Adjustment of Observations. 221 



may be neglected ; these corrections are now the quantities 

 whose most probable values we require. Substituting the 

 above values in equations (1) and expanding by Taylor's 

 Theorem we obtain 



/ 1 fX x ,Y 1 ...A,B...)+|£-A* 1 +... ^l&a+... = 0, 

 /,(X 2 . Y 3 . . . A, B . . .)+ gjf.A*+ • • ■ +^4 A «+ • • • = 0. 



or putting 



etc., 



f x (X„Y 1 ...A,B...) = -fc etc., 



axi -6 ' 3x 2 - f2 etc " 



a*i 



= *?i, 



d/ 2 



: ^2 





= «,, 



d/ 2 



: a 2 





= A, 



d/ 2 



dB~ 



A 



the equations become 



fcAtfi+^Ay, + . . . + «!Aa.+ ftA6 + . . . =:</>!, "1 



f 2 A<r 2 + ?? 2 A v 2 + . . . -h * 2 Aa 4- & A6 + . . . = <f> 2 , I 



K3) 



b-^^'x -h ^-Ayx -{- • • • 4- a^Aa r /3^Ab+ . . . = N . j 



Substituting the same values oi: a? 1? y,, ? b etc. in equa- 

 tions (2) they become 



A*! = #,' -X, Ay, = y,' -Y, etc, *] 

 A# 2 = x 2 ' — X 2 A// 2 = y 2 ' — Y 2 „ ] 



: : : : : : I • W 



A.r x =^'-Xx Ay N =y N '— Y N „ J 



The equations (3) and (4), N?i + N in number, and 

 involving N» T 7 unknown quantities, are all linear, and the 

 coefficients and absolute terms are all known numerical 



