These numbers will be regarded as distinct although two or more may be equal in value. Let ni (i= 1, 2, , k) be any fixed set of k positive integers whose sum is n, let C= Hi=, ni! and let R=n!/C. We shall consider partitions P*= {B *, B*, * * , B * } of the set B into disjoint subsets B?*, each B* containing ni elements of B (i= 1, 2, , k). By a partition here is meant an ordered k-tuplet of subsets, i.e. we consider two partitions P*, P** to be the same (or equal) if and only if B =B ** (i= 1, 2, , k). The number of such partitions is given by the multinomial coefficient R. If we let BI dedenote the first n1 elements of B, B the next n2 elements of B,2 Bl the last nk elements of B, then pl = fBl, Bl , Bl } is a particular partition of B. Similarly if we let BR denote the last n1 elements of B, B2 the next n2 elements of B, , BR the first nk elements of B, then pR = {BR B, B BR } is another particular partition of B. Let A,=Ai (i=l, 2, * * *, k) be defined similarly for the set A, except that we shall regard the A as ordered subsets of A, the order being that given in (1). For convenience we define N= { 1, 2, * * * , n and the subsets N, = Ni (i = 1, 2, * * , k) exactly as was done for B. The theorem that follows is concerned with the n! cross products EJ=i aibji where (j, j2, * * *jn) is a rearrangement of (1, 2, * * *, n). Corresponding to any fixed partition Pr (r= 1, 2, * * * , R) we consider the set Vr = { Vcr I c = I, 2, * * , C } of the C cross products obtained by associating the elements of B' with Ai (i = 1, 2, * k), all possible rearrangements within the subsets B, being allowed. The n! cross products are thus divided in R sets with C cross products in each set. Let pr (r = 1, 2, * * *, R) denote an arbitrary enumeration of the R partitions except that pl and pR are defined above. We shall introduce (see The partial ordering below) a partial ordering (written P*>P**) among the partitions such that