  We prove that the modules of differential operators of order 2 on the
classical Coxeter arrangements are free by exhibiting bases. For this purpose,
we use Cauchy-Sylvester's theorem on compound determinants and Saito-Holm's
criterion. In the case type $A$, we apply Cauchy-Sylvester's theorem on
compound determinants to Vandermond determinant. By using the Schur
polynomials, we define operators which form a part of a basis of modules of
differential operators on the classical Coxeter arrangements of type $A$. In
the cases of type $B$ and type $D$, the proofs go similarly to the case of type
$A$ with some adjustments of operators and determinants.
