J. F. ENCKE ON THE METHOD OF LEAST. SQUARES. 341 



symmetrical function of any given magnitude. The compound 

 probability, if every error is actually squared, will be 



Ul e- /'= { m X- - 2 [«] X + [«2] } , 



i m 

 IT* 



to which expression it is easy to give the form 



iie-'"{M-^'+-('-'?y}- 



Consequently the negative exponent will be the smallest for 



^=W, (13.) 



and the minimum of the squares of the remaining errors is 



= \n^-\ - W!^ (14.) 



This form leads at once to the estimation of the exactness of 

 this determination of x. If we take 



m 

 then the probability of this hypothesis becomes 



p L mi 



V 



But any other value of x, 



^ = W + A', 

 m 



has the probability 



__e I ™ J. 



TT 



Consequently, according to Proposition II., the probability of 

 the arithmetical mean being the true value, is to the probability 

 of its being erroneous by the magnitude A', as 



1: e ; 



or, according to the above proposition (IV.), the value of H, 

 which is deduced from m equal observations, and which belongs 

 to this determination of x, is 



ll = hV m, (15.) 



so that the function <$ A for this determination of x becomes 



In some cases, instead of expressing the relative exactness of 



