Mr. ELLIS, ON THE METHOD OF LEAST SQUARES. 



211 



From the n equations (B) we deduce a new system of p equations. To obtain the first of these, 



a, O2 

 we multiply equations (B) by 7; > T? > &c. respectively, and add the results. For the second, we 



employ instead of the factors — - , &c., the factors — , , -5, &c. and then proceed as before. And 

 ' ■ A:; ft, k:^ 



similarly for the others. 



In consequence of the relations 



2^1=1, l,/ih = 0, &c. = 0, 



the new system of equations will be 



ab 



1 = \, S — + X2 2 — + &c. 

 k- k- 



^ ab ^ b^ 

 0=X,2-7 + \22- + &c. 



0= &c. 



(C). 



These p equations determine X,, Xj... X^, and thus in virtue of (B) the values of fxt, fi,... fx^ 

 become known. Finally as 



cc = n^V^ + ti.^V^+ ... + n^V„, 



X will be completely determined. 



Now let us recur to the original equations of condition stated in the last paragraph. 



e, = a^x + h^y + &c. - F, 

 €2 = a^a: + b^y + &c. - V^ 

 &c. = &c. 



e„ = a„a; + b„y + &c. - V„ 



From this system we deduce a new one, containing p equations. The first of these is got by 



fl, a. 



kVt 



'-. Xrr. : fliiH sn nn ns hpf 



ah 



(a). 



fl, (It) 



multiplying equations (a) by p ? 7:2 ' ^^'' ^"*^ adding the results : the second by using the factors 

 7i' i3' ^^' ■ ^"*^ ^^ °" ^^ before. The resulting system will be, neglecting all errors, 



a flu a \ 



ab b'^ b 



-2-^ + 2.2^ + &c. = E^F 



&c. 



= &c. 



(/3')- 



The system (/3') contains as many equations as there are unknown quantities x, y, &c. I pro- 

 ceed to show that if x be determined from this system, its value will be the same as if it had 

 been obtained from the most advantageous system of factors, namely, that which is determined 

 by means of (/?) and (C). In order to prove this, we multiply equations (/?') by X,,Xj, 8cc., 

 and add the results. Then, in virtue of (C) 



Vol. VIII. Part II. 



k- Ic' 



Ee 



