Numerical methods are given for solution of axisymmetrical problems involving the partial differential equation∂2&psgr;∂z2+∂2&psgr;∂&rgr;2+K&rgr;∂&psgr;∂&rgr;=0,where &rgr; is the radial coordinate andzthe coordinate parallel to the axis. The various values ofKwhich occur in physical situations are discussed, and common iteration methods for handling these problems are given. For Laplace's equation,K= 1. For the Stoke's stream function,K= − 1, but it is pointed out that for numerical work a new function, called theflow‐disturbance function, havingK= 3, is more tractable. Similarly a new function withK= 5, thestress‐concentration function, is much easier to compute than the usual stress function (K= − 3) for the case of the torsion of a circular shaft of varying diameter. The methods are illustrated by computation of the equipotentials for an electron lens, and by a complete computation of the stresses and strains in a particular grooved circular shaft under torsion.