441 



We have here, he said, the results of three distinct series 

 of experiments, conducted upon different principles, and by 

 different processes ; and, as we observe, the mean values of the 

 coefficient thus deduced present the most complete agree- 

 ment, the greatest difference amounting only to '00011. It is 

 almost indifferent under these circumstances, which of these 

 results be adopted ; but in order to do complete justice to 

 the subject, we shall here investigate the most probable value 

 of the final mean, as given by the calculus of probabilities. 



In order to do this, it is necessary to deduce, in the first 

 instance, the probable error of each mean, as derived from 

 the results of its own series. This error, it is well known, 

 is expressed by the formula 



_ -455Sfo~a) 2 

 E - n (n - 1) ' 



in which S (x — a) 2 denotes the sum of the squares of the 

 differences of each partial result and the mean, and n the 

 number of observations. The results of this calculation are 

 given in the last column of the annexed Table. 



Series. 



n 



m 



E 



1 

 2 

 3 



11 



19 

 24 



•01151 

 •01150 

 •01140 



•00031 

 •00005 

 •00006 



The most probable value of the final mean, will now be 

 given by the formula 



m l _i_ ^ _i_ H^ 



2 *T" „ 2 "!"„ 2 



m — 



El 



E2 E3 



El 2+ E 2 2 + E 3 2 



from which we find m — *01145. 



In the preceding deduction we have supposed that the 

 only errors to which the separate values of m are liable are 

 the errors of observation, in which case the positive and 



