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LII. On a Method of Discriminating Real from Accidental 

 Coincidences between the Lines of Different Spectra. By 

 C. Runge, Professor of Mathematics at the technische Hoch- 

 schule, Hanover*. 



IN the Philosophical Magazine for January 1888 E. F. J. 

 Love gives a method of discriminating real from acci- 

 dental coincidences between the lines of different spectra. 

 The method is as follows : — The differences between the 

 wave-lengths of the lines compared are arranged in groups, 

 each group containing those observations the errors of which 

 lie within certain narrow limits. The number of observations 

 in each group is then plotted as an ordinate of a curve, the 

 average error of the group being the abscissa. It seems 

 allowable to assume that this curve will have the form of the 

 curve given by the law of error y = ae~ c2x2 in case the coin- 

 cidences are real and not merely accidental. Any serious 

 divergence from the form of the latter curve will therefore 

 indicate that the coincidences are accidental. So far, I 

 think, one may agree with the author. But he argues 

 further: — If the plotted curve resembles the curve given bv 

 the law of error, the coincidences are not accidental. I d*o 

 not see the necessity to draw this conclusion. On the con- 

 trary, I am able to show that for a certain distribution of lines 

 in one spectrum the plotted curve must always resemble the 

 error curve for any lines that one pleases to take as lines of 

 the other spectrum. Let X , X x , . . . \ n be the wave-lengths of 

 any spectrum, X the smallest and X n the largest. What then 

 is the probability that an arbitrary number X between X and 

 \ n does not differ by more than x from the nearest of the wave- 

 lengths X , X l5 . . . \ n ? To find this probability one must add 

 together all the parts between X and X„ that differ by not more 

 than x from one of the numbers X , X x . . . X». This sum divided 

 by \ n — X is equal to the probability in question. Let 

 di, d 2 . . . d n signify the differences of two consecutive wave- 

 lengths in such order that d\ < d 2 < d d . . . < d n , and let 

 d v+1 be the first of these quantities, that is not smaller than 

 2x, so that dv < 2# < d v+1 . Then all the intervals d h d 2 , . . . 

 d v have to be included in the sum, whilst for each of the 

 intervals d v+h d v+2 , . . . d n only a part equal to 2x has to 

 be added. We find therefore the probability in question 



* Communicated by the Author. 





