20 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 56 



the radius, his melting point curve would have reduced to a straight line. 

 Such a restriction in the circumstances is not objectionable and corre- 

 sponds to the limitation of the problem of refrigeration as solved by 

 Kelvin, whose formulas, strictly speaking, represent the cooling of a solid 

 of indefinite extent bounded by a plane surface and not the cooling of a 

 sphere. It can be and has been shown that the error introduced by neg- 

 lecting the earth's curvature is insignificant. 



It will be sufficient, therefore, to assume that the initial temperature 

 of the earth increased in simple proportion to the distance from the sur- 

 face, and this simplification renders it easy to modify the Fourier equa- 

 tion employed by Kelvin to satisfy this condition. In his solution an 

 infinite homogeneous solid is supposed divided by a plane on one side of 

 wliich, at the initial instant, the temperature has one uniform value, while 

 on the other side it has another uniform value. The object of the second 

 mass is to replace outer space and preserve a constant temperature in the 

 dividing plane. This device may seem at first sight too artificial, but 

 Kelvin proved tliat after a comparatively brief period the temperature of 

 the surface of the globe would be affected to an entirely negligible extent 

 by internal heat. Eadiation and convection, or briefly " emissivity," ac- 

 complish substantially the same end as the hypothetical conducting solid, 

 in that they dispose of the heat as fast as it reaches the surface excepting 

 for a period of possibly a few thousand years after the surface of the 

 globe solidified.' 



Fourier's partial differential equation for the linear conduction of 

 heat is 



dv _ d}v 



where v is temperature and x distance from a plane, while k is the dif- 

 fusivity which is assumed to be constant and known. Any value what- 

 ever of V which will satisfy this equation is a solution of some problem in 

 conduction. In general the problem of finding a value of v which satis- 

 fies given boundary conditions is difficult, but in the particular case under 

 discussion the appropriate form of v is easily arrived at. The equations 



dv V 



and 



dx ^/^^Kt 



v — Vo=V - ■-- \ e-^'dz + CX ( 2 ) 



V ~ Jo 



1 So far as the earth is concerned, the evaporation of water is the most important factor in the 

 removal of heat from the surface to the upper regions of the atmosphere, or in what may be re- 

 garded as the emissivity of the globe. According to Sir John Murray's figures (Geochemistry, 

 p. 53), the average annual rainfall less the run-off is about 70 centimeters, and substantialh' all of 

 this is evaporated. The evaporation of 1 cubic centimeter absorbs about 591 gram calories and 

 thus the total evaporation removes from the earth's surface some 13 times as much heat as the 

 earth emits, the large residue being of course derived from the sun. 



