572 LAPLACE. 



In a system of observations the quantity given by tbe observa- 

 tion is in general not the element which we wish to determine, 

 but some function of that element. We suppose that we already 

 know the approximate value of the element, and that the required 

 correction is so small that we may neglect its square and higher 

 powers. Let the correction be represented by u ; let A^ be the 

 approximate value of the function at the ^*^ observation, and 

 Ai 4- uq^i its corrected value. Let Bi be the value of the function 

 given by observation, e^ the unknown error of this observation. 

 Then we shall have 



^i + 6i = u4, + uq^. 



Put 8i for Bi — Ai, so that Bi is the excess of the observed 

 value above the approximate value of the function ; thus we 

 have 



ei = uqi — 8i. 



A similar equation will be furnished by each of the s observa- 

 tions. All the quantities of which qi and S^ are the types will 

 be known, and all those of which e^ is the type will be unknown. 

 We wish to obtain from the system of equations the best value 

 of u. 



Form the sum of all such equations as the preceding, each 

 multiplied by a factor of which 7^ is the type. Thus we obtain 



tyiei = u^yiqi-t%Si (1). 



Then by equation (4) of Art. 1002 the probability that 2)7^6^ 

 will lie between I— t] and Z -f ?; is 



., _v^ 



^"^ dv, 



where I and /c have the values assigned in that Article. 



Put 27-5 = ^ ; thus the probability that Xyi€i will lie between 

 I — 2tk and I + 2tk is 



~ I'e-^'dt (2). 



