1881.] On Harmonic Ratios in the Spectra of Gases. 341 



of which is very insignificant, the number of coincidences for the 

 smaller limits is in excess. 



Table III. 



Observed for Calculated from 



smaller limit. larger limit. 



0—10 48 43 



10—20 180 167 



20—30 329 314 



30—40 478 444 



40—50 625 610 



50—60 777 778 



60—70 886 882 



70—80 924 894 



80—90 667 662 



90—100 253 263 



5167 5057 



The fact that the number of coincidences, though falling short of 

 the calculated values for both limits, is relatively greater for the 

 smaller, suggests the possibility that still narrower limits might give 

 results which are still more favourable to the theory of harmonic 

 ratios. This indeed is the case. I have counted for all fractions, the 

 denominator of which is smaller than 30, the number of coincidences 

 for a series of 8 limits. The results are embodied in Table IV, and 

 show that there is a tendency of the fractions to aggregate into the 

 compartments for smaller limits. With the exception of the first and 

 last numbers, there is a gradual decrease of coincidences as we 

 recede from the harmonic ratios. 



Table IV. 



Limits ± •0000. Number of coincidences. 



000—095 71 



095—195 85 



195—205 78 



295—395 70 



395—495 68 



495—595 66 



595—695 56 



695—755 41 



We have now to reconcile two apparently opposite results of our 

 calculations. On the one hand it was found that the coincidences 

 with harmonic ratios are fewer than we should expect from the theory 



