  We give explicit formulas for Hadamard's coefficients in terms of the
tau-function of the KdV hierarchy. We show that some of the basic properties of
these coefficients can be easily derived from these formulas. The first
immediate corollary is the symmetry of Hadamard's coefficients about the
diagonal. Another well known fact, which follows from this approach, is that on
the diagonal Hadamard's coefficients determine the right-hand sides of the
equations of the KdV hierarchy. The proof of the main result uses Sato theory
and simple properties of Gegenbauer polynomials.
