  The classical infinitesimal model is a simple and robust model for the
inheritance of quantitative traits. In this model, a quantitative trait is
expressed as the sum of a genetic and a non-genetic (environmental) component
and the genetic component of offspring traits within a family follows a normal
distribution around the average of the parents' trait values, and has a
variance that is independent of the trait values of the parents. In previous
work, Barton et al.(2017), we showed that when trait values are determined by
the sum of a large number of Mendelian factors, each of small effect, one can
justify the infinitesimal model as limit of Mendelian inheritance.
  In this paper, we show that the robustness of the infinitesimal model extends
to include dominance. We define the model in terms of classical quantities of
quantitative genetics, before justifying it as a limit of Mendelian inheritance
as the number, M, of underlying loci tends to infinity. As in the additive
case, the multivariate normal distribution of trait values across the pedigree
can be expressed in terms of variance components in an ancestral population and
probabilities of identity by descent determined by the pedigree. In this
setting, it is natural to decompose trait values, not just into the additive
and dominance components, but into a component that is shared by all
individuals within the family and an independent `residual' for each offspring,
which captures the randomness of Mendelian inheritance. We show that, even if
we condition on parental trait values, both the shared component and the
residuals within each family will be asymptotically normally distributed as the
number of loci tends to infinity, with an error of order 1/\sqrt{M}.
  We illustrate our results with some numerical examples.
