344 Mr. H. Bell on Atomic Number and 



the upward curvature. The tables contain values calculated 

 on this assumption, and it will be seen that an improvement 

 results, especially for large values of N. 

 This exception is to be expected, for, if 



V^=m(N-N )=mN(l-N /N) 



be the actual locus of the given points, i. e. 



1 log v = log m + log N-N /N, 



then, on a logarithmic diagram, the points will be above the 

 logarithmic straight line (N /N being negative), i. e. the loga- 

 rithms curve downwards relatively to the actual observations. 

 This is seen to be the case from the tables, the logarithmic 

 set of values being worse than the others. We are therefore 

 compelled to regard Runge and Precht's law as being of the 

 nature of an empiricism, especially since the improvement, 

 where it exists, is not great. The logarithmic method fails 

 to show graphically the branching relation of the columns, 

 as is clearly shown by a reference to the papers already 

 cited. A further empirical improvement would plainly be 

 effected by assuming z/ = A(N — N ). 



The question of doublet and triplet differences has recently 

 been gone into extensively by Sommerfeld *. If we write 

 the equation to a series as 



\a 2 (??i + fju) V 



where m has integral values and fx, a, and A are curve-fitting 

 constants, then his theory ascribes the constant doublet dif- 

 ferences to the term l/d\ The above expression for n is in 

 fact proportional to the loss of energy for a revolving electron 

 when falling from an outer to an inner Bohr ring. If the 

 inner ring for the series be in reality double (of different 

 eccentricities according to Sommerfeld) then a has two 

 possible values, and we have the constant frequency 

 difference 



_ A /1 1 \ A / 1 _J_ 



F-m-fi.-A^ (m+/ , )2 ;- A ^ 2 2- ( , m+ia)2 



=A W~~v)' 



Now Moseley's equation 



n = R(N-l) 2 (l/l 2 -l/2 2 ), 



where R is Rydberg's constant and in which Sommerfeld 



* Ann. d. Phys. li. (1916). 



