A finite-difference method, using the transverse field components as variables, is developed for calculating the properties of waveguides containing nonuniform dielectrics. It is shown how an essential consistency condition in the discretisation may be satisfied. Results are given for the dispersion of the quasi-TEM waves and the cutoff frequencies of higher waves in some typical problems. Other methods of calculation are discussed, and, in particular, it is shown that the use of the axial field components as variables complicates the problem.