﻿no Strain within a Solid Earth. 133 



u = the like when sphericity is taken account of ; 



e = the coefficient of linear contraction of those portions of 



the globe which have sensibly cooled ; 

 k = the conductivity measured in terms of the capacity of 

 rock for heat ; 

 then the differential equation for the diffusion of heat in 

 the sphere will be 



d(zu) _ d 2 (zu) 

 dt ~ K dz l ' 



Professor R. S. Woodward, U.S.A., gives the solution of 

 this, which is suitable to the case of a sphere initially at a 

 uniform temperature throughout, and cooling into a medium 

 such that its surface is maintained at a constant temperature 

 considered to be zero. The solution is * 



2r y »=«/_!)«+! - K (^) 2 t t z 



zu= 2, e sinn7r-. 



7r n=1 n r 



This solution meets the objections raised by Professor Blake 

 to Lord Kelvin's solution, in which the radius was assumed 

 infinite. 



Prof. Woodward, with great ingenuity, transforms the above 

 expression into one or other of two rapidly converging series, 

 in either of which he says the first term is sufficient in the 

 case of the earth, if the time since the commencement of the 

 cooling is less than 100,000,000,000 years. The second of 

 these series (no. 20) written in our symbols is 



J x 2r+x 



WJ 2r-x ) 



^ The first term of this series being sufficient, if we differen- 

 tiate it with respect to t we get 



du V r x -* 2 



dt VTrr—xt^kKt 



* < Annals of Mathematics/ vol. iii. June 1887. 



t The convergency of this series is evidently due to the rapidity with 



which the definite integral 1 e-M 2 dp approximates to the limiting value 



Jo 

 i Vtt as the upper limit increases. When that is no greater than 2-17, 

 the first six places of decimals are the same as for the limit, and when 

 it is 4 the first ten. 



