The instability of a cylindrical tube of uniform crosssection, made of a homogeneous, isotropic Bell constrained material and subjected to end thrust parallel to its axis, is examined. The theory of small deformations superimposed on an assigned finite homogeneous deformation is applied to derive equations for the stability analysis. The solution presented by Wilkes for the incompressible case is extended to study the system of equations for the infinitesimal superimposed displacements. In contrast to Wilkes, the roots of the characteristic equation are examined for three possible situations in which these roots may be real, pure imaginary, or complex conjugate. The solutions of the system of ordinary differential equations corresponding to these cases are discussed. The buckling equation is thus derived, and conditions imposed by the nature of the characteristic roots are presented.