574 LAPLACE. 



observation the quantities of which hi is the type are all equal, 

 and so are those of which Ic^ is the type. Thus (4) becomes 



and the limits of error become 



2t;i 



+ 



V(2^,^) • 



If we suppose also that positive and negative errors are 

 equally likely, we have ^ = 0, as in Art. 1004. Thus (5) be- 

 comes 



u = 



(6). 



This agrees with Laplace's result. 



Laplace also presents another view of the subject. Suppose 

 that i/r (x) dx represents the chance that an error will lie between 



X and x-\-dx\ then I x^jr (x) dx may be called the mean value 



of the positive error to be apprehended — la valeur moyenne de 

 Verreur a craindre en plus. Laplace compares an error with a 

 loss at play, and multiplies the amount of the error by the chance 

 of its happening, in the same way as we multiply a gain or loss 

 by the chance of its happening in order to obtain the advantage 

 or disadvantage of a player. Laplace then examines how the 

 mean value of the error to be apprehended may be made as small 

 as possible. 



In equation (4) of Art. 1002 put c — rj', and suppose positive 

 and negative errors equally likely, so that 1=0: then the proba- 

 bility that 27^6^ will lie between and 2?; 



Thus the probability that Syi€i will lie between and t is 



e~^' dv, 



■j^ z*^ «2 



^k^JttJ^ 



