586 LAPLACE. 



■ 



This cannot hold for all values of - and a^ — a^^ unless i/r' {z) be 



s 



independent of z ; say i/r' (z) = c. 



Hence '^ {z) = cz + c, where c and c are constants. 



Thus the function of the facility of error is of the form Ce~''^^ ; 

 and since an error must lie between — oo and oo , we have 



CJ e-'''dz = l', 



J —CO 



therefore C = —r- • 



The result given by the method of least squares, in the case 



of a single unknown quantity, is the same as that obtained by 



taking the average. For if we make the following expression a 



minimum 



{x-ay+ {x-a^y-\- ... + (x-a^y 



we obtain 



a +a +... + as 



x = — . 



s 



Hence the assumption in the preceding investigation, that 

 the average of the results furnished by observations will be the 

 most probable result, is equivalent to the assumption that the 

 method of least squares will give the most probable result. 



1015. Laplace devotes his pages 310 — 312 to shewing, as he 

 says, that in a certain case the method of least squares becomes 

 necessary. The investigation is very simj)le when divested of the 

 cumbrous unsymmetrical form in which Laplace presents it. 



Suppose we require to determine an element from an assem- 

 blage of a large number of observations of various kinds. Let 

 there be s^ observations of the first kind, and from these let the 

 value a^ be deduced for the unknown quantity; let there be s,^ 

 observations of the second kind, and from these let the value a^ be 

 deduced for the unknown quantity ; and so on. 



Take x to represent a hypothetical value of the unknown quan- 

 tity. Assume positive and negative errors to be equally likely; 

 then by Art. 1007 the probability that the error of the result 

 deduced from the first set of observations will lie between x — a^ 



Q 



and x + dx — a.i^ ^ e-^i'^C^- a)'' dx. 



