PROFESSOR W. THOMSON ON THE SECULAR COOLING OF THE EARTH. 161 



is to reduce them to nearly half, or to increase them by rather more than half. 

 A reduction of the Greenwich underground observations recently communicated 

 to me by Professor Everett of Windsor, Nova Scotia, gives for the Greenwich 

 rocks a quality intermediate between those of the Edinburgh rocks. But we are 

 very ignorant as to the effects of high temperatures in altering the conductivities 

 and specific heats of rocks, and as to their latent heat of fusion. We must, there- 

 fore, allow very wide limits in such an estimate as I have attempted to make ; 

 but I think we may with much probability say that the consolidation cannot 

 have taken place less than 20,000,000 years ago, or we should have more 

 underground heat than we actually have, nor more than 400,000,000 years 

 ago, or we should not have so much as the least observed underground incre- 

 ment of temperature. That is to say, I conclude that Leibnitz's epoch of " emer- 

 gence" of the " consistentior status" was probably between those dates, 



12. The mathematical theory on which these estimates are founded is very 

 simple, being in fact merely an application of one of Fourier's elementary solu- 

 tions to the problem of finding at any time the rate of variation of temperature 

 from point to point, and the actual temperature at any point, in a solid ex- 

 tending to infinity in all directions, on the supposition that at an initial epoch 

 the temperature has had two different constant values on the two sides of a 

 certain infinite plane. The solution for the two required elements is as follows : — 



03" 



dx ^nxKt 



nr. 



2V 



^='^o+^y2VKf j^^- 



where k denotes the conductivity of the solid, measured in terms of the thermal 

 capacity of the unit of bulk ; 



V, half the difference of the two initial temperatures ; 



tj„, their arithmetical mean ; 



t, the time ; 



x, the distance of any point from the middle plane ; 



V, the temperaturp of the point a; at time t ; 



dv 

 and, consequently (according to the notation of the differential calculus), — the 



rate of variation of the temperature per unit of length perpendicular to the iso- 

 thermal planes. 



13. To demonstrate this solution, it is sufiicient to verify — (1), That the 

 expression for v fulfils the partial differential equation, 



dv _ d^v 

 dt ~ dx^' 



Fourier's equation for the " linear conduction of heat;" (2) That when i = 0, 



VOL. XXXIII. PART I. 2 Y 



