THE STRUCTURE OF THE SERIES OF LINE- AND BAND-SPECTRA. 575 
This is probably the most convenient form for computation. It becomes identical with 
BaLMeEr’s formula for «= 0 and 6=0, because we have then tan a=m~* and 
—2 
V = Ve —Qo. + tan a = Ve — An > mM ¥ 
If only 8 =0, we find RypBere’s formula 
v= Ve —o+(m+p)* 
Hence we see that both BaLmer’s and Rypsera’s equations suppose the transversal 
to be parallel to the ray 0»), whereas, according to the more general RypBERG-THIELE 
formula, the two lines may form any angle 8 with each other. 
Now, an important result is arrived at if we investigate more closely these angles . 
We notice that while in some cases (6 is small, and hence the transversals are nearly 
parallel to Ov), there are also instances where this angle is considerable. The extreme case 
B 5) 
Fie. 3. 
Seems to be represented by Czesium, where 6 is about 64°. Naturally the question 
arises, what would happen if 8 should become still greater and should finally be +90”. 
In this case the transversal representing our series would be parallel to the ray Ova, and 
hence a» would become infinite. But from the equation (116) we find that 
v=Vv)F4(m+p)*. (13). 
Now obviously this equation is a more general form of Destanpres’ formula for the 
band-spectra into which it converges for u = 0, viz. 
v=V) Fam, 
where », represents the wave-frequency of the first line (the “head” of the band), while 
the upper or the lower sign indicates that the band “shades off” towards the red or 
the violet side respectively. We perceive, then, that the Rypperc-TH1eLe formula 
ineludes as a special case also the DesLanpres formula, and thus opens the prospect of 
TRANS. ROY. SOC. EDIN., VOL. XLI. PART III. (NO. 24). 86 
