THE STRUCTURE OF THE SERIES OF LINE- AND BAND-SPECTRA. 557 
while the second column contains, in the same units, the residuals computed from the 
Rypserc-THIELE formula : 
SEM esate an 8 sacl gahe} ey? 
-36. | -04 | 0-0 0-0 
Some 0°38 0-0 0-0 
+ 1:3 +0°1 0-0 -01 
+0°8 +02 +03 +01 
—0°3 20: % + 0°2 0-0 
— 0-4 00 | O(n i) waeOs4: 
Po 10:5 Ost" | +1:0 +08 
2 0s O11, amecets! +11 
| | 
A comparison of the two columns shows the considerable superiority of the 
Rypserc-THiELte formula. Indeed, the representation of the observed wave-lengths 
by Kaysrr’s formula appeared to Professor Kayser himself so little satisfactory that 
he abandoned this method of computing the constants. In order to obtain more reliable 
values of these, he disregarded the first line and computed his equation from the second, 
third and fourth lines only. The residuals in this case are given in the third column. 
In another calculation, the result of which is represented in the residuals of the fourth 
column, he computed the constants from all the lines except the first. We observe 
that in both cases his endeavour to improve the shorter wave-lengths was defeated, 
notwithstanding the sacrifice of the first line, which shows an enormous discordance. The 
computations demonstrate clearly that KaysEr’s formula cannot be made to represent 
the observed wave-lengths nearly so well as the RypBERG-THIELE equation, although the 
discrepancies of this latter, in the special case considered, are much greater than usual. 
But it may be urged that the introduction of another unknown in Kayssr’s formula 
might improve matters. To decide this point, let us consider the more general 
equation 
10®’“!=a+bm-2+em-*+dm-6, 
I have computed the numerical forms of this equation for the principal series of 
Li, Na and K, by using first the 1., 3., 5. and 7. lines of the series, and obtained the 
following equations : 
Li: 108A-1 = 43552-5 — 130802 m-2 — 1156838 m-4+ 124237 m-6 
Na: 108A-1=41516°7 — 128378 m-2— 829617 m-4 — 36437 m-§ 
K: 108\7!=35075°6 — 126522 m-2— 618583 m4 — 287275 m-6 
The residuals, expressed in units of wave-frequency, are shown in the subjoined 
table under the heading K-R, while those of the RypBERG-THIELE sougul: are repeated 
from the preceding tables and given under Ry-Th. 
