406 Professor Carslaw, The cooling of a solid sphere 



ra rb 



Thus I L\Vi dr and U.^V^ dr are both positive. 



- J a 



But we have just shown that 



(a^ - /32) (JJ U^V^dr + ^ [' U^V^ dr^ = 0. 



If follows that our equation cannot have roots of the form 

 i ± ir), when neither f nor rj vanish. 



[Added June 3, 1921]. 



6. My attention has been called to the fact that the problem 

 with which this paper deals was set in the Mathematical Tripos, 

 Part II, 1904, 4 June, 2-5 p.m. In that paper Question 7 reads as 

 follows : 



A solid sphere of conductivity k and dififusivity a^ and of radius b is enclosed 

 m a spherical sheU of conductivity k' and diffusivity a'2 and of internal and 

 external radii b and c. Initially the whole system is at uniform temperatm-e lu 

 and from the epoch t = onwards the surface r = c is kept at zero temperature 

 Prove that at any subsequent time t the temperature at a distance r from the 

 centre is given by equations of the form 



. sin A„r 1 , ,, 



^ . sin A/ (c-r)l 



^ = ?^« sin a7(c^6) r '-'''"''' ^^'^ '>^> ^' 



where \/a' = \a, and X^is the sth root in order of increasing magnitude of the 

 equation 



further 



4 ^ 2mo {{k - k') sin A/ (c - b) + k'c\/} sin2 Xfi sin" A/ (c - b) 

 '" V A-6 sin-^ A/ (c - 6) + {aVa'^) k' (c - ftUin^^ft '' 



Explain the bearing of a numerical solution of this problem on the calcula- 

 tion of the age of the Earth. 



This solution of our problem is wrong. The equations which the 

 temperatures (w, u') must satisfy at r = 6 are 



u = u' \ 



dr dr \ 



so that, if we take terms of the type given in the summation, the 

 second of these equations leads to 



b {kXs cot X,b + Jc'X/ cot A/ {c - b)) ^ k - k' . 

 It is clear that the mistake arose from taking 



^Wr^''') = ^'Tr^^'')' 

 mstead of the proper equation at this surface. 



