Wilh Collins diffraction integral formula in the paraxial approximation, the analytical expressions for the linearly polarized Bcssel-Gauss beam and the azimuthally polarized Bessel-Gauss beam through any axisymmetrie optical system that can be described by an ABCD ray transfer matrix are giyen in this paper. For example， the propagation characteristics of these two kinds of Bessel—Gauss beams through an optical system with a lens are compared by a numerical calculation. And the results are also analyzed and discussed.