  By a theorem of Wahl, for canonically embedded curves which are hyperplane
sections of K3 surfaces, the first gaussian map is not surjective. In this
paper we prove that if C is a general hyperplane section of high genus (greater
than 280) of a general polarized K3 surface, then the second gaussian map of C
is surjective. The resulting bound for the genus g of a general curve with
surjective second gaussian map is decreased to g >152.
