Previous researches on the helix have made simplifying approximations about the nature of the boundary conditions. In this paper the field equations and boundary conditions are formulated exactly. They are then solved by an expansion in powers of the ratio of the thickness of the wire to the distance between turns. The method used consists of introducing a new coordinate system, which is such that a helical wire of circular cross section is a surface in which one coordinate is constant. Maxwell's equations and the electromagnetic boundary conditions are expressed in terms of this system. Since nonorthogonal coordinates are involved, the equations cannot be solved exactly; but a perturbation procedure may be applied as indicated above. The result of the analysis is to show that there is a principal mode, which propagates with the free space velocity of light in the direction of the wire. The characteristics of this mode are studied, and they are compared successfully with experiment.