Ixxxviii THEORY OF ERRORS. 



Secondly, take the case, also alluded to above, of the observed sum and the 

 observed difference of two numbers. Denote the numbers by ^ and rj, the latter 

 being the smaller. Let the observed values of the sum ($ -|- 7;) be denoted 

 by Xi, X., . . . x,„ and their weights Pi, p^, . . . /„, respectively. Likewise, call 

 the observed values of the difference (^ — ■>;), }\^ j.,, • • • J>'«, and their weights 

 ^\, q-i' ■ • Qn respectively. Then there will be vi -\- n equations of the type (4), 

 namely : — 



f + ^ — ip^i + ^-'*'^i) = o, 



i-\-^ — (a\> + Ax,) = o. 



i^t] — {x„, + Ax„,) = o, 



i — -n — O'l + Ajt'j) = o, 

 i — V— (j'2 + Ar.) = o, 



(a) 



^ - V — (j'n + Aj'„) = o ; 



and the minimum equation is 



« =p, {^x,y + A (Ax,)- + . . . + <^, (Aj,y 4- ^, (A^,)2 + ...:= a min. (b) 



The equations of group (a) give 



Axi = i -\- rj ~ Xi, 

 Axo = ^ -|- 77 — x.>, 



(c) 



Aj'i = i — ■>] — }\, 

 • ' • ') 



and these values in (b) give 



u =p, (^i + 7; - x,)'^ + . . . + ^1 a - -r)-}'^' + . . . (d) 



Thus it appears that all conditions will be satisfied if i and 7; are so determined 

 as to make u in (d) a minimum. Hence, using square brackets to denote sum- 

 mation of like quantities, the values of ^ and 77 must be found from 



ll = [/ + ^] ,^ + [/ - ^] ^ - [At + qy^ = o, 



-^ = [p - q] ^ -\- [p + q] V - [/-^ - ^y] = o. 



Equations (e) give i and 7/, and these substituted in (c) will give the corrections 

 to the observed quantities. 



b. Relation of probable, mean, and average errors. 



The introduction of the law of error (2) in equations (3) furnishes the following 

 relations, when it is assumed that the limits of possible error are -co and -}- °o : 



ep = 0.6745 e,„ = 0.8453 'a- (6) 



