1881.] On Harmonic Ratios in the Spectra of Gases. 339 



deviation from "25 in the fourth column. The term probable in pro- 

 bable deviation is here used in the same sense as in " probable error." 

 It has been calculated by means of the approximate formula — 



2 , /SsS 

 p = — — ertc. a / — - 



where a = *25, 



5= the number of lines in each spectrum, 



p is the probability that the mean value lies between + 8 ; for p equal 

 fo> one-half, c is the probable deviation. 



It will be noticed that the actual deviation never differs much from 

 the probable one, but that it is greater for the two elements having the 

 greatest number of lines. If, therefore, any deduction is to be drawn 

 from the preceding table, it is that the ratios formed by two given 

 lines rather seem to avoid harmonic ratios. 



The method just explained, and which has given us such decidedly 

 negative results, I believe to be very well adapted for the discussion 

 of spectra which have a comparatively small number of lines; but 

 the iron spectrum may be examined by a more direct and complete 

 method. We may directly calculate how many fractions ought to 

 agree within certain small limits with harmonic ratios if no law exists, 

 and counting how many do thus coincide. I have found, for instance, 

 twenty-eight pairs of lines which coincide within limits so narrow 

 that they can be easily due to errors of measurements with fractions, 

 the denominator and numerator of which are both smaller than 10. 

 This number might appear large at first sight, and some support for 

 the law of harmonic ratios might be derived from it. But the cal- 

 culation gives the larger number 32 as the one we ought to expect, if 

 all the lines were distributed at random ; so that here, also, the frac- 

 tions seem to avoid rather the harmonic ratios. 



A little difficulty is experienced in fixing the limits within which 

 we may consider a coincidence to have taken place. They must 



o 



depend, of course, on the accuracy which we assign to Angstrom's 

 measurements. 



I thought it best to work out the results with two different limits, 

 one of which was half as large again as the other. We gain a decided 

 advantage in classifying the results for two limits. It is in fact 

 equivalent to using a third method of discussion, for supposing the 

 spectral lines to be distributed at random, the number of coincidences 

 found should be proportioned to the limits chosen. If, on the other 

 hand, the law of harmonic ratios is correct, the narrower limit should 

 relatively show the greater number of coincidences. The limits taken 

 were — 



± -0000505 



and + -0000755. 



