  We generalize Schwinger's original mass formula to the case of an additional
isosinglet mixing with the nonet mesons. We then make further generalization to
either (i) an arbitrary number of additional isosinglets mixing with nonet
mesons, or (ii) an arbitrary number of mesons mixing with an additional
isosinglet. In the former case, we present an explicit relation, while in the
latter case, we show by numerical example that the new mass formula is only
weakly affected by the inclusion of additional mesons, and hence holds with
good accuracy for each of the 3x3 mass sub-matrices.
