OAT THE METHOD OF LEAST SQUARES. 23 



or 



where we have supposed 



u = 2 



It is evident, that whatever I may be, this expression for P 

 is a maximum when 



2/^ 2 & 2 is a minimum. 



Hence we get the following remarkable conclusion : When the 

 number of observations increases sine limite the most advantage- 

 ous system of factors are those which make 



a mnmum. 



It remains to determine JJL from the condition of the minimum 

 taken in connexion with those already stated, viz. 



2yua = l, 2/^ = 0, &c. = 0. 

 We have 



(A). 



= 



&c. = J 

 Let X 15 X 2 ... \ be indeterminate factors, then we may put 



............ (B). 



&c. = &c. J 



From the n equations (B) we deduce a new system of p 

 equations. To obtain the first of these, we multiply equations 



(B) by % , j5, &c. respectively, and add the results. For the 



&1 KZ 



second, we employ instead of the factors yf , &c., the factors 



#i 



j\, T| , &c. and then proceed as before. And similarly for the 



K! K% 



others. 



