An analytical study is made of the potential distribution about an electrode on a stratified earth, from the point of view of developing special formulas appropriate for computation at small and large distances from the electrode. Power series in the distance for small distances and simple asymptotic expansions for large separations are derived first for a two layer earth, from integral expressions for the potentials. For graphical integration a transformation is derived which makes the integrals converge very rapidly. For a many layer earth a general method for deriving the power series for small distances is indicated. For large distances, general expansions are given in closed (Eq. (23)) and asymptotic forms (Eq. (24)) applying to all types of conductivity variation. Explicit expressions for the power series are given for a three layer earth with the middle layer of thickness equal to and twice that of the upper layer. Closed and very rapidly converging series of Hankel function expansions are then developed for the third layer having zero or infinite conductivities, with the thickness of the middle layer being equal to, twice and three times that of the upper layer. Thirty numerical cases have been computed and plotted as curves to illustrate the fundamental formulae developed. In an appendix general proofs are given for the asymptotic expansions of not only the type of integrals occurring in the present problem but also for integrals involving Weber and Hankel functions as well as the Bessel functions, and where these functions are of nonhyphen;zero orders.
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