328 J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 



then 



al! = ^{r {^ {s - c), C) = ^{r {s, c), 



= t (M* - ^)' *) = ^ (^' *)' 



= i^ {^ (J? -«),«) ='»/^ (*,«). 



But these three formulae, from what has been said above, should 

 give a symmetrical form to x in reference to a, b, c, which, as s 

 is already in itself a symmetrical form, can only be if c, b, a dis- 

 appear in the development by the powers of s ; consequently, in 

 the same manner, from all three, 



X = yjr {s). 



If now in a given case a = b = c, the only possible value of 



X would he x = a; consequently, 



a = yfr {3 a), 



or the function sign ■f would signify the dividing by 3. Hence 



follows, 



a + b + c 



^= 3 — 



for three observations. 



In like manner it follows generally, that if for n observations, 



the value to be chosen is 



a + b + c . . . . + n 



X = ; 



n 



then, if another observation j9 is added, for [n + \) observations, 



a-\-b-{-c....-\-n+p 



X = :; 



« + 1 

 ought to be chosen ; for the equahty of the observations re- 

 quires that if we put 



a + 6 + c....+«-f-jo = «, 

 then 



X = ^ {^-{s-p),p\&c. 



should be a symmetrical function of all the n + l values. Now, 

 as this form is good for three values, it follows that it is so also 

 for any number of observations, great or small. 



This proposition, — that in any number of equally good obser- 

 vations of an unknown quantity, the arithmetical mean of all 

 gives the value which is to be preferred, and which consequently 

 must be regarded as the most probable value, — has been re- 

 ceived as a fundamental proposition ever since combinations of 



