in the theory of Conduction of Heat 419 



Now here o^JkI = 8-4 roughly, and so 



2g-aV«< _ 4.5 X 10-4 nearly. 



The corresponding correction to the value of t will be of the 

 relative order 9 x 10-^, and so will change t to about 9-37 x 10^ 

 (in years). 



The correction on account of the terms in P2P^> ^zP^-: ••• will 

 lead similarly to an estimate of the relative orders le"^"-''"^, 26-^*'/"*. 

 which are entirely unimportant in the present problems. 



§ 3. Numerical tests of the fortnulae of § 2. 



In view of the difference in form between the series of § 2 and 

 the corresponding Fourier-expansions, it seemed desirable to 

 compare the results of numerical calculation with values not very 

 different from those of § 2. Prof. J. Perry* had given the results 

 of some calculations in connexion with a problem which may be 

 regarded as the limit of that of § 2, when the thickness of the 

 shell (l) tends to zero, the value of s remaining fixed. However 

 some discrepancies were found (see below) and I decided to recal- 

 culate with slightly different constants so as to reduce the labour 

 of calculating the Fourier-expansion; I had not then| the 

 advantage of Prof. Carslaw's results with which to compare my 

 work. 



If we make /• tend to zero in § 2 we find that 



A ■-= 1/(1 — s -}- sqa coth qa), 



and then if we replace V by (Vq — v)/vo we obtain the formula 



» i\, — v 1 a sinh qr 



Vq {i — s -\- sqa coth qa) r sinh qa ' 



This represents the symboHcal solution for the temperature v of 

 a sphere initially at temperature Vq, radiating into a medium at 

 zero temperature ; this problem was solved by Fourier J in the form 



V _2a ySind sin {drja) e-''«'«/«' 

 Vq rs 6 {d — sin 6 cos 6) ' 



where the summation refers to the roots of the equation 



1 - s -f- s^ cot ^ - 0. 



* Nature, vol. 51, 1895, p. 225. 



t These calculations were made at intervals in the latter part of 1916 and in 

 1917. 



J Theory of Heal (Freeman's translation), § 293. 



