418 Dr Bromivicli, Symbolical methods 



On reduction this formula gives 



t = 2-95 X 10^7 



or in years* t = 9*36 x 10^. 



Finally we shall estimate the error which may be due to re- 

 placing coth qa by 1 in the formula for A and g used above. It is 

 easy to see that 



1 — s + sqa coth qa 1 — s( \l—pj} 



where p = e"^***. 



Thus on expansion we obtain 



+ «V^= (1 + 6, + ...)_... 



1 



+ :^ — : — {nqa — n\^a^ + n^q^d^ 



1-s ' 1-s 



+ ^ {nqa — 2n^q^a^ + 3n^q^a^ — ...) p + ... 



= Po + P^p + P,p^ + Psp'+ ... 



where the first term in P^ is 2nqa/{l — s). 



The series P^ leads to the asymptotic series already used; and 

 similarly we can obtain series for P^p, P2P^, etc. Clearly the most 

 important term (with the numerical values under consideration) 

 is the first term in P^p, which is 



2nqa „ 2n a ,, ^ 



1 — S 1 — S-\/{TTKt) 



^y § 4 (vii) below. Thus in comparison with the term nqajil — s) 

 in Po, the relative order of this term is 2e-"'''''K 



* On comparing this result with the value 9-02 x 10° found by Prof. Carslaw 

 from the first term of the Fourier-expansion, I was led (by comparison with the 

 numbers given in the first example of § 3 below) to the conjecture that the dis- 

 crepancy must be due to the neglect of the second and higher terms in the Fourier- 

 formula. 



Prof. Carslaw has kindly re-calculated his formula for the above value of t, 

 and obtains (using the first and second terms) a gradient 



g = -00035092 -i- -00001348 = -0003644. 

 This is equal to 1/2744; and so agrees with the value assumed for g to one jDart 

 in 3000, which is roughly the same order as the correction on accoimt of neglecting 



