In 1888 Dr Thomas Muir gave the following theorem:—

Theorem A.—If any two determinants A and B of the nth order be taken, and from these two sets of determinants be formed, namely, first, a set of nCr determinants, each of which is in r rows identical with A and in the remaining rows with B; and, secondly, a set of the same number of determinants each of which is in r columns identical with A and in the remaining columns with B, then the sum of the first set of determinants is equal to the sum of the second set.