SCIENCE 



[N. S. Vol. XXV. No. 627 



the spectra of the elements are arranged 

 in series such that the wave lengths of any- 

 one series are functions of only two con- 

 stants and the successive whole numbers? 



So far as I am aware, no answer which 

 is even approximately satisfactory has ever 

 been offered in reply to this question. The 

 fundamental difficulty here has been shown 

 by Lord Rayleigh to lie in our measure of 

 force, so to speak; in the fact that force is 

 a second derivative of displacement with 

 respect to time. Describe any djmamical 

 system you please in terms of a differential , 

 equation; integrate it under conditions 

 which yield a periodic solution; solve for 

 the frequency, and you will find its value 

 always entering to the second power. 



But the difficulty under which the Cam- 

 bridge atom here suffers is not peculiar to 

 it alone. 



Ritz,^^ in a doctor's dissertation of ex- 

 traordinary merit, offered at Gottingen, 

 has succeeded in devising a formula which 

 contains fewer constants than that of 

 Kayser and Runge, yet represents the ob- 

 served wave lengths with a distinctly high- 

 er accuracy. And it might, at first glance, 

 appear that we have here a truly dynamical 

 explanation of the series phenomenon. But 

 on closer inspection one finds that the fun- 

 damental picture— the mechanism, if you 

 please— from which Ritz derives his differ- 

 ential equation is one having properties 

 which are purely hypothetical and, in na- 

 ture as we know it on a larger scale, quite 

 impossible. 



His vibrating body is a square (some- 

 times a plate, sometimes a membrane), 

 whose behavior he studies under different 

 boundary conditions. But it has this re- 

 markable property that the effect of any 

 one element of the membrane upon any 

 other element is not merely a function of 

 the distance separating the elements, but 



''Ritz, Ann. der Physih., 12, 264-310, 1903. 



varies directly as this distance. A device 

 so artificial could at most be called quasi- 

 kinematieal or purely mathematical. How- 

 ever, in each of the particular cases which 

 he has integrated the frequency expression 

 turns out to be practically identical with 

 Rydberg's formula. The only dynamical 

 justification for the entire proceeding ap- 

 pears to lie in the fact that he has chosen a 

 two-dimensional body to yield a double in- 

 finite number of spectral lines. 



Garbasso^* has made an interesting at- 

 tempt to obtain the Kayser and Runge 

 series from certain combinations of Hertz- 

 ian oscillators. And his solutions have 

 this special merit, namely, they all refer to 

 physically realizable models. But the 

 number of frequencies which he has com- 

 puted is too small to furnish even an ap- 

 proximate test as to whether they satisfy 

 the law of Kayser and Runge, much less 

 do they point out a general dynamical sys- 

 tem from which the law of the series may 

 be derived. 



In spite of the fact that no satisfactory 

 explanation has been obtained, one can 

 hardly avoid the conclusion that Rydberg's 

 formula is something more than a con- 

 venient expression for interpolation. The 

 fact that his second constant, N^, is the 

 same for all elements, while another has a 

 characteristic value for each particular 

 element, and that a third constant locates 

 the particular series in any one element, 

 would seem to indicate that these three 

 quantities are in some sense parameters of 

 matter. Yet I am aware that this view is 

 a mere suspicion and is not at present 

 capable of proof. 



When other types of spectra, such as 

 that of iron, are brought under the 'reign 

 of law' we may find a simpler view; or 

 what is more likely, we may feel, with 



" Garbasso, ' Theoretiselie Spectroskopie,' pp. 

 130 and 180. 1906. 



