  We reexamine the external field problem for $N\times N$ hermitian one-matrix
models. We prove an equivalence of the models with the potentials
$\tr{({1/over2N}X^2 + \log X - \Lambda X)}$ and $\sum_{k=1}^\infty t_k\tr{X^k}$
providing the matrix $\Lambda$ is related to $\{t_k\}$ by $t_k=\fr 1k
\tr{\Lambda^{-k}}-\frac N2 \delta_{k2}$. Based on this equivalence we formulate
a method for calculating the partition function by solving the Schwinger--Dyson
equations order by order of genus expansion. Explicit calculations of the
partition function and of correlators of conformal operators with the puncture
operator are presented in genus one. These results support the conjecture that
our models are associated with the $c=1$ case in the same sense as the
Kontsevich model describes $c=0$.
