404 Professor Carslaiv, The cooling of a solid sphere 



The equation (9) is 

 F (a) = a cos aa sin /xa (6 — a) + sin aa cos fxa (b — a) 



-\ — sin aa sin aa (b — a) = 0. 



ixaa 



The roots of this equation will be the common roots, if any, of 



sin aa = 0) 



and the roots of 



sin ^a (6- a) = Of ^^^^' 



a cot aa + cot jxa (b — a) -\ = (14). 



LLCCCt 



Since (b — a)la is small, the values of aa, if any, given by (13) 

 will be large. Thus for our solution we require only the earlier 

 roots. of (14), which, with the above constants, reduces to 



2M cot « + cot (2-35 x lO'^ x) = '^ (15), 



where x = aa. 



It will be found* that the first two roots of (15) are 



a;i = 2-9871 or 180° - 8° 51', 



x^ = 5-980 or 360° - 17° 22', 



and that x^ lies between mr — ^ and mr. 



Taking the first term only, the value of t is required for which 



, Iff cot a-.a cot jxai (h - a)-\ cot uai (h - a) - 1 \ 



\ aa cosec2 aia -{-aih- a) cosec^ uai {h-a) -\ -— / 



This gives the equation 



2^ = 8 X 3-742 X 10-^ x l^ x e— (sifk^^ , 



since the numerator and denominator of the fraction on the right 

 of (16) are, respectively, - 2-0298 x 10^ and 5-9763 x 10^. 



Beducing the answer to years, this leads to 9-02 x 10^ yearsf. 



* I am indebted to Mr R. J. Lyons for the solution of this equation. 



t Heaviside {J,oc. cit. p. 19) gives 9-03 x 10^ but adds that he has not taken 

 special pains to get the third figice right. 



As Heaviside compares his result with Perry's for the problem when the 

 capacity of the skin is neglected, it may be worth while to point out that some 

 arithmetical errors have crept into Perry's solution. (Cf. Nature, 51, p. 225.) 



The first two roots of the equation 19 tan a +a =0 are 



a^ =2-985676 or 180° -8° 56', 



02 = 5-9783345 or 360° - 17° 28'. 



The first term in the series for the temperature is 138-13 instead of 142-7, and the 

 second term is 4-82 instead of 5-65 given by Perry. 



