Adjustment of Observations, 225 



the form 



#1 — x i •> 



X% = X<i , 



#3 = tfa'i 



.r 4 = # 4 , 



So^ + XxAS+Trr^', 



S # 2 + X 2 AS + T = y 2 ', 



S .i' 3 + X 3 AS + T=^', 

 S a?4+X 4 AS + T = .y 4 ', 



from which the following normal equations are obtained : 

 (S 2 + 1> 1 + S X 1 AS + SJ = S# 1 '+* 1 ', 

 (S 2 + l).i' 2 +S X 2 AS +S T ^Sqys'+W, 

 (S 2 4- l).r 3 + S X 3 AS + S T - S y 3 ' + *,', 

 (S 2 + l) t r 4 +S X 4 AS + S T «Stf/.+ * 4 ', 

 S .SX.i- + SX 2 .AS + 2X.T=2Xy, 

 S .2a +XX.AS + 4T = 2/; 



or eliminating ^ <r 2 , <r 3? # 45 



sx 2 .as+2x.t = xxy-SoXxy, 



SX.AS+ IT = 2/ -SpSaK. 



As a first approximation we assume X 1 =#i', 'X 3 =a i l J etc., 

 and obtain the values 



S = 1-339, 



T - --029, 



j?i = '397, 



x 2 — *612, 



x z = -780, 



a? 4 = *911. 



Using these in turn as approximate values and re-solving, 

 we obtain 



AS = -010, 



S = 1-349, 



T = --035, 



which are correct to within one unit in the last place *. 



* The same result would of course have heen arrived at by the 

 methods of analytical geometry, assuming ?/=S.r+T as the equation of 

 the line required, and imposing the condition that the sums of the 

 squares of the perpendicular distances from it of the points (.fi'^/i'), 

 (#2'ya'), fa'pa), (**'y/) should be a minimum; the solution by this 

 method would have been direct. 



