The previous papers in this series have dealt with the differential equations governing the reflexion of long and very long radio waves from the ionosphere. In the present paper these equations are cast in a simpler form by using a generalized matrix admittance function A, and the corresponding equations are obtained in a form suitable for numerical integration by a step-by-step process from properly chosen initial solutions. From the value of A so obtained for a point below the ionosphere the reflexion coefficient matrix R is obtained, whose elements include the familiar reflexion coefficients. The equations are given in full for one case. The equations, and the computational procedure, are checked by using them to calculate the reflexion coefficients of a sharply bounded homogeneous ionosphere and comparing the result with that calculated by other methods. The new method of calculation is considerably faster than the old and will make possible calculation for higher frequency waves where many more steps are needed in the numerical integration. A theorem on the equivalence of certain directions of propagation is stated and proved.