  In this paper we study the addition formulae of the KP, the mKP and the BKP
hierarchies. We prove that the total hierarchies are equivalent to the simplest
equations of their addition formulae. In the case of the KP and the mKP
hierarchies those results had previously been proved by Noumi, Takasaki and
Takebe by way of wave functions. Here we give alternative and direct proofs for
the case of the KP and mKP hierarchies. Our method can equally be applied to
the BKP hierarchy.
