in the theory of Conduction of Heat 417 



To estimate the numerical values of these terms is easy; take 

 the data adopted in Prof. Carslaw's paper (following Perry and 

 Heaviside), 



Z = 4 X 105, a = 6-38 x 10«, k = -1643, Kj_ = -0117. 



Then s = -04953, n = -05211. 



After a little trial it was found that a value of t about 3 x 10^' 

 (that is, about 9-36 x 10" years) would fit the data for g and Vq 

 used by Lord Kelvin (see below) : and then it appears that 



\PJK^t = 2-3 X 10-", 



so that this term will be negligible in our calculations (and naturally 

 the same inference can be made with reference to the terms already 

 neglected in Z^ P, ...). 



On the other hand it is found that 



\n''a'-lKt = -0114; 



and accordingly this term, and other terms in n'^a'^JKH^, will probably 

 affect the conclusion. 



We shall accordingly complete the formula for g, by including 

 higher powers of nqa; on rejecting q^, g^, q^, ..., the result is 



^ = ^7 j — '^ + :, {nqa + n^q^a^ + n^q^a^ + ...)> 



As already explained, we have rejected the terms in Z^, Z^ ...; 

 and the interpretation of q, q^, q^, ... follows from § 4 (i)-(iii) below. 



The series now obtained is not convergent ; but it possesses the 

 asymptotic property that the error in stopping at any stage in the 

 series is less than the following term of the series (see § 4 (ix) 

 below). 



Inserting the values given above for a, k, t, the series in the 

 bracket ( ) becomes 



1 - -0114 + -0004 = -9890, 



the error being less than the following term (roughly -00002). 

 Thus our formula becomes (approximately) 



l_na \ a (-9890) 



Vq bl \ (1 — S)\/{7TKt)j 



If we now assume the values assigned by Lord Kelvin 

 g = 1/2743, Vo = 4000, 

 the corresponding value of t can be estimated by writing our 

 formula in the shape 



,/*- , (-9890) = (1 - s) f 1 + ^ --) = -9505 + -6653 = 1-6158. 

 Vi^rKt) ^ ' ^ \ Vq na) 



27 2 



