Frequency Differences in Spectral Series. 345 



substitutes N— 3*5 for N — 1, gives a good account of the 

 X-ray spectra (K a -line). This leads us to expect that the 

 constant A contains (N — N ) 2 as a factor, where N may be 

 interpreted as the shielding effect on the outer electrons by 

 the inner ones, the nuclear positive charge being N*. Hence 

 by the above equation, since a may be a constant for a given 

 family, we might expect an equation of the form 



i/v=m(N-N ), 



as is found. Against this point of view stands the fact that 

 the limiting frequencies for the series, v^—A/a 2 , decrease 

 in a given column with increasing atomic number. 



In discussing the X-ray series Sommerfeld finds that the 

 electron's increase of inertia with speed has to be taken into 

 account. Bohr * had shown that the divergence of the 

 observed frequencies, in the case of hydrogen, from 

 Balmer's formula 



n = RQ/2 2 -l/m 2 ) 



could be explained in this manner and corrected the 

 formula to 



n = R(l/2 2 -l/m 2 ){l+« 2 /8(l/2 2 + l/m 2 )}. 



Here « 2 is the small quantity (27re 2 /hc) 2 , (where e is the 

 electronic charge, h Planck's constant, and c the velocity of 

 light) occurring again as a universal constant in the discus- 

 sion of the fine structure of lines. Paschen f measures it 

 more exactly and finds, in the case of helium, that it has the 

 value a 2 = 5-30 x 10 ~ 10 in our units. Sommerfeld % develops 

 this idea in the case of X-rays and shows, inter alia, that the 

 divergence of the observed frequencies for the K and L series 

 can also be explained in this way. If the frequencies of the 

 K series be written 



n = A(l/a 2 -l/n 2 ), 

 he finds that 



A/a Hit) L 1+ *(— ) +«(—) + ••• •]' 



where k = l'6 and p=l. We may therefore surmise a com- 

 plete expression for Vv of the form 



*/- =n ( N _ No )[l + W (N~N O ) 2 +<N-N ) 4 + ....], 



where u,*v, .... are decreasing small quantities and u is of 

 the order of 10~ 5 or less. 



* Phil. Mag. xxix. p. 332 (1915). 

 t Ann. d. Phys. 1. p. 901 (1916). 

 \ Loo. cit. 



Phil. Mag. S. 6. Vol. 36. No. 214. Oct. 1918. 2 A 



