Abstract : In reference to any solution of a conservative dynamical system with two degrees of freedom, Hill's equation is generalized to encompass non- necessarily isoenergetic displacements as well as the isoenergetic displacements caused by a variation of a parameter. This new variational equation is made the foundation of a methodical procedure for continuing numerically natural families of periodic orbits. The method consists of two steps-- an isoenergetic corrector and a tangential predictor. Although the algorithm makes no assumption of symmetry on the periodic orbits to be continued, special attention is paid to the symmetric orbits, but only to show how in these cases the method can be simplified substantially.