AucusT 17, 1899] 
NATURE 
369 
longer be considered as expressing the law connecting 
the mutual relationships between the wave-lengths of 
lines in a spectrum. 
Liveing and Dewar! next called attention to the fact 
has extended. They have attacked the question mathe- 
matically from different standpoints. In the following 
table I give the formula employed by Kayser and Runge, 
and that employed by Rydberg. : 
Fic. 3.—Spectrum of the Cleveite gases. 
that the distance between two consecutive lines of these 
groupings decreases with diminishing wave-lengths, so 
that eventually the lines asymptotically approach a limit. 
“Harmonic” was the term they used to express such a 
series of similar groups of lines. 
It was, however, the work of Balmer which gave the 
subject the impetus by which it has of late years made 
great progress. 
Balmer? published a formula by which the positions of 
the hydrogen lines could be calculated with wonderful 
accuracy. The formula is as follows :— 
y 
ae 
A, 
Re” — 4 
in which A is the wave-length in vacuo of the line to be 
calculated, A a constant common for all the lines, and 
z one of a series of numbers from 3 to 15. 
The constant A, according to Cornu’s measurements, is 
3645°42 Angstrom units, or, using Ames’ more correct 
value, 3647°20 Angstrom units. 
Simultaneously with Balmer’s discovery, Cornu? pointed 
out that the lines of aluminium and thallium, which are 
readily reversible, bear a definite relation to those of 
hydrogen, while at a later date Deslandres* published a 
formula from which could be calculated the wave-lengths 
of the lines composing the bands of numerous elements. 
The above brief history brings us down to the year 
1887, in which Kayser and Runge ® began their series of 
minute investigations dealing with a great number of 
Formulae for Calculating Series. 
Kayser and Runge Rydberg. 
T=A+Bn-2+€n-4 n=n,- __No 
r (70 +m)? 
where where 
A= wave-length | 2=wave frequency 
(or Hh PHS Gc 
8 me frequency) No= 1097216 (a constant 
R | applicable to all 
ail Bae series of every ele- 
A, B, C= constants calculated ment) 
for each series. 
characteristic con- 
°=-~ stants varying with 
each series, 
In the above formula, when 
M=n, N=N 3 OF My is the 
limit which the number of 
| waves 2 approaches when 7 is 
infinite. 
The value of N, is assumed 
| by Rydberg to be constant, as 
it varies only slightly, and this 
variation may be due to un- 
| certain data. 
The constants for the principal 
series are different from those 
used in the subordinate series. 
For sub-series of Na, K, Rb, 
Cs, Cu, Ag, Al, In, and TI, the 
constants Band Care identical. 
For all series the constant B 
does not vary by more than 22 
per cent. This constant B cor- 
responds to Rydberg’s No. 
You will see that they are not by any means identical, 
but both deal with wave frequency, that is to say, the 
Fic. 4.—Spectrum of the Cleveite gases sorted out into six regular series. 
elements. 
commenced to take up the subject. A f 
I will state generally the ground over which their work 
1 Phil. Trans., 1883, p. 213, and previously. 
2 Wied. Ann. (1885), 25, 80. 
8 Comptes rendus (1885), 100, 1181. 
4 bid. (1886), 103, 375 3 (1887). 104, 972- 
5 ** Ueber die Spectren der Elemente” (Aéhandlungen d. K. Akad. 
Berlin, 1880, 1889, 1890, 1891, 1892, 1893. 
6 Svenska Vetenskat. Akad. Handlingar, Stockholm (1890), 23, No. 11; 
Wied. Annailen (1893), 50, 629 ; (1894), 52, 119 
NO. 1555, VOL. 60] 
It was also about this time that Rydberg® | n a 
| they employ a certain sign, 7, to represent the 
Then 
succes- 
certain 
number of waves in a given unit of length. 
sive integers which have to be used to define 
of their terms, and in addition to this we get certain 
constants which are calculated for each series. The 
most interesting consideration from this point of view is 
that Rydberg found that there was one constant which 
he could use in order to search for the series of lines in 
the spectra of all the chemical elements with which 
he worked. There was no common constant similar 
