the Lines of various Spectra. 39 



" Given n quantities distributed arbitrarily between 1 and 

 A, what is the probable number of quotients of two of these n 

 quantities, which will be equal to a + 8, where 8 is small ?" 



He finds* the probable number to be 



n(«-l)(AV-l)g 



where A< — . Applying this result to the spectrum of iron 



he finds, on the whole, fewer harmonics actually present than 

 the formula demands. His conclusion, however, is that the 

 law of distribution, although a complicated one, may in special 

 cases resolve itself into the harmonic one. Since this investi- 

 gation of Schuster's, the search for harmonic ratios has nearly 

 ceased ; but, giving a broadened meaning to the word, Liveing 

 and Dewarf have called attention to certain "harmonic" 

 relations in spectra. These are, the repetition of similar 

 groups, a certain sequence in distance, and a law of physical 

 quality such as intensity and clearness. 



A more recent attempt to apply dynamical reasoning and 

 formulas to the spectra of the elements has been made by 

 M. V. A. Julius J. On dynamical grounds there should be, 

 he thinks, present in spectra lines whose frequencies are the 

 sums or differences of the frequencies of other lines (the com- 

 binational tones of Helmholtz). So he deduces the probable 

 number of coincidences, within limits + S, if the differences 

 are taken between each two of n quantities lying at random 

 between P and Q. He then actually calculates the differences 

 in the case of eight spectra, and finds more coincidences than 

 his formula says mere chance would give him. This theory 

 of combinational rays would account also for the regularity of 

 doublets and triplets, which was first noticed by Hartley, but 

 which M. Julius apparently rediscovered. Until the wave- 

 lengths of the lines in the various spectra are known much 

 more accurately than they are at present, such work as this is 

 almost thrown away. The objective existence of combina- 

 tional tones is now generally acknowledged, however ; and we 

 may certainly expect corresponding phenomena in light-waves. 

 This idea is not a new one ; but M. Julius is the first to 

 elaborate it. In doing so, however, he neglects too much, I 

 think, the physical properties of the lines. We would expect 

 the strongest lines, if any, to be the ones to produce combina- 



* Proc. Roy. Soc. xxxi. (1881). 



t Phil. Trans, clxxiv. (1883). 



X s Ann. de VEcole Polyt. de Delft. 1889. 



