0/V THE METHOD OF LEAST SQUARES. 31 



Gauss, adopting this way of considering the subject, pointed 

 out that it involved the postulate that the importance of the 

 error S^e, i.e. the detriment of which it is the cause, is propor- 

 tional to its arithmetical magnitude. Now, as he observes, the 

 importance of the error may be just as well supposed to vary 

 as the square of its magnitude : in fact, it does not, strictly 

 speaking, admit of arithmetical evaluation at all. We must 

 assume that it is represented by some direct function of its 

 magnitude, such that both vanish together. One assumption is 

 not more arbitrary than another. Let us suppose, therefore, 

 that the importance of the, error is represented by (2/^e) 2 . That 

 is, that (S/u-e) 2 is the function whose mean value is to be made a 

 minimum. I now proceed to find it. 



(^6) 2 =S/,V+2S^ 2 6^ 2 ......... (13). 



The mean value of e 2 is /^ e 2 $ede = 2& 2 . 



Hence, that of 2/&V is 2 2^k\ 



The mean value of S/u^e^e, is zero, positive and negative 

 errors of the same magnitude occurring with equal frequency on 

 the long run. 



Consequently, 



mean of (2/*e) 2 = 2 S^ 2 ......... (14) ; 



and therefore, as before, 2/u. 2 & 2 is to be made a minimum. The 

 rest of the investigation is of course the same as that of La- 

 place. 



Nothing can be simpler or more satisfactory than this 

 demonstration. It is free from all analytical difficulty, and 

 applicable whatever be the number of observations, whereas that 

 of Laplace requires this number to be very large. 



Recurring to equation (11), differentiating it for m 2 , and then 

 making m 0, we find 



r x * = r " *, *i f 



^ -00 ^ -03 J - 



and as the first member of this equation is evidently the mean 

 value of w 2 or of (S/*e) 2 , this is a new verification of our analysis. 



