THE MATHEMATICAL THEORIES OF THE EARTH. 193 



Eartli unless we sapi)ose the epoch of transit through the liot region 

 exceedingly remote and the temperature of that region exceedingly 

 high. The continuity of geological and paleontological phenomena is 

 much better satisfied by the Leibuitzian view of an earth long subject 

 to comparatively constant surface conditions but still active with the 

 energy 'of its primitive heat. 



Notwithstanding the indefatigable and admirable labors of Fourier 

 and Poisson in this field, it must be admitted that tliey accompbshed 

 little more than the preparation of the machinery with which their suc- 

 cessors have sought and are still seeking to reap the harvest. The dif- 

 ficulties which lay in their way were not mathematical but physical. 

 Had they been able to make out the true conditions of the earth's store 

 of heat, they would undoubtedly have reached a high grade of perfec- 

 tiou in the treatment of the problem. The theory as they left it was 

 much in advance of observation, and. the labors of their successors have 

 therefore necessarily been directed largely towards the determination 

 of the thermal proi)erties of the earth's crust and mass. 



Of those who in the i>resent generation have contributed to our 

 knowledge and stimulated the investigation of this subject, it is hardly 

 necessary to say that we owe most to Sir William Thomson. He has made 

 the question of terrestrial temperatures highly attractive and instructive 

 to astronomers and mathematicians, and not less warmly interesting to 

 geologists and paleontologists. Whether we are prepared to accept his 

 conclusions or not, we must all acknowledge our indebtedness to the 

 contributions of his master hand in this field as well as in most other 

 fields of terrestrial physics. The contribution of special interest to us 

 in this connection is his remarkable memoir on the secular cooling of 

 the Earth.* In this memoir he adoi)ts the simple hypothesis of a solid 

 sphere whose thermal properties remain invariable while it cools by con- 

 duction from an initial state of uniform temperature, and draws there- 

 from certain striking limitations on geologic time. Many geologists 

 were startled by these limitations, and geologic thought and opinion 

 have since been widely influenced by them. It will be of interest there- 

 fore to state a little more fully and clearly the grounds from which his 

 arguments proceed. Conceive a sphere having a uniform temperature 

 initially, to cool in a medium which instantly dissipates all heat brought 

 by conduction to its surface, thus keeping the surface at a constant 

 temperature. Suppose we have given the initial excess of the sphere's 

 temperature over that of the medium. Suppose also that the capacity 

 of the mass of the sphere for the diffusion of heat is known, and known 

 to remain invariable during the process of cooling. This capacity is 

 called diffusivity, and is a constant which can be observed. Then from 

 these data the distribution of temperature at any future time can be 

 assigned, and hence also the rate of temperature increase, or the tem- 



* Transactions of the Royal Sockti/ of Fdhiburgh, 18G,;. Thomson aufl Tait's Natural 

 Philosophy, vol. i, Part 2, Appendix D. 

 H. Mis. 129 13 



