Let A and B denote polynomials of degrees m and n in a variable x over an integral domain D. As usual, if M = M _{A , B} is the corresponding Sylvester matrix, then the kth subresultant is the polynomial where M_k ^{( i )} denotes the submatrix of M consisting of the first n - k and m - k rows of the A and B coefficients, respectively, and the first m + n - 2 k - 1 columns as well as the m + n - k - ith column. For any polynomial C in x over D, let A ○ C and B ○ C be the polynomials obtained from A and B by replacing x with C. The author investigates the effect of such composition upon the subresultants and shows that sres_k ( A ○ C , B ○ C ) = p _μ sres _μ ( A , B ) ○ C when k = ℓ μ, sres_k ( A ○ C , B ○ C ) = q _μ sres _{μ - 1} ( A , B ) ○ C when k = ℓ μ - 1, and sres_k ( A ○ C , B ○ C ) = 0 otherwise. Here ℓ is the degree of C, and p _μ and q _μ are specified constants. This result generalizes the well-known fact that the resultant res_k ( A , B ) = det M _{A , B} is invariant under translation.