342 



Dr. A. Schuster. 



[Jan. 27,. 



of probability, and on the other hand the results obtained with 

 different limits showed that the smallest always gave the most favour- 

 able result. The regularity with which this latter fact appears in 

 Tables III and IV, proves it not to be accidental, and if not 

 accidental, it can only mean that the law of harmonic ratios is at 

 least partially a true law. 



The following explanation has occurred to me as possibly account- 

 ing for the facts. We may suppose the harmonic ratios really to 

 exist in appreciable numbers, but to be chiefly confined to fractions, 

 the denominator and numerator of which are larger than those we 

 have taken into account. The fractions, for instance, formed by 

 integers between 100 and 200, if arranged in order of magnitude 

 in our auxiliary tables, would fall generally about midway between 

 the fractions formed by the smaller numbers. Any coincidence with 

 the fractions formed by the higher numbers would reduce the number 

 of possible coincidences with the fractions formed by the smaller 

 numbers, and hence we should have the effect which actually exists, 

 of a number of coincidences smaller than that given by the theory of 

 probability. If, now, in addition to these coincidences with fractions 

 formed by higher numbers, we should have a small quantity of real 

 coincidences with the fractions which we have taken into account, the 

 increased quantity of coincidences for small limits over those of larger 

 limits, would be explained. 



This explanation might be supported by the fact that, for fractions 

 formed by numbers between 70 and 100, the coincidences observed are 

 more numerous than those calculated on the supposition that the lines 

 are all distributed at random. It must, however, be remarked that a 

 similar effect might be produced, if any unknown law existed, con- 

 necting the lines together, a law which in special cases reduced itself 

 to a law of harmonic ratios. 



That some law hitherto undiscovered exists I have no doubt, for 

 just in the cases where we have reason to suppose that different lines 

 belong to one system of vibration, we cannot find any coincidences 

 with harmonic ratios. The lines of sodium, for instance, are all 

 double ; yet in the set of lines given by Thalen the two components 

 approach each other much more rapidly as we pass to the more re- 

 frangible end of the spectrum than they would if the lines were con- 

 nected together by the harmonic law. In the additional sets described 

 by Professors Liveing and Dewar no regularity exists in the distance 

 of the two components. 



A similar remark applies to the four triplets of magnesium lines. 

 The triplets resemble each other in so far as the middle line is always 

 nearest to the most refrangible line ; but the resemblance is only a 

 general one, and there is no absolute relation between the relative 

 distances in each triplet. 



