576 LAPLACE. 



that the value of u in (4) is the same as we should obtain by 

 seeking the minimum value of 



that is the minimum value of 



1008. It is very important to observe how much is demon- 

 strated with respect to the results (4), (5), and (6) of the preceding 

 Article. There is nothing to assure us that we thus obtain the 

 most prohable value of u, in the strict sense of these words ; neither 

 Laplace nor Poisson makes such an assertion : they speak of the 

 method as the most advantageous method, as the method luhich 

 ought to he 'preferred. 



Let us compare this method with another which would perhaps 

 appear the most natural, namely that in which each of the factors 

 7j, 72' ••• ^^ taken equal to unity. 



In the preceding Ai'ticle we arrived at the following result, 



Now suppose that instead of giving to the factors 7^, 73, ... the 

 values assigned in the preceding Article we take each of them 

 equal to unity ; then the quantity I of the preceding Article be- 

 comes SA^i, that is sh if we suppose the function of the facility of 

 error to be the same at each observation. Hence instead of (5) we 

 shall have 



"=2^ + % • W- 



Now (5) is preferable to (7) because it was shewn in the pre- 

 ceding Article that, corresponding to a given probability, the limits 

 of the error in (5) are less than the limits of the error in (7). In 



2T7^ 

 fact the limits of the error in (5) are + ..^ g, , and in (7) they 



are + — ^ — - ; and the result that the former limits are less than 

 Mi 



