The analysis presented relates to conductors parallel to the plane boundary of two semi-infinite, homogeneous media and to exponential wave propagation, but does not involve any of the previously necessary additional, simplifying assumptions. Generality of the solution is ensured by specifying the vector potential as the sum of complementary transverse magnetic and transverse electric-wave components, in each of the two regions. The formulation is presented initially with reference to a single, lossless, infinitely long conductor above a lossy ground. The analysis is, however; completely general and is extended to the lossy, multiconductor case. The field components are obtained by satisfying the boundary conditions at the ground and conductor surfaces. The resulting E field components yield the propagation constants, the modal current distribution, the line voltages, and hence the the line equivalent circuit constants, in that order. The transverse magnetic-vector potential above ground is seen to comprise three components; a ‘primary’, an ‘image’ and a ‘ground’ wave. The last mentioned is entirely due to the imperfect ground, inserts the appropriate interdependence between the four field components and is central to the higher accuracy and deeper physical understanding which the new method offers. Numerical results are published separately in a companion paper.