PROCEEDINGS 



OF THE 



Cambritrgc ^bilosopbicitl ^acietg. 



The Cooling of a Solid Sphere with a Concentric Core of a Different 

 Material. By Professor H. S. Carslaw. 



[Read 2 May 1921.] 



1. Fourier himself remarked* that the present temperature 

 gradient near the surface might be used to obtain an estimate of 

 the time that has elapsed since the earth began to cool from its 

 molten state. And in a paperf which attracted much attention at 

 the time of its pubHcation, and to which, even towards the end of 

 his life, Kelvin attached considerable weight, he based his estimate 

 of the Age of the Earth upon the simple mathematical problem of 

 the distribution of temperature in a solid bounded by the plane 

 a; = and extending to infinity in the direction of x positive; the 

 initial temperature is constant and the boundary a; = is kept 

 at zero. 



In this case, with the usual notation, the temperature v at the 

 time t is given by ^ 



2l'o r2vuo ._ 

 V =: ~ I e-'^'da, 



and ^= -7T™e i^t, 



ox y^TTKt) 



the initial temperature being Vq. 



In 1895 Perry reopened the question in a series of papersj the 

 aim of which was to show that other possible internal conditions 

 would give greater ages than Kelvin's estimate of 10^ years, which 

 was regarded by the geologists as quite inadequate. Heaviside§ 

 made important contributions to this discussion and the problem 

 solved in this paper is one to which he refers. He mentions that 

 he had obtained its solution by his "operational method," but his 

 work has not been published. 



* Cf. CEuvres de Fourier (Darboux's Edition), T. 2, p. 284. 



t Trans. R. Soc, Edinburgh, 23, p. 157, 1864. 



% Nature, 51, pp. 224, 341 and 582, 1895. 



§ Heaviside, Electromagnetic Theory, vol. 2, ch. v. 1899. 



VOL. XX. PART IV. 26 



