LAPLACE. 573 



If in (1) we put I for X^e^ we obtain 



and there is therefore the probability assigned in (2) that the 

 error in the value of u will lie between 



2tk , 2tk 

 and 



Supposing then that r remains constant, the error to be ap- 



/c 



prehended will be least when -^^ is least : and therefore the 



factors of which 7^ is the type must be taken so as to make 

 this expression as small as possible. Put for k its value; and 

 then the expression becomes 



We then make this expression a minimum by the rules of the 

 Differential Calculus, and we find that the factors must be deter- 

 mined by equations of which the type is 



^' hi ' 

 where z^ is a coefficient which is constant for all the factors. 

 With these values of the factors, equation (3) becomes 



„ = _jf + _5 (4); 



.hi ^ hi 



and the limits of the error for which there is the probability 

 assigned in (2) become 



If the function of the facility of error is the same at every 



