  In a recent work, we proved that a fundamental result of Gauss related to
quadratic reciprocity is equivalent to a special case of a well known result of
Sylvester related to the Frobenius coin problem. In this note, we appropriately
generalize Gauss's result so that it becomes equivalent to the general version
of Sylvester's result. This generalization of Gauss's result naturally leads us
to another proof of Gauss's Lemma and Eisenstein's Lemma for Jacobi symbols.
Using our work, we also obtain a new formula for the Jacobi symbols.
