32 ON THE METHOD OF LEAST SQUARES. 



As an illustration of Gauss's principle, let the fourth power 

 of the error be taken as the measure of its importance : 

 (2/-te) 4 = 2//V -f GS/^'V/e/e/ + terms involving odd powers of e. 

 Therefore, 



mean of (2/*e) 4 = 2V * 4 + 24 S/O^&i'V (15) 



and /Lt r .. jJb n must be so determined that this may be a minimum. 



I have already said that the results given by what Laplace 

 called the most advantageous system of factors are not strictly 

 speaking the most probable of all possible results. 



As the distinction involved in this remark seems to me to be 

 essential to a right apprehension of the subject, I will endeavour 

 to illustrate it more fully. 



Kecurring to the equations of condition, as given in p. 18, 

 we see that the values Laplace assigns to the factors HJL^ &c. 

 are independent of V^V Z &c. They depend merely on the 

 coefficients ab &c., which are quantities known a priori, i.e. 

 before observation has assigned certain more or less accurate 

 values to the magnitudes F t F 2 &c. All we then can say is, 

 that if we employ Laplace's system of factors, and also any 

 other, in a large number of cases (the coefficients ab &c., being 

 the same in all) we shall be right within certain limits in a 

 larger proportion of cases when the former system of factors is 

 made use of than when we employ the latter. And this conclu- 

 sion is wholly irrespective of the values of V l F 2 &c., and conse- 

 quently of those which we are led in each particular case to assign 

 to xy &c. The comparison is one of methods, and not at all one 

 of results. But when V l F 2 &c. are known, another way of con- 

 sidering any particular case presents itself. We can then com- 

 pare the probability of different results. For, let us consider a 

 large number of sets of equations of condition (in each of which 

 not only are ab &c. equal, as in the former case, but also F 1 V 2 

 &c.). The true values of the elements xy &c. may be different 

 in each. But in affirming that 77 &c., are the most provable 

 values of xy &c., we affirm that the true values of xy &c. are 

 more frequently equal to fy) &c. than to any other quantities 

 whatever. Here we have no concern with the method by which 

 the values rj &c. were obtained. The comparison is merely one 

 of results. 



