18 ON THE METHOD OF LEAST SQUARES. 



The two notions are heterogeneous : the conditions implied 

 by the one may be fulfilled without introducing those required 

 by the other : and we have already seen that by losing sight of 

 this distinction, we are led to the inadmissible conclusion, that a 

 principle recognised as true ci priori necessarily implies a result, 

 viz. the universal existence of a special law of error, not only 

 not true a priori, but not true at all. 



Having stated what seem to me to be the objections in point 

 of logical accuracy to this mode of considering the subject, 

 I will briefly point out the manner in which, from the law of 

 eiTor already obtained, the method of least squares is to be 

 deduced. 



Let 



e = ac + ~b + &c. ~l 



e = ax + + &c. F 2 



&c. = &c. 



e n = a n x + b n y + &c. - V n J 



be the system of equations of condition, which are to be com- 

 bined together so as to give the values of x, y, &c. The error 

 committed at the first observation is v at the second e 2 , and so 

 on ; each observation corresponding to an equation of condition. 

 The probability of the concurrence of all these errors is, 

 (according to the law of error already arrived at) propor- 

 tional to 



e - h2[(aix + liy + &c. - Fi) 2 + (a z x + Izy + &c. - F 2 ) 2 + &c.] 



and it is to be made a maximum by the most probable values of 

 x, y, &c. These values will therefore make 



(a 1 x + b l y + &c. - FJ 2 + (a z x + b 2 y + ... - F 2 ) 2 +..., 



a minimum: that is to say, they will make the sum of the 

 squares of errors a minimum. 



Hence the method of least squares. The conditions of the 

 minimum give the linear equations : 



............ 09), 



&c. =& c J 



