The conductivity tensor is obtained directly from the perturbed Vlasov equation for a plasma in an inhomogeneous magnetic field. The conductivity tensor elements are obtained consistently to all orders in the 'perpendicular wavenumbers' and to first order in the equilibrium magnetic field gradient. Conditions on the parallel wavenumber and the magnetic field gradient for which such a method is valid are given. The wave differential operator is obtained from the conductivity tensor using Maxwell's equations. The coupled differential equations are then truncated to second order to model to case of minority and second harmonic ion cyclotron heating. The forms of the electromagnetic and kinetic power flows and the minority and majority cyclotron damping are obtained simply. The differential equations are solved numerically as a linear combination of boundary value problems for the perturbing electric fields using standard NAG library routines, and the transmission, reflection and mode conversion are calculated for a range of kyand kzvalues. On setting kyto zero, excellent agreement is found between the results obtained from the authors' code and published results from codes by other authors.
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