﻿" Densities in the Earth's Crust." 417 



Mr. Fisher leaves his guide his equations become wrong. 

 The formula given by Prof. Woodward for the temperature 

 u becomes indeterminate when we make #==r, i. e. when 

 we seek to know the temperature at the centre, as the expres- 

 sion in brackets becomes zero ; and to this there is no objec- 

 tion. But when Mr. Fisher takes only the first two terms 

 ard differentiates, he gets 



du V r x 



iKt 



dt Virr—xttJ^Kt 



du 

 If in this we put x=r, and r is not infinite, then -*- 



becomes infinite — that is, the rate of cooling at the centre is 



infinite- -while in reality it must he quite small. In like 



du 

 manner -j- also becomes infinite, L e. the temperature at the 



centre is infinitely greater than elsewhere. As Mr. Fisher's 

 " results depend on the values of these differential coefficients, 

 it does not seem possible that they can be correct. 



If the mathematics is to teach us anything we must keep 

 the meaning of our equ r tions well in mind throughout. As 

 I have shown, each term of the series which satisfies the 

 fundamental equation 



d(zu) _ d 2 (zu) 



~dT~ K ~W 



gives the cooling greatest at the centre, and the sum of a 

 number of maxima must itself be a maximum. Hence, if 

 we do not discriminate in some way, it would follow that 

 throughout the whole cooling from £ = to £ = oo the cooling 

 is greatest at the centre ; whereas we know very well that 

 at the beginning the cooling will be greatest at the surface. 

 This must be accounted for in some way, whether my ex- 

 planation of it be correct or not ; in fact, in the cooling of 

 every body there must be two stages — the first, a temporary 

 one, when the exterior cooling is most rapid, and the second, 

 one which lasts till all the heat is lost, during which time 

 the surface has so little heat to lose that the centre cools 

 quickest ; and the question is whether the earth has now, and, 

 if so, when it, entered on this second stage. 



Only when the temperature condition of the earth is settled 

 can we enter into the question of its results as regards con- 

 traction and a level of no strain. There appears to me to be 

 an error also in Mr. Fisher's investigation of this point, as 

 he deals with the circumference of the earth at the level he 

 discusses rather than with the spherical area ; but as we do 



Phil. Mag. S. 5. Vol. 38. No. 233. Oct. 1894. 2 F 



