PROF. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 61 



may be a positive or negative fraction, or may be > 1. If we know that the first 

 observed line of a series is actually the first member m = 1 for that line and ju. is 

 definite. If we assume, for instance, that the first doublet of a principal series has 

 the same separation as those of the associated series, the question is settled ; or if 

 such a line has not been observed, then the degree of convergence of the separation of 

 any P doublet will give its order at once, e.g., the fact that the D lines in Na have 

 the same separation 17 '21 as the sharp series, would show that the D lines are 

 actually the first members of the series. If these had happened to have been outside 

 the region of observation, however, and the doublet (3302, 3303) had been the first 

 observed lines of the P series, their separation (5 '50) would have at once shown that 

 for them m = 2, and again /x would be definitely determined. In spectra with no 

 doublets or triplets, as in the singlet series of He or O, this criterion would not be 

 available. Recourse might then be had to the fact that in all known cases the first 

 line of a P series is not very different from the first line calculated from any approxi- 

 mate formula of the sharp series witli sign reversed. It was this consideration which 

 led RYDBERG to postulate a value for p. > I in the principal series of the alkali 

 metals. In all cases there is nothing to distinguish the order for a diffuse series. If, 

 therefore, we wish to discuss how the value of /u. changes from element to element, 

 RYDBEUG'S formula can only give an approximation to its fractional part alone. As 

 soon, however, as a more approximate formula, such as that used in this paper (/i) 

 or that used by RITZ (a/m 2 ), is applied, /u, becomes definite at once e.</., in the 

 principal series of the alkalies the denominator is (/' denoting a fraction) 

 ni+l+j'ajm anc l not m+foi/m, for the latter will not reproduce the series within 

 the limits of observational error, and consequently /A > 1 ; in the sharp series \L < 1. 

 In the sharp series, however, of Group II. or the Zn group /A > 1, from which it may 

 be remarked incidentally it follows, using RYDBERG'S law, that the wanting principal 

 series must be looked for in the ultra-red. The unique determination of ju is a matter 

 of the first importance for comparative study. It only fails when the lines are so few 

 or the measurements so bad that afm or a./(m+ l) will either of them reproduce them 

 within the observational limits. 



In order to draw safe conclusions as to any relationships between the constants for 

 various elements, or between the different series of the same element, we need to 

 know their limits of possible variation. This is possible, as mentioned above, when 

 the data are based on the measurements of KAYSER and RUNGE or of SAUNDERS. It 

 is not proposed, in what follows, to give complete expressions for these variations in 

 terms of observational errors, but the method can, perhaps, be understood best by 

 taking an actual example say that of K.S. The wave-numbers of K.S. (3.4.5) 

 reduced to vacuum from the wave-lengths given by K.R. are 14407 '80, 1723072, 

 18721 '20, with possible errors, as deduced from wave-length errors of T04, "15, '53 

 respectively. The true values are taken to be 14407'80 + r04p, 1723072 + -15</, 

 18721 '20 +'53r, where p, q, r may have any values from 1 to +1. The lines as 



