«)|[ptr+2(u+2)] 
—— (uta) Pat Wee =9 
The positive root of this quadratic equation is the desired value of (u+2). Let us take 
as an example the principal series of Sodium and select the 1st, 3rd, 5th, and 7th 
observed line for our computation. From the observations we have therefore : 
c, and 
(m+2)? 
Vy = 16960'19 z2-ae2=4 p=2 
Vy = Va49 = 35051°93 w—y=4 q=4 
yp = vps o8E50780 y -2=2 AG 
Vp = Veug= 39805°27 waee 
4.4 18091°74 x 1254°47 
4:4, 1800174 x 1254'47 _ o.ggurnge 
°= 99 * 9159001 x 475334 ° 
(u +a)? + 6(4+2)-17:992=0; p+a=2°195. 
