353 



Denoting the values of /?,, /?.^ and n in the initial stale and in the final slate of a 

 given transition by n\, n',, n' and /j", ;i'J, n" respectively, we get according to (1) for 

 the freciuency of the radiation emitted during this transition 



.„ = /fiV^(-i.,_M, ^-'%:^'"'(u! (120) 



V. =— . A«-iV' „,— ,, , a - , (121) 

 4 \ n ' n ' / i7C 



1^2 = Ka-N' ( ,}. „~ }. .\. (122) 

 \ n '/J., n /},/ 



The expression for v„ coincides with the simple formula (105) for the fre- 

 ({uencies of the lines of the hydrogen spectrum {N ^ 1) and of the analogous 

 helium lines (iV = 2), which holds when the relativity modifications are neglected. 

 The expression for v., is determinative for the frequency differences of the fine 

 structure components of a spectral line corresponding to given values of n and n", 

 while which contains only n and /i", has influence only on the absolute values 

 of the fre(juencies of these components. In the following a hydrogen line which 

 corresponds to a transition from an initial state n = n' to a final state n = n'' will 

 again be characterised by the symbol (n' n"). In the same way a transition be- 

 tween an initial state /i, = n'j, n.^=- n\ and a final state /ij=/ij, n'„ will be 

 denoted by {n\, n', n", np. The ensemble of components corresponding to all imagin- 

 able transitions between stationary states for which n' and n" have the same values 

 will be obtained by letting n[, assume each of the values 1,2, .../j' and n'J each 

 of the values 1, 2, ... ii". If all transitions between stationary states were possible, 

 the fine structure of a given line would therefore consist of a set of n' x n" compo- 

 nents. On account of n" being smaller than n', this set may conveniently be des- 

 cribed as consisting of n" congruent groups each containing n' components. Thus 

 the line (3 -> 2) would show two congruent triplets, the line (4 -3) three congruent 

 ([uartets, a. s. o. It must, however, be remarked that these groups will in general 

 partly overlap each other (compare fig. 10, 11, 13 on Plate IV). Due to the small 

 value of the constant a appearing in (122), the frequency differences between the 

 components of a given line are so small that it must be expected that in general 

 they cannot be separated entirely by the instrument used for the observations. It 

 is easily seen, however, that this will hold to a less degree for the helium lines 

 (i'V = 2) than for the hydrogen lines (A' -1), because,, due to the factor A'- in (120) 



M Tlie faAor y^'„, ■ wliicli is of importance if tlie tiicory is compared with measurements of the 

 atjsolute values of tlie frequencies of the fine structure components, does not appear in (118), because 

 in the calculations of § 2 the mass of the nucleus was considered as infinite compared witli that of the 

 electron (compare § 5, page 40). 



