﻿IN THE METHOD OF LEAST SQUARES. 199 



The following tests of the correctness of all the numbers entering into the final equa- 

 tions will be found useful. They apply equally to the three methods. 

 Let the sums S, S' , s, :s' be formed as follows : — 



a +h -\- +m =S, „-|-/3+y+ =2, 



(31-) a' + i'+ + m' = S', „' _|_ /3' _(_ y _|_ ^ v/ , 



We shall then have the following test controlling at once all the computed quan- 

 tities, Pi,P', , 'Qi, Q, , A, M, 



2 (A, Q>, -Ri L„M„N, ) = 2: (S^). 



For we have 



P, + P'l + + Li = a a +«&+... . 



+ «' a' + «' &' + } = S « + S' «' + 



+ J 



(32.) ^ 13' a' + 1^' h' -\- I =S/3 + S'/3'4- 



"-Ri + 'B. + + N,^y a+y h -{- ^ 



+ y«' + v'S' + j> -Sy + S'y + 



The sum of all the equations (32) is 



(P. + P\ + + ii) + CQ, + Q. + + M,)-\- = S^+S'^' + 



or 



(33.) ^ (P„ Q,,R, L„M„N, ) = ^ (S ^). 



When this is not fulfilled, the particular equation at fault may be detected by using 

 the partial tests, 



Pi + P'x + i. = S « + S' «' + 



(^^•) 'Qi+Q. + M,= S/3 + S'/3'-}- 



If all the numbers are not carried out to the same number of figures beyond the 

 decimal point, it will be advisable, in order to apply the test to the best effect, to alter 

 arbitrarily the decimal point, making, for the time being, the number of figures to the 

 right hand of it the same in all the equations. 



The successive stages in the computations by (II.) and (III.) will be as follows : — 



a) To assign the values of the equivalent factors a, /3 , which can readily be 



done by simple inspection. The factors once adopted must be used rigorously in mul- 

 tiplying every term of the equations to which they belong. 



