Let $(Pu)(t)=-\\frac{d}{d t}\\left({\\omega}^{2}(t)q(t)\\frac{d\nu(t)}{d t}\\right)$ be a degenerate non-selfadjoint elliptic differential\noperator, defined on the space $H_{\\ell}=L^{2}{(0,1)}^{\\ell}$,\nwith Dirichelet-type boundary conditions, where $\\omega(t) \\in\nC^{1}(0,1)$ is a positive function with further assumptions that\nwill be specified later and $q(t) \\in C^{2}([0,1],\\;End\\;\n{\\bf C}^{\\ell})$ is a matrix function.In this article we investigate some spectral\nproperties of the operator $P$.We estimate the resolvent of $P$ , prove the limit argument Theorem and we will find a formula for the distribution of the\neigenvalues of the operator $P$ acting on $H_{\\ell}$ .