n^>7 



LAPLACE. 577 



the latter is equivalent to the known algebraical theorem that 



{%q^^ is less than s'Sql. 



Moreover suppose that we neglect the second temi on the right- 

 hand side of (5) and of (7); and thus arrive at 



"=%/ (*5)' " = 2^, (^)' 



then there is another reason why (6) is preferable to (8) ; for, by 

 virtue of the algebraical theorem just quoted, the term which is 

 neglected in arriving at (6), is less than the term which is neg- 

 lected in arriving at (8). 



» 



1009. It was shewn in Art. 1007 that there is the probability 



(2) that the limits of the error in (6) are + ,,^ ^, . This involves 



^ " V(S?i ) 

 an unknown quantity h. Laplace proposes to obtain an approxi- 

 mate value of h from the observations themselves. It is shewn in 

 Art. 1006 that there is a certain probability that the sum of the 

 squares of the errors will lie between l—y and l-\-rj. Assume I 

 for the value of the sum of the squares of the errors ; thus 



t€,' = l = s[ \y {2) dz = 2sh\ 



J 



Therefore approximately 



72 _ ^_ ^ (uqi - ^iY . 



^ ~ 2s ~ 2s 

 and with the value of u from (6) of Art. 1007, we obtain 



^^ 2sXqr 



Thus the mean value of the positive error to be apprehended, 



7 



which was found in Art. 1007 to be ., ^ ^. , becomes 



tqNi^'-rrs) 



This agrees with Laplace's page 322. 



37 



