Based on two properties of the discrete eigenvalues of the one-speed neutron transport equation with anisotropic scattering, we discuss the number and mathematical character of these discrete eigenvalues when the scattering function can be expanded into a finite series of Legendre polynomials. Such special cases as linearly and quadratically anisotropic scattering are studied in detail, and our results are plotted in parameter space. We also investigate the two physically interesting problems of isotropic scattering and linearly anisotropic scattering in the center-of-mass system. The discrete eigenvalues of these two problems are obtained numerically.