  It is well known that in a small P\'olya urn, i.e., an urn where second
largest real part of an eigenvalue is at most half the largest eigenvalue, the
distribution of the numbers of balls of different colours in the urn is
asymptotically normal under weak additional conditions. We consider the
balanced case, and then give asymptotics of the mean and the covariance matrix,
showing that after appropriate normalization, the mean and covariance matrix
converge to the mean and variance of the limiting normal distribution.
