  We continue our program to define and study $n$-point correlation functions
for a vertex operator algebra $V$ on a higher genus compact Riemann surface
obtained by sewing surfaces of lower genus. Here we consider Riemann surfaces
of genus 2 obtained by attaching a handle to a torus. We obtain closed formulas
for the genus two partition function for free bosonic theories and lattice
vertex operator algebras $V_L$. We prove that the partition function is
holomorphic in the sewing parameters on a given suitable domain and describe
its modular properties. We also compute the genus two Heisenberg vector
$n$-point function and show that the Virasoro vector one point function
satisfies a genus two Ward identity. We compare our results with those obtained
in the companion paper, when a pair of tori are sewn together, and show that
the partition functions are not compatible in the neighborhood of a two-tori
degeneration point. The \emph{normalized} partition functions of a lattice
theory $V_L$ \emph{are} compatible, each being identified with the genus two
theta function of $L$.
