  We define the $n$-point function for a vertex operator algebra on a genus two
Riemann surface in two separate sewing schemes where either two tori are sewn
together or a handle is sewn to one torus. We explicitly obtain closed formulas
for the genus two partition function for the Heisenberg free bosonic string and
lattice vertex operator algebras in both sewing schemes. We prove that the
partition functions are holomorphic in the sewing parameters on given suitable
domains and describe their modular properties. Finally, we show that the
partition functions cannot be equal in the neighborhood of a two-tori
degeneration point where they can be explicitly compared.
