Schwinger's equations for the propagation functions of quantum electrodynamics are redefined in a way to give the finite (renormalized) propagation functions without reference to divergent integrals or infinite renormalization constants. This is achieved by incorporating in the equations themselves a limiting process which is an extension of that introduced by Dirac and Heisenberg. The formulation is given independently of the power-series expansion, but the cancellation of singularities is established only in terms of such an expansion. The method is illustrated first by considering the lowest-order approximations. The lowest-order electron self-energy and vertex-part expressions are worked out, and the compensation of the singularities corresponding to the 'b' divergences is indicated in the fourth order. In the power-series expansion, the prescriptions are in a one-to-one correspondence to those of Dyson. Their formulation independently of this expansion sums up the rules obtained in the different approximations.