﻿IN THE METHOD OF LEAST SQUARES. 193 



When the final equations obtained by (II.) are solved in an analogous manner, and 



the notation is changed so as to indicate the coeiRcients of Xi, ^,, &c., Pj, P', , 



which replace P, P', &c., we have 



(6) 'Q.a;, + Q, y, + Q', r, + -f M, =ol Final 



(c) "K, a-, + 'JJ. !/, + K, 2, + -j- JVi = ( Equations. 



(22.) (K) Q,.y.+ Q',. ^1 + + iif,. =0] 



/ \ /DID I 1 Tvr n [ Equations formed by 



i eliminating .t'l. 



(c^y) Rixy Zi -j- + -ZVi ij, = ") Equations formed by 



i eliminating x, and yi. 



1 Ip- 



Probable error of the equation (22) (a) = (1 ± - ^) /t — , 



2 N^ 



(c) = (1 ± 



^^'-Jf- 



(23.) " " " {K) = (1 ± 2 ^) /' Jf"^ 



('^,.) =(i±-2»)/'J|^^ 



To demonstrate these results, (23), it is to be observed that the probable errors of the 



second members of the equations (22), (a), (6 J, are the probable sums of the 



second members of the equations 



P, (-To — -^-i) + i". (yo — ?/,) + = «£„ + «'<;'„ + 



Qix (2/0 - y,) + = (^ - ^ «) fo + (/3' - ^ «') «'o + 



The probable value of a^el is, by (13) and (14), 



A a 



.2 /.a — 



«'e: 



2 ^ f . 8 u\ It a 2/11 s » a 



uc; = w ell 1 -\ ) — = A* ( 1 ± g) — , 



° ' \ awj A '^ ^ A 



