  We consider a multivariate generating function F(z), whose coefficients are
indexed by d-tuples of nonnegative integers: F(z) = sum_r a_r z^r where z^r
denotes the product of z_j^{r_j} over j = 1, ..., d. Suppose that F(z) is
meromorphic in some neighborhood of the origin in complex d-space. Let V be the
set where the denominator of F vanishes. Effective asymptotic expansions for
the coefficients can be obtained by complex contour integration near points of
V.
  In the first article in this series, we treated the case of smooth points of
V. In this article we deal with multiple points of V. Our results show that the
central limit (Ornstein-Zernike) behavior typical of the smooth case does not
hold in the multiple point case. For example, when V has a multiple point
singularity at the point (1, ..., 1), rather than a_r decaying on the order of
|r|^{-1/2} as |r| goes to infinity, a_r is a polynomial plus a rapidly decaying
term.
