Formulas are derived for the vector potential and inductance of a circuit comprised of a long linear cylinder enveloped by a return conductor of eccentric‐annular cross section, the two cylinders and the surrounding medium each being of different permeability. The restriction to very long cylinders rendering the problem two‐dimensional, a familiar scheme of analysis can be employed. A plane cross section is mapped conformally on a rectangle of infinite length and of breadth 2&pgr;, the perimeters of the conductors going over into line segments parallel to the short sides. In each of the three resulting rectangular regions of different permeability the vector potentialAis expressed as a sum of appropriate infinite double series of trigonometric functions with constant coefficients, these latter being determined in accordance with the known boundary conditions—continuity of the normal component of induction and of the tangential component of field intensity on the perimeters of the conductors. The magnetic field energy is obtained by evaluatingW=½∫SwAdS; whence the inductance follows fromW=½LI2. The resulting expression for the inductance yields known formulas for certain special cases of the general problem, thus verifying the analysis. As the symbol for the vector potential is capable of physical reinterpretation as temperature, the foregoing analysis also furnishes solutions to certain heat problems of technical interest.