214 SEE— TEMPERATURE, SECULAR COOLING [April 20. 



When the diffusion of heat takes place in every direction the state 

 of the solid is represented by the integral 



e = 



If the initial heat is contained in a determinate portion of the solid 

 mass, one must take account of the limits which include this heated 

 part, and the quantities a, /S, y, which vary under the integral sign, can 

 not take values exceeding these limits (cf. " Theorie Analytique de 

 la Chaleur," Chap. Ix., p. 445). In the case of a symmetrical body 

 like the terrestrial spheroid, it is usual to take the internal distribu- 

 tion of heat to be symmetrical about the center of gravity, which 

 greatly simplifies the general problem. Observations made in many 

 lands indicate that the isothermal surfaces are about equally near the 

 surface at all places, except for the effects of unequal conductivity 

 in the crust ; and hence the symmetrical distribution of heat assumed 

 to hold true within the globe seems to be justified. The probable 

 mode of formation of the earth, and the long period during which it 

 has existed, give other grounds for the hypothesis of a symmetrical 

 distribution of the primordial heat about the center of gravity; the 

 unequal temperatures near the surface, due to unequal conductivity 

 in cooling, being confined to a shallow layer of very small extent 

 compared to the globe as a whole. In treating of the heat flow- 

 ing outward from the center of a symmetrical body like the earth, 

 it thus becomes sufficient to consider the propagation of heat in a 

 single direction normal to the surface. The triple integral is thus 

 reduced to a single integral corresponding to the single variable 

 coordinate, as in the differential equation (14). 



The form given the equations by Lord Kelvin and Rev. O. Fisher 

 is but slightly different from that originally used by Fourier, who 

 was occupied with the subject for more than a quarter of a century; 

 but as Lord Kelvin determined the constant of conduction with great 

 accuracy, and also carefully investigated the observed average rate 

 of the increase of temperature with the depth, we shall generally fol- 

 low his notation, which will also facilitate the comparison of his 

 results with those here obtained. 



The rate of variation of the temperature per unit of length 

 perpendicular to the isothermal plane is 



