64 



square of the ratio of the velofily of the electron and the velocily of light c, is 



h^n^ + n.y 





(68) 



where, as in the calculations in § 1, the charge and the mass of the electron are 

 denoted bj' — e and m, and for sake of generalit}' the charge of the nucleus hy Ne. 

 Further n^ and n^ are the integers appearing on the right side of the conditions 

 (16) as factors to Planck's constant. Wliile n^ may take the values 0, 1, 2, . . ., it 

 ■will be seen that n.^ can only take the values 1, 2, ..., because in the present case 

 there will obviouslj' not correspond any stationary state to n^ = 0, since in such 

 a state the electron would collide with the nucleus. Introducing the experimental 

 values for e, h and c, it is found that e- /,c is a small quantity of the same order as 

 10~S; and, unless N is large number, the second term within the bracket on the 

 right side of (68) will consequentlji be very small compared with unity. Putting 

 /jj -j- /j^ = n, it will further be seen that the factor outside the bracket will coin- 

 cide with the expression for W^ given by (41) in § 1, if we look apart from the 

 small correction due to the finite mass of the nucleus. Due to the presence of the 

 second term within the bracket, we thus see that, for any value of;;, formula (68) gives 

 a set of values for E which differ slightly from each other and from — W^. Sommer- 

 feld's theory leads therefore to a direct explanation of the fact, that the hydrogen 

 lines, when observed by instruments of high dispersive power, are split up in a 

 number of components situated closely to each other; and, by means of formula 

 (68) in connection with relation (1), it was actually found possible, within the limits 

 of experimental errors, to account for the frequencies of the components of this 

 socalled fine structure of the hydrogen lines. Moreover the theory was supported 

 in the most striking way by Paschen's-) recent investigation of the fine struc- 

 ture of the lines of the analogous helium spectrum, the frequencies of which are 

 represented approximateh' by formula (35), if in the expression for A', given by (40), 

 we put N = 2. As it should be expected from (68), the components of these lines 

 were found to show frequency differences several times larger than those of the 

 hydrogen lines, and from his measurements Paschen concluded, that it was possible 

 on Sommerfeld's theorj' to account completely for the frequencies of all the com- 

 ponents observed. 



We shall not enter here on the details of the calculation leading to (68), but 

 shall only show how this formula may be simply interpreted from the point of 



') A. Sommerfeld, Ann. d. Phj's. LI, p. 53 (1916). Compare also P. Debte, Phys. Zeitschr., XVII, p. 512 

 ,1916). In the special case of circular orbits (iii = 0), this expression coincides with an expression 

 previous!}- deduced by the writer (Phil. Mag. XXIX p. 332 (1915)), bj' a direct application of the condition 

 I = nh to these periodic motions. 



2) F. Paschen, Ann. d. Phys. L, p. 901 (19161. See also E. J. Evans and C. Croxson, Nature, XCVII, 

 p. 56 (1916). 



