518 VAN ORSTRAND AND WRIGHT: MINERAL ANALYSES 



m =- 3 _|_ 3 "T X^3 • • \V\-^\ • • Pn-^nJ- • {^J 



Xi -\- X2 'T' • • • "t" ^^n 



A more general solution is obtained by adopting the observa- 

 tion equations: 



a + mxi = 2/1 (weight pi) ] 

 a + 771X2 = y\ ' (weight pa) 



\ (9) 



a + mXn = Vn (weight p„) 



The least square solution of these equations gives the normal 

 equations : 



(pi+ . . . +Pn)a+(pi.Ti+ . . . +2)„a;„)m = p,2/i+ . . +p„?/„ 1 



r (10) 



(PlXi-h . . . PnXn)a-\-(piXl+ . . . +PnXl)m = PiX:yi+ . . Pn^nyn i 



from which the most probable values of a and m are readily- 

 determined. In this case the residuals satisfy the two equations 



PlV + P2V2 . . + Pn'Vn = 

 PlXiVi + P2^2^2 + . . + Pn^nVn = 



The condition that the computed values (y'l, y'2, • • • y'n) 

 satisfy the relation 



y\ +y'2 + . . . + y'n = 100 . 



can be imposed by writing in addition to the n observation 

 equations (9; the observation equation 



na -h (x -{- X2 . . + xj m = ICO (weight 00) . . (11) 



and then solving the system of n + 1 equations in the usual 

 manner; or we may substitute the value of a obtained from (11) 

 in (9) and then solve equation (2) for 7n. The general result of 

 the adjustment of the n + 1 equations (9) and (11), is repre- 

 sented graphically in figure 1. 



The coordinates of the point P are 



x = Xi + X2 + . . . + x,„ y = 100 



and the assignment of different weights, in the least square so- 

 lution, to the points Pi, P2, . . . Pn has the effect of rotating 

 the line O'P through a small angle ± a about P as center. If the 



