﻿188 ON THE USE OF EQUIVALENT FACTOES 



Used in connection with (3) and the limit defining the uncertainty of s, 



£ 1 



y. > 0.477 , , or usually x > — t . 



ij wUl thus be comprised between the limits i-\-h'x and s. — hy.. when ri" has such a 

 value that 



(n* 



10 n — n' ' 



The relative accuracy of x and Xi will now be investigated for some special exam- 

 ples of deviation from a strict compliance with the method of least squares. 

 Let the equations proposed for solution be the following : — 



a X -j-^ y4" ~\- m ^e, weight = w, 



(7.) a' x' + J' y + -f ni' = e', " = ?/?', 



where e is the diflference between the obseiTcd and computed value of m ; m being the 



element derived from observation. 



In soh-ing these by least squares, the final equation for x is formed by taking the 



sum of all the equations after multiplying the first by a lo, the second by a w', and 



so on, and then making 



a w e -\- a' w' e' -\- =0) 



and for y 



I w e -{- b' ic' e' -{- =0, 



continuing in succession to form new equations until a final equation is obtained for 

 each unknown quantity. 



We shall compare the results of two solutions of the above equations (7), in one of 



ivhich (I.) the factors a w, a iv conform strictly to the method of least squares. In 



the other (II.), these factors are replaced respectively hy a., a! ; a being that one of 



the numbers 0, 1, 2 9, or of the fractions \,^ -i-, o/- o/ their jn'oducts by an in- 

 tegral power of 10, which approaches most nearly to a given ratio with a w, and a that 

 which approaches most nearly to the same ratio with a' w', 8fc. In a similar manner, 

 /8, 7 ai'e used in the place of bit; cw 



The true values of x and y we will indicate by Xq, y^ Those deduced by 



(I.) will be denoted by a', j^ , and those deduced by (II.) will be denoted by cT,, 



y^ For the final equation for x, we make 



ate e -\- a! xe' e' -\- =0. 



