1^07 ] AND CONTRACTION OF THE EARTH. 215 



and the temperature at any point {x, t) in the soHd is 



X 



2 V r^yft 



e = ©, + —. e-'dz. (i6) 



y ttJo 



In these equations V = half the difference of the two initial temper- 

 ature and ©o = their arithmetical mean. 



It may easily be proved by differentiation that the expression for 

 © satisfies Fourier's equation (14). 



When t = o, the expression for 0, x being positive, becomes 



®o + ^ r '"'"^^ = ®o + ^- ^vv = @, + F. (17) 



I/ttJo Vtt 



For all negative values of x, © = ©o — V. 



did 

 By differentiating (16) we obtain -|—; and it is easy to see that 



for all values of t, the second term of the right member of this equa- 

 tion has equal positive and negative values for corresponding values 

 of X. Taking Lord Kelvin's experimental value of the conductivity, 

 K = 400, equation (15) is reduced to the form 



— — = -e i6oo<. (18) 



^^ 3S'4Vt 



Lord Kelvin remarks that if ^=1,000 million years, and x> 

 3,000,000 feet, the exponential factor becomes less than e ~^^ , or less 

 than 1/270, which may be neglected as insensible. This indicates 

 that at depths greater than 568 miles the rate of variation of temper- 

 ature does not become sensible in 1,000 million years. A temperature 

 gradient thus exists only within a thin crust, and the influence of 

 curvature of the surface may be neglected ; so that the solution in the 

 case of Fourier's infinite solid becomes immediately applicable to the 

 cooling of the earth. 



If we take t= 100 million years from the beginning of the radia- 

 tion, 



d& I a;2 



—I— = i Ve 1600x108, (iq) 



^;tr 35 4 X 10^ ^ 



