An explicit formula is given for the number of representations of a pair of positive integers by a representative set of inequivalent pairs of integral positive-definite binary quadratic forms with given invariants.

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Journal Proceedings of the American Mathematical Society
Hardy, K. (Kenneth), Kaplan, P. (Pierre), & Williams, K.S. (1989). The representation of a pair of integers by a pair of positive-definite binary quadratic forms. Proceedings of the American Mathematical Society, 105(4), 847–855. doi:10.1090/S0002-9939-1989-0960644-0