ON OUR KNOAVLEDGE OF SFECTEUM ANALYSIS. 



123 



For the full explanation of Table II. we must refer to tbe paper 

 which has already been quoted. The second column gives the number 

 of fractions investigated for each element. The third column gives a 

 number which ought to be nearly '25 (probably within the limits of the 

 values of the fourth column), if the lines are distributed at random aud 

 decidedly smaller than this niimber if the law of harmonic ratios is true. 



It will be seen that three out of the five elements considered, 

 including the two containing the greatest number of lines, give a mean 

 value greater than '25, and that in the two remaining cases the number, 

 though smaller than '25, falls within the limits into which we must 

 expect it to fall, on the supposition of a distribution at random. 

 Table III. shows the results of a more detailed examination of the iron 

 spectrum, over 10,000 fractions having been calculated and compared with 

 ratios of integers smaller than 100. In order to calculate the number of 

 coincidences which we might expect on the theory of probability, the 

 limits had to be fixed within which we may consider a coincidence to 

 have taken place. These limits must of course depend on the accuracy 

 which we assign to the measurements of the lines. The results were 

 worked out for two different limits, which were ± '0000505 and ± -0000755. 

 When, therefore, two lines had j^eriods the ratio of which fell within the 

 indicated limits of some ratio of two integer numbers smaller than 100, 

 this was called a coincidence. In Table III. the columns headed ' Observed ' 

 and ' Calculated ' give the number of these coincidences as actually found, 

 and as calculated from the theory of probability. In the first row all 

 fractions were taken into account the denominator of which is smaller 

 than 10 ; in the second row the denominator is between 10 and 20, and so 

 on for the other rows. 



Table III. 



The result seems, again, decidedly against the theory of harmonic 

 ratios. For all fractions with denominator smaller than 70 the calculated 

 coincidences are in excess of the observed ones. There seems, however, 

 to be a greater number of ratios than we should expect which agree 

 nearly with fractions the denominators of which lie between 70 and 100. 



If we compare the results given for the two different limits we find 

 that the smaller limit gives results decidedly more favourable to the 

 theory than the larger ones ; and this is an important fact which cannot 



