82 



SCmNOK 



[N. S. Vol. XI. No. 264. 



Kelvin is being attacked from the mathe- 

 matical side.* My own views on this sub- 

 ject were expressed somewhat at length ten 

 years ago, in the address already referred 

 to, and it seems unnecessary here to go 

 into the matter any further than to reaffirm 

 my conviction that the geologists have ad- 

 duced the weightier arguments. Beautiful 

 as the Fourier analysis is, and absorbingly 

 interesting as its application to the problem 

 of a cooling sphere f is, it does not seem to 

 me to aiford anything like so definite a 

 measure of the age of the earth as the visible 

 processes and effects of stratification to 

 which the geologists appeal. In short, the 

 only definite resultswhich Fourier's analysis 

 appears to me to have contributed to knowl- 

 edge concerning the cooling of our planet 

 are the two following, namely : first, that 

 the process of cooling goes on so slowly that 



Kelvin's address on the age of the earth as an abode 

 fitted for life,' Science, June 30, 1899 ; also 

 Sir Archibald Geikie's presidential address to Geo- 

 logical Section of the British Association for the Ad- 

 vancement of Science, Dover meeting, 1899. 



* Notably by Professor John Perry. See Nature, 

 January 3, and April 18, 1895. 



1 1 have recast this problem of Fouiier in two 

 papers published in the Annah of Mathemntics, Vol. 

 III., pp. 75-88 and pp. 129-144. The solution there 

 given is the only one, so far as I am aware, which 

 lends itself to computation for all values of the time 

 in the history of cooling. A point of much mathe- 

 matical interest on which this solution depends is the 

 equivalence of the two following series : 



1- ,1=1 \ n J 



= r«o —^ 2 Je dz. 



Yn " = l2n + 1)mo — m 



In these u is the temperature at a distance r from the 

 center of the sphere at any time i ; Wq is the initial 

 uniform temperature of the sphere ; r„ is the radius of 

 the surface of the sphere ; a* is the diffusivity, supposed 

 constant ; and m= r j {2 y' t ), mo= r,,/ (2ai/ t ). It 

 will be observed that when the first series (which is 

 Fourier's) converges very slowly, the second converges 

 very rapidly, and vice versa. It will be seen also, 

 that the series refuse, as they should, to give values of 

 the temperature corresponding to negative values of 

 the time. 



nothing less than a million years is a suit- 

 able time unit for measuring the historical 

 succession of thermal events ; and secondly, 

 that this process of cooling goes on substan- 

 tially as if the earth possessed neither oceans 

 nor atmosphere. 



It was the well-founded boast of Laplace 

 in the early part of the century that astron- 

 omy is the most perfect of the sciences;* 

 and expert contemporary opinion, as we 

 have seen in the case of no less a personage 

 than Green, agreed that the ' Mecanique 

 Celeste ' left little room for further advances. 

 Indeed, it would appear that the complete- 

 ness and the brilliancy of the developments 

 of celestial dynamics during the half century 

 ending with 1825 (the period of Laplace's 

 activity) completely overshadowed all other 

 sciences and retarded to some extent the 

 progress of astronomy itself. The stupend- 

 ous work of Laplace was chiefly theoretical, 

 however, and he was content in most cases 

 with observational data which accorded 

 with theory to terms of the first order of 

 approximation only. He was probably not 

 so profoundly impressed as men of science 

 at this end of the century are with the ne- 

 cessity of testing a theory by the most 

 searching observational means available. 

 In fact, in elaborating his methods and in 

 applying his theories to the members of the 

 solar system, it has been essential for his 

 disciples to display a degree of ingenuity 

 and a persistence of industry worthy of the 

 master himself. But the prerequisite to 

 progress in celestial mechanics consisted 

 not so much in following up immediately 

 the lines of investigation laid down by 

 Laplace, as in perfecting the methods and 

 in increasing the data of observational as- 

 tronomy. 



* " L'Astronomie, par la dignity de son objet et 

 par la perfection de ses theories, est le plus beau 

 monument de I'esprit bumain, le titre le plus noble 

 de son intelligence." Systeme du Monde, Ed., 1884, 

 p. 486. 



