5S2 LAPLACE. 



Tims we have completed one mode of arriving at the result, 

 and we shall now pass on to the other. 



If we proceed as in the latter part of Art. 1007 we shall 

 find that the probability that the error in the value of x, when 

 it is determined by (5), lies between t and t-\-dt\^ 



^e ^^dt (11). 



For put c = ?; in equation (4) of Art. 1002. Then the proba- 

 bility that SYi^i will lie between and 2?; 



ZksIit J ^r, ZkisItt J 



Thus the probability that S^ej will lie between t and r + dr is 



1 



e ^"^ dr, 



and therefore the probability that ^Yi^i will lie between Z + t and 

 Z + T + dr is 



1 _ j1 



This is therefore the probability that the error in the value of 

 X when determined by (5) -will lie between 



T 1 T + dr 

 ;s— and -= . 



And therefore the probability that the error in the value of x 

 when determined by (5) will lie betv/een t and t^dt\^ given by (11). 



The mean value of the positive error to be apprehended in the 

 value of X will be obtained by multiplying the expression in (11) 

 by t and integrating between the limits and oo for t. Thus, since 



^Ji^i — Ij we obtain -^- for the result : and therefore if we pro- 



ceed to make this mean error as small as possible we obtain the 

 same values as before for the factors y,^, %, %, -" 



It will be interesting to develop the value of k. Multiply 

 equation (7) by 7^, and sum for all values of z; thus by (6) we 

 obtain 



/c^ = X. 



