LAPLACE. 575 



and therefore the probabiHty that Xyid will lie between t and 

 T ■]- dr is 



1 



2iK sjlT 



e ^^'dr. 



This then is the probability that the error in u will lie 



between =; • and -;=i ; and therefore the probability that 



the error in u will lie between x and x-\-dx \^ 



P^e ^-^ dx. 



This then is what we denoted above by i|r (x) dx ; and we 

 obtain therefore 



/.oo 



I x^lr (x) dx = 



which is least when r^ is least. This leads to the same re- 



suit as before. The mean value of the positive error to be ap- 

 prehended becomes — — ^^-^;r . 



Since 6i = uqi — 8i we have 



If we were to find u from the condition that the sum of the 

 squares of the errors shall be as small as possible, we should obtain 

 by the Differential Calculus 



which coincides with (6) ; so that the result previously obtained 

 for u is the same as that assigned by the condition of making the 

 sum of the squares of the errors as small as possible. It will 

 be remembered that (6) was obtained by assuming that the 

 function of the facility of eri'or is the same at every observation, 

 and that positive and negative errors are equally likely. The 

 result in (4) does not involve these assumptions. It Tvdll be found 



