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The problem of the conduction of heat in a solid sphere with a concentric core of a different material, the surface kept at a constant temperature, and the initial temperature of the whole zero, has already been solved in these Proceedings.
* Carslaw, , “The cooling of a solid sphere with a concentric core of a different material”, Proc. Cambridge Phil. Soc. 20 (1921), 399–410.Bromwich, , “Symbolical methods in conduction of heat”, Proc. Cambridge Phil. Soc. 20 (1921), 411–27. In the first of these papers contour integration is used; in the second Heaviside's operational calculus as developed by Bromwich.
† Carslaw, , Math. Gaz. 22 (1938), 264–80.Carslaw, and Jaeger, , Phil. Mag. (7), 26 (1938), 473.Carslaw, and Jaeger, , Bull. American Math. Soc. (in the Press). Also the work of Doetsch quoted on p. 396 and papers by Churchill, Lowan and others.
‡ Carslaw, , Conduction of heat, 2nd ed. (Macmillan, 1921), § 64. We quote this book below as C.H.
* We use this bar notation throughout for the Laplace transform. Thus
* See, for example, Doetsch, , Theorie und Anwendung der Laplace-Transformation (Berlin, 1937), p. 126, Satz 2.Churchill, , Math. Z. 42 (1937), 569, Theorem 1.
* Cf. C.H. § 106.
* If k < 1 the result is true for ξ ≥ 0, t > 0.
* C, C 1, … are used for different constants, and ρ0, ρ1, … for fixed values of ρ.
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