256 Mr. W. Sutherland on the 



summarized simply by defining a notation to express them as 

 follows : — In the two Principal series n Q has the one value p n , 

 .and /J, has the value p fii in the first and p fi 2 in the second. The 

 Diffuse and Sharp series taken together may be called the 

 Usual series in contrast to the Principal series, which have 

 been clearly shown only in the Alkali spectra. 



In the first Diffuse and the first Sharp series the value of n Q 

 is nearly the same and will be denoted by u n ; but if a 

 difference has to be expressed, a n and s r> will be used. 

 Similarly in the second Diffuse and second Sharp series n 

 becomes u n + v 1? and in the third it is u n + j/ 2 . 



In all the Diffuse series //, has the value «*/*, and in all the 

 Sharp series s /jl. 



The most important of Rydberg's laws is that the difference 

 between the values o£ n for a Principal series and for a 

 Usual series of the same order is equal to the wave-number of 

 ihat line which is the first member of the Principal series. 

 In symbols this is 



p^ -„w = p w — B/(1+ P p 1 ) 2 .... (7) 

 ... w n = B/(l+ pMl ) 2 , (8) 



probably we can write similarly tt ra -f v = B/(l-f-pU 2 ) 2 . Ryd- 

 berg gives the data establishing this law in Wied. Ann. Iviii. 

 p. 674 (1896). He holds that there is a reciprocal relation 

 to go along with (8), namely, 



,n = B/(H-„ / , 1 ) 2 ; (9) 



ibut this has not yet been as well established as (8). Rydberg 

 compresses (8) and (9) into the following terse expression: — 

 The wave-numbers of the Principal and Sharp series are given 

 by the formula 



n _^ 1 1 /in\ 



± B ~ ('mVufif ~ (m" + p/*) 2 ' * " { } 

 where for the Principal series m has always the value 1 in the 

 first term on the right, while in the second term it has any 

 integral value ; and for the Sharp series m has the fixed value 

 1 in the second term but any integral value in the first. Thus 

 a Principal and a Sharp series may be regarded as having their 

 first line in common and as being branches of a single series. 

 The value of Rydberg's discoveries can be best appreciated by 

 following his reasonings concerning the hydrogen spectra. 

 If we write out the wave-numbers for Balmer's series in 

 hydrogen and take the differences of the successive members, 

 we get a series of numbers almost the same as the correspond- 

 ing differences in the lithium Diffuse series and in the sodium 

 Diffuse series, so that probably the Balmer series in hydrogen 

 is a Diffuse series, and B for Li and JSTa must be nearly the 



