Reduction of error by linear compounding

Ifu1 , u2 , u3 ,... are observed values of a single quantity U —if, in other words, there are observed values of U containing errorsu 1 —U,u 2 —U,u 3 — U,... —, we may take as an approximate value of U any function of then’s which, if each of the u's in it were replaced by U, would itself become identically U. Usually this function is a linear compound (linear function) of the u’s, and is called a weighted mean: and we regard as the best, weighted mean that in which the constants of the function are chosen so that it shall have the least possible mean square of error. The principle applies also when theu ’s, instead of being observations of a single quantity U, are observations of a set of quantities U1 , U2 , U3 ,...; with the modification that the effect of replacing (say)u 1 by a function of theu 's will be that, if theu 's in the function were put equal to the corresponding U’s, the function; would not become identically equal to U1 , but only approximately equal to it. In the particular case in which the differences of theu 's successively diminish, a permissible form of the function is the sum of and a linear compound of the differences of theu 's of Sufficiently high order.