﻿200 ON THE USE OF EQUIVALENT FACTORS 



b) After the multiplications have been peifoiraed, and the sums taken, the numbers 

 adopted iu the final equations are to be tested by (33). 



c) The solution of the final equations. 



d) The determination of Aveights. 



If changes have been made in the decimal pointing, or otherwise, by introducing 



the constants A, B , it must be remembered that, although the final equations 



thus formed will give the same values of a', , 3/1 , «S:c. that would have been obtained 

 if no such alteration had been made, the determination of the weights and probable 

 errors of c-r, , 1/1 , &c. requires that the correct pointing be restored in the coefficients, 

 or else that the probable errors be computed in conformity with the formulae (23). 



When the number of indeterminates is considerable, it will be advisable, in solving 

 the final equations, to eliminate .v, 1/, z, &c., in succession, and then to repeat the 

 operation, commencing the elimuiation in the reverse order, z, 3/, .r, &c. One of the 

 advantages of so doing is a complete check upon the work by the comparison of the 

 value of that indeterminate which is obtained last by both eliminations. It is, however, 

 mostly recommended from its facilitating the computation of the weights. In this case, 

 the following formula; may be used, if the number of mdeterminates does not exceed 

 six. Let these be a?, 1/, z, |, ■??, ?, and their weights, TF(^) , TF^^j , &c. The ordinary 

 formulae for computing the weights give 



(35.) Tfto = t/x y z ; , , W(.) = Po 5 z J, . 



W^Q is the coefficient of f in the equation resulting from the elimination of x, 1/, z, f, 

 and r), by the process indicated in (18), (19), and (20), and W^^y the coefficient of x 

 after ^, rj, f, z, and y have been eliminated. We have, also, 



(36.) Wc) = ^^^^ TTco , W^, = ^-IJ-^ TF(.> . 



The factors and divisors required in (36) will have been already computed during 

 the eliminations which have preceded. 



From the equations containing only ^, 77, f, the latter is to be eliminated ; and from 

 the equations containing z, y, and x, x is to be eliminated. We then have 



J-x y z C, Hi A I X 



For the weight of .j;,, ^,, &c., when (II.) or (III.) is used, we shall have, from (23), 



Weight oixi — A Wf^^-) , 

 (37.) " y, = B W^^ , 



