in the theory of Conduction of Heat 421 



so that the two formulae agree as well as could be expected with 

 four-figure accuracy. 



In Prof. Perry's calculation, the value 1/s = 20 was taken, 

 and the value of k - k/c as in § 2 above, while t was taken as 

 96 X 108 years; then with Vo = 4000 the first and second terms 

 are stated to be equal to 



142-7 + 5-65 = 148-4. 



I did not succeed in confirming this value ; and Prof. Carslaw has 

 recalculated the Fourier-formula with the above data. His result is 



138-13 + 4-83 = 142-96, or say 143-0. 



The corresponding value of the asymptotic series is found to be 



1 - -01131 ^ -00038 - -00002 = -98905. 



Hence 1 = ^t (_ i + 1-6783) = -03570, 



and so v = 142-8, 



which agrees sufficiently closely with Prof. Carslaw's result*. 



Perry has also given numerical results for the same Fourier- 

 expansion when the constants are adjusted so that s = l; the 

 asymptotic series used above will clearly fail under this condition, 

 and a fresh formula becomes necessary. 



When 5 = 1, the equation for 6 becomes 



cot ^ = 0, 

 thus the values of d are ^tt, §77, pr, ... ; and the Fourier-expansion 

 simplifies to 



"" = 2 L-«^'/«= = ^, (e- + ie-«- + ^6-25-+...) 



where ^ = -n'^Ktlia^. 



In Perry's actual calculation, the value of w is nearly equal 

 to 1 ; and so the corresponding value of 



a2/,cf - ^TT^ = 2-47 nearly. 

 Thus the method of approximation adopted in § 2 needs recon- 

 sideration; and it turns out that the new formulae are not very 

 convenient for numerical work in this special case (see below). 

 However it is easy to recalculate this simple Fourier-expansion 

 for other values of w ; and to select values which are suitable tor 

 purposes of comparison. 



* The correction on account of the first term in er"'l^\ is of the same relative 

 order of magnitude as in the calculation of § 2; and this gives 

 /1^6782\ ^^^ ^ jQ_,j ^4QQ()) ^ ,^rj^ 



which accounts for the small residual difference. 



