XCIV THEORY OF ERRORS. 



There will be as many equations of this type as there are independent relations 

 which the quantities .Yj -|- A.Ti, x.y -\- Ajvo, . . . must satisfy. Suppose there are k 

 such relations, and let the differential coefficients SF/dx^, QF/Sxo, ... for the sec- 

 ond relation be denoted by di, b.,^ b;,., • . - ; for the third relation by c^, r,, c^, . • . ., 

 etc. Then all of the conditional equations may be written thus : 



Ux^Xi -\- a-i^x., -{- a^^x-i -]-... -\- q^^z^o, 



bx -\- b.> -\- bs -\- . . . ^g, = o, (a) 



i\ 



+ ^2 + ^3 + • • • + ^3 = O, 



the number of these equations being /:. 



Call the weights of the observed quantities x^, x,, . . . p\, p-2-, .... Then, sub- 

 ject to the conditions {a) we must have (in accordance with (5)) 



« =p^x^ + A(Ax,)^ + . . . = [/(Aaf ] ib) 



a minimum. 



Equations {a) and (8) contain the solution of all problems falling under the 

 present case. Obviously, the number of conditions {a) must be less than the 

 number of observed quantities x, or less than the number of Aa-'s in {f) ; in other 

 words, if ;;/ denote the number of observed quantities, m > /', for if m ^ k the 

 minimum equation {l>) has no meaning. 



The question presented by {a) and (b) is one of elimination only. Two methods, 

 the one direct and the other indirect, are available. Thus, by the direct method 

 one finds from {a) as many Aa:'s as there are equations (a), or k such values, and 

 substitutes them in (J)). The remaining {in — K) values of Ax in ib) may then be 

 treated as independent and the differential coefficients of ?/ with respect to each 

 of them placed equal to zero. Thus all of the corrections Aa" become known. 



By the indirect process, one multiplies the first of equations id) by a factor Q^., 

 the second by Q^, the third by Q.,... and subtracts the differential (with respect 

 to the Ax's) of the sum of these products from half the differential of {f). The 

 result of these operations is 



\ du = {AA^i - (a,Q, + b,Q, + ^,Q, + •••)} ^'''^^-^1 



+ {A>A.r2 - («.<2l + ^Q-2 + ^2<23 + . . .)} ^^'^''^"2 



+ . . . 



+ {p„Ax,„ — (a,„Q, + b,„Q. + c,„Q:, + ...)} ^^^m 



Now we may choose the factors <2i, Q2, • • • Qk '" such a way as to make k of the 

 coefficients of the differentials in this equation disappear; and after thus elimi- 

 nating k of these differentials we are at liberty to place the coefficients of the 

 remaining {nt — k) differentials equal to zero. Thus all conditions are satisfied 

 by making 



d Qi + b, Q2 -\- r,Q., -\- . . . — pAxi = o, 



Oo -\- bo 4" ^o -(-...— /oAa's ^= o, 



.'. . . ' (0 



«m + b,„ + r,„ 4- ... — p„Ax„, = o ; 



and the values of the corrections will be given by these equations when the fac- 

 tors Q], ^2) • • • ^re known. To find the latter it suffices to substitute the values 



