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Progress in modern industrial technology is aided to a great extent by improved heat treatment of ferro-alloys, the most widely used basic construction materials in all kinds of machines and mechanisms. Wide use is now made of localized surface hardening of working regions of machine tool components following induction heating. This localization is achieved by the control of the frequency of an alternating current carried in an induction coil of shape suitable to the work piece so that the eddy current may be confined to the desired surface layer. In this paper is treated a two-dimensional boundary value problem associated with the surface hardening of a metallic cylinder. Under conditions which are not unduly restrictive, and which are usually satisfied in the physical situation, a representation theorem for the solution is proved. Some approximations to this solution are provided for numerical calculation of results. It is hoped that the method used will also be useful for analyzing other problems havin...

On a Certain Boundary Value Problem and Its Relevance in the Theory of Surface Induction Hardening