This paper deals with the problem of computing the number of representations of one bilinear form by another mod \(p^a\), where \(p\) is a prime and \(a\) is any positive integer. Expressed in matrix language, this is the number \(N(A, B, a)\) of mod \(p^a\) incongruent solutions \(U, V\) of the matrix congruence \(UAV\equiv B \pmod{p^a}\), where \(A, B\) are given rectangular matrices with integral entries. The author has given an explicit formula for \(N(A, B, 1)\) in an earlier paper (see [Duke Math. J. 22, 497--509 (1955; Zbl 0066.26803)]). In the present paper he considers the case \(a>1\). He derives a complicated recurrence relation which he uses to prove the bilinear analogs of Lemmas 12 and 13 of \textit{C. L. Siegel}'s paper [Ann. Math. (2) 36, 527--606 (1935; Zbl 0012.19703)].