A new treatemant has been given for solving Dirac’s equations for hydro. genic atoms, and the radial functions are expressed in terms of a combination of Bonine’s polynomials T(µ)n (p), Tk(µ)n-1 (p) of only two Mnsecutive degrees n, n-l; and the elementary properties of such polynomials have enabled us to tMkle the Zeeman effed problem in general (homogeneous field) leading to the standard quadratic equation in energy for the effect. © 2009 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.