THEORY OF ERRORS. Ixxxix 



c. Case of a single unknown quantity. 



The case of a single unknown quantity whose observed values are of equal or 

 unequal weight is comprised in the following formulas : — 



Xi, X,, . • . x,n = observed values of unknown quantity, 



/i, poy . . . A. = the weights of x^, oto, . . . 



z\, z'2, . . . v,n == most probable corrections to Xi, x,, . . . 



X = most probable value of the unknown quantity, 

 fu = the number of indepe^ident observations. 



Then the conditional equations (4) are 



X — Xy ^^^ Z'jj 

 X — vV^ ^^^^ 7^2> 



X X„^ t'„, , 



the minimum equation (5) is 



P<v^ ^-p-ivi + . . . = [/z'^] = lp{x - x,y] = a min., 

 where i = i, 2, . . . m, and 



"^"~ A+A + ---A. [/]' 



When the weights are equal, A =p2 = . . . =/,„> ^"^ 



.= M 



or the arithmetic mean of the observed values. 



Weight of x = [/>] when the p's are unequal, 

 = m when the p's are equal. 



Mean error of an observed value of weight unity = \/ — ^— for unequal weights, 



for equal weights. 



\ m — I 



Mean error of an observed value of weight/ = V/ ,- — — — \— for unequal weights. 



/ \pvv\ , . , 



Mean error of x =. V/ -, — ^ — \\ j,-\ ^^^ unequal weights, 



— i/ — J — =! — ^ for equal weights. 



— ym {m — i) ^ "^ 



The corresponding probable errors are found by multiplying these values by 

 0.6745. See equation (6). 



