340 



Dr. A. Schuster. 



[Jan. 27, 



so that two lines were said to have the ratio of 3 : 4, for instance, if in 

 the first case the ratio lay between '7500505 and '74995195, and 

 similarly for the second limit. 



If the measurement of the least refrangible line is correct, an error 

 of 1 in 20,000 made in the measurement of the most refrangible line 

 would correspond to the narrower limit. 



The results are given in Table II. 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. The columns headed " calculated," give 

 the number of coincidences which we should expect on the supposi- 

 tion that the lines are distributed at random. The formula employed 

 will be proved in the Appendix. 



Table II. 





Limits ± 



•0000505. 



Limits ± 



•0000755. 



Observed. 



Calculated. 



Observed. 



Calculated. 



0—10 



48 



52 



64 



77 



10—20 



180 



206 



250 



308 



20—30 



329 



363 



469 



544 



30—40 



478 



521 



664 



779 



40—50 



625 



679 



912 



1015 



50—60 



777 



837 



11(53 



1251 



60—70 



886 



968 



1318 



1447 



70—80 



924 



896 



1337 



1340 



80—90 



667 



629 



989 



940 



90—100 



253 



2 a 



393 



361 



Total 



5167 



5392 



7559 



8062 



At first sight the result seems again decidedly against the theory of 

 narmonic 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 that, as has been explained, is an 

 important fact which cannot be left out of account. In the following 

 Table (III), I have compared the number of coincidences for the 

 smaller limit with those calculated from the larger one, on the sup- 

 position that the coincidences are proportional to the limits, as they 

 ought to be if no connexion exists between different lines of the same 

 spectrum. It will be seen, that with the exception of two cases, one 



