It is shown that the equations governing oblique propagation in a horizontally stratified ionosphere with an oblique magnetic field can be cast into a form suitable for solution by successive approximations. In the general case this ‘coupled’ form consists of four first order differential equations, each being associated with one characteristic wave. In the special cases (a) horizontal magnetic field, (b) plane of incidence perpendicular to the magnetic meridian, two second order coupled equations of a particular type can be derived, each of which is associated with a pair of corresponding upgoing and downgoing characteristic waves. These latter equations are similar to, and include, those already given by Försterling(7) for vertical incidence.

The cases in which there is no coupling are briefly considered from the point of view of the first order equations.

The general formulation provides a basis for assessing the validity of the standard ‘ray’ approximation, alternative to that developed by Booker (2), and brings out the nature of its breakdown in coupling and reflexion regions.

Specific applications and extensions of the theory are left for later consideration.

In the theoretical treatment of finite elastic deformation for isotropic, incompressible materials, a series of relations may be obtained, which, in the most general case, can be reduced to four partial differential equations for the determination of four unknown quantities. For these dependent variables we may conveniently choose the three components of displacement in any given system of coordinates, and an arbitrary parameter which arises from the assumption of incompressibility. The equations thus obtained are non-linear in form, and exact solutions subject to specified boundary conditions are usually not readily obtainable by orthodox methods of approach. The successful treatment of a number of problems by Rivlin (see references), and by other workers (1,2), has depended essentially upon the imposition of a restriction upon the form of the deformation, so that by an appropriate choice of initial and final reference frames in which to specify the configuration of the elastic body, partial differential equations can be avoided.