6 Prof. W. Thomson on the Secular Cooling of the Earth. 



founded is very simple, being in fact merely an application of one 

 of Fourier's elementary solutions 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 extending to infi- 

 nity 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 : — 



dv V __J* 



e 4jtf t 



dx VlTKt 



dze~", 



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 ; 



v their arithmetical mean; 



t the time ; 



x the distance of any point from the middle plane ; 



v the temperature of the point x at time t ; 



and, consequently (according to the notation of the differential 



dv 

 calculus), — the rate of variation of the temperature per unit of 



length perpendicular to the isothermal planes. 



13. To demonstrate this solution, it is sufficient to verify 

 (1) that the expression for v fulfils the partial differential 

 equation 



dv _ d 2 v 



Jt~ fC da? > 



Fourier's equation for the "linear conduction of heat;" (2) that 



when / = 0, the expression for v becomes v -\-Y for all positive, 



and v — V for all negative values of as; and (3) that the expres- 



dv 

 sion for -=- is the differential coefficient with reference to x, of 

 dx 



the expression for v. The propositions (1) and (3) are proved 



directly by differentiation. To prove (2), we have, when £ = 



and x positive, 



2 V f °° 



V = V n -\ 7 1 dZ€~ z * 



r A 



or, according to the known value, \^iT i of the definite integral 

 Jo v=v hV; 



