Dr BromwicJi, Symbolical methods in Conduction of Heat 411 



Symbolical methods in the theory of Conduction of Heat. By 

 Dr T. J. I'a. Bromwich, F.R.S. 



[Received 29 April. Read 2 May 1921.] 



In 1914, I communicated a paper "Normal Coordinates in 

 Dynamical Systems" to the London Mathematical Society*; and 

 I explained there (§§ 2, 3, 4) the relation of my methods to the 

 symbohcal methods used by Heaviside for various Electrical 

 problems. But I reserved! any general application of the corre- 

 sponding methods in Conduction of Heat and Difiusion; owing to 

 the pressure of the War and other difficulties I have not had 

 leisure to arrange my results for publication until now. 



I have given here only the special points which are suggested 

 in connexion with problems arising out of the question of the 

 "Age of the Earth" as handled by HeavisideJ and Perry; reference 

 is also made to the paper by Prof. H. S. Carslaw on the same 

 topic, communicated at the same time as this paper. 



Not much importance is attached to the estimate made here 

 (9-37 X 10^ years) for the age of the Earth; but the same data 

 have been adopted as in the original suggestion of Perry, that the 

 internal conductivity and heat capacity affected the estimate more 

 than the corresponding constants of the skin. It would be easy 

 to estimate new values of k and k with which the formula of § 2 

 (for ^r/^o) would yield almost any value of t from 10^ to 10^° years§. 



The ease of manipulating the constants in the formula of § 2 

 is in marked contrast to the labour involved in solving the same 

 problem by means of a Fourier-expansion; and mistakes are far 

 more quickly detected in the numerical work. Some comparisons 

 of the work will be found in § 3. 



The contents of the paper are as follows: 



§ 1. General consideration of a method for solving Conduction 



of Heat problems. 

 § 2. Application of Heaviside's method to the problem of a 



sphere surrounded by a shell of different material ||. 



* Proc. Lond. Math. Soc. ser. 2, vol. 15, 1916, p. 401. 



t L.c. p. 402. 



X Electromagnetic Theory, vol. 2, §§ 227-237. 



§ Compare Heaviside's estimates (I.e. §§ 232, 236). On radio-active gromids, 

 the "most recent estimates appear to be from 3 x 10' to 5 x 10^ years (H. N. Russell, 

 Proc. Roy. Soc. A, vol. xcix. (1921), p. 84). 



II This problem appears to have been solved originally by Heaviside himself 

 (see his remarks in Electrotnagnetic Theory, vol. 2, § 230), but his solution was not 

 published. A solution with an erroneous result was set as a question in Part II 

 of the Mathematical Tripos, 1904. 



