﻿212 ON THE USE OF EQUIVALENT FACTORS. 



If we compute the limits of these values of /a, we find that it is only an even 

 chance that ^, for the best solution, is comprised within the limits 



,1= ± 69' .3, and F = ± 106".7. 



Such is the extreme uncertainty of the only clement by which the question of pref- 

 erence between I., II., and III. can be decided. Compared witli it, the inconsiderable 

 differences which we find between the values of fi in I., II., and III. will admit of 

 but the single inference, that either of the systems of values of di, dSl, dtp. Sec, 

 presented in (46), notwithstanding their great disagreements with each other, actu- 

 ally fulfils the criterion of accuracy proposed in the method of least squares so 

 nearly, that it is impossible to give a decisive reason for adopting one rather than 

 another as the most probable solution. It is worthy of notice, moreover, that the 

 arithmetical mean of the above residual errors, irrespective of their signs, is lesK 

 in II. and III. than in I. Thus in this instance the latter would rank lowest of 

 the three, if we were to compute the relative probabilities according to a process 

 recommended by the highest authorities* as the most suitable for ordinary use, in 

 which the probable enor is directly proportional to the arithmetical mean of the errors 

 irrespective of their signs. 



* Laplace, Theorie Andlytique des Prolabilites ; Gauss, Zeitschrift fur Astronomic, B. I. ; Peters, Astr. 

 Nach., No. 1034. 



