Two methods are described for obtaining numerical solutions of the differential equations which govern the reflexion of long and very long radio waves from the ionosphere at vertical or oblique incidence. Both methods have been used with the EDSAC, the automatic digital computer at the University Mathematical Laboratory, Cambridge, but the paper is not concerned with the details of the EDSAC programming, nor with the large series of results that has been obtained. In the first method the first-order simultaneous equations, derived from Maxwell's equations and the constitutive relations for the ionosphere, are integrated by a step-by-step process proceeding downwards. The integrations are started from properly chosen initial solutions. From the resulting field variables at the bottom of the ionosphere a reflexion coefficient matrix R is derived, whose elements include the familiar reflexion coefficients. Two integrations are needed for each derivation of the elements of R. For the second method, it is shown that the formulae for R for a level below the ionosphere can be applied also within the ionized medium, and define a more general matrix variable whose elements are the dependent variables in a new set of differential equations. These are integrated by a step-by-step process as in the first method. The solution obtained below the ionosphere gives the required set of reflexion coefficients without further calculation. Only one integration is required for each derivation. The equations are given in full for certain important special cases.