Cause of the Structure of Spectra. 253 



the formula 



___ 107250 



n = 3 9 7 .50 — > ~ t-, 



(3-2'343/s) 2 ' 



we find the experimental numbers reproduced with a maxi- 

 mum error of 8 in 1000. This of course is a very large 

 •error in investigations connected with wave-numbers of light, 

 but, nevertheless, although our final result is not an accurate 

 representation of the peculiar Mg series, our analysis proves 

 conclusively enough that we have to do with a phenomenon 

 of the nature of nodal subdivision of a vibrating body. The 

 series of numbers marked A, above, proves that the modes of 

 vibration analogous to harmonics possess the harmonic periods 

 1/3, 1/4, 1/5, 1/6, 1/7, 1/8 within one per cent. 



3. A Kinematical Analysis of Buhners Formula. 



It will simplify the rest of our work if we now investigate 

 a theory of Balmer's formula in the somewhat more general 

 form given to it by Rydberg for elements other than hydrogen, 

 namely, 



n = n — B/(m + ti) 2 . 



If X is the wave-length in free sether, where the velocity of 

 light is Y, then t the time of vibration being equal to \/V 



__i r i i_ 



2Y*/n \ s /n + vB/(m + ^J + </n — */B/(m+ p) 



}, (3) 



and t appears as the sum of two times. This suggests the 

 following line of thought. Consider the atom simplified to a 

 circle which is to represent the closed path round which a dis- 

 turbance travels. Let it travel in the two opposite directions 

 with angular velocity v for the radius-vector from centre to 

 disturbance, and let another radius-vector, which we may 

 call the reference-vector, travel with angular velocity u, so 

 that the one disturbance-vector has an angular velocity v + u, 

 and the other v — u, relative to the reference-vector. Now 

 the one disturbance meets the reference-vector after time 

 2tt/(v + u), and, if instantaneously reflected with the same 

 velocity, will meet it again after time 27^/(l , — ?*), when it is 

 again instantaneously reflected and starts to repeat the 

 movement. The other disturbance would meet the reference- 

 vector after time 27r/(r — u), and on reflexion again after 

 time 27r/(i* + ^), when the whole motion would be repeated. 

 Thus, after a time 



2w{l/(t> + i*) + l/(»-i0} 8 



both disturbances would be ready to repeat their movements. 



