20 ON THE METHOD OF LEAST SQUARES. 



in our determination of cc, viz. 2/Lte, lies within the limits Z, 

 than if we had made use of any other set of factors whatever. 



On this principle Laplace determines what he calls the most 

 advantageous system of factors. 



It does not follow that the value thus obtained for x is the 

 most probable value that could be assigned for it. But if we 

 consider a large number of sets of observations, (the quantities 

 a, b, &c. being the same for all) then the error which we commit 

 by using Laplace's factors will in a greater proportion of cases 

 lie between I, than if w r e had used any other system of factors. 



The investigation has reference merely to the different ways 

 in which by the method of factors a given set of linear equations 

 may be solved. 



We now enter on the analysis requisite to determine P. 



Let the probability that 2/^e will be precisely equal to u, 

 be pdu. Then manifestly 



P = 



and we have therefore only to determine p. 



Let e t 2 ...e n be the errors which occur at the first second &c. 

 observation; (f> i e 1 d6 1 , $ 2 e 2 e?<- 2 ...</> n e n de n be the probabilities of 

 their occurrence: the form of the function <f> determining the 

 law of probability of error, which, for greater generality, we 

 suppose different at each observation. The probability of the 

 concurrence of these errors is of course 



and the first principles of the theory of probabilities show that 

 the value of pdu will be obtained by integrating (1), e r ..e n 

 being subjected to the condition /ze = u. 

 Thus 



with the relation 



