34t) Atomic Number and Frequency Differences, 



Now it is not possible with so few points to determine 

 such an expansion with certainty, but we can obtain an 

 approximation as follows : — Solving the equation 



(1) N = N + <? v 1 / 2 -r^' 2 



for s]v we obtain 



(2) 



The constants in (1) may be obtained from three points. 

 If, for example, we take the values for Ga, In, Tl, we find 

 that the series obtained from 



N = -6843 + 1-07367 ^ 2 -2-1024xl0- 5 ^/ 2 , 

 i.e. v/v=N'(l + r9581xl0- 5 N' 2 + l-1503xl0- 9 N' 4 + ....), 



where N' has been written for / n ) , is satisfied by 



all three points. Substituting N = 13 for Al we obtain 138*5 

 instead of 112-07 cm. It is plainly possible to obtain an 

 expansion of the form (2) passing through the Al point as 

 welL What interests us is the fact that the first coefficient 

 is of the same magnitude as Sommerfeld's calculation (10~ 5 ) 

 ascribes to the inertia effect. The exact values obtained have 

 no particular interest at this stage, being so dependent on 

 the particular function chosen, but a calculation shows that 

 the same order of magnitude for r/q corrects the points for 

 Au and Hg. It is noticeable, on the other hand, that Bi 

 falls into line with Sb and As without this correction, but we 

 are dealing here, perhaps, with a phenomenon of a different 

 kind. 



Summary. 



The law of Bydberg, and Kayser and Runge that the 

 square root of the doublet and triplet differences is propor- 

 tional to the atomic weights, has been subjected to numerical 

 tests, substituting, however, atomic number for atomic 

 weight. 



A similar relationship has been found among Kayser and 

 Runge's frequency intervals in the nitrogen column, and 

 a frequency difference of 41*8 cm." 1 in the phosphorus 



