The analytical theory is given for the potentials and gradients to be found on the surface of a horizontally stratified earth in which the current source is an infinitely extended cable or line electrode lying on the surface. Power series expansions in the distance from the electrode are derived for small distances and asymptotic expansions for large distances. Detailed treatments are given for the two layer earth, for all thicknesses of the surface layer, and for the three layer earth in which the thickness of the second layer either equals or is twice as great as that of the surface layer, as well as formal expansions for general many layered earths. Simple closed hyperbolic function expressions are derived for the surface gradients for three layer systems in which the lowest layer is either of infinite or zero conductivity. The solution for the direct interpretation problem in which the continuous vertical conductivity variation may be determined from a knowledge of the surface gradients is briefly outlined, following the method of Slichter and Langer for the case where the current source is a point electrode. The analytical results of the indirect interpretation problem for a stratified earth are illustrated by thirty numerical and graphical examples.