326 Mr. C. Davison. On Secular Cooling and the [May 5, 



ness a, the change of radius of the inner surface of the shell A^ is 

 proportional to 



+ 0' + a) s (* 2 -* 1 )+r^]- . . (1.) 



i.e., if the shells be infinitely thin, to 



e — 



t being proportional to the rate of cooling of any shell. 



If this expression be positive for any shell, the shell is stretched ; if 

 negative, it is crushed or folded. 



To apply this problem to the case; of the earth, the law of cooling 

 taken is that which follows from Sir W. Thomson's solution, in his 

 memoir on the secular cooling of the earth. The expression in the 

 form (2) proves unserviceable, and therefore the expression (1) is 

 made use of as follows : — 



Taking the time since solidification provisionally at 174,240.000 

 years, it is shown that the rate of cooling (dv/dt) is practically insen- 

 sible at a depth of 400 miles. The radius of a sphere equal to the 

 earth in volume being about 3959 miles, the earth is supposed to be 

 constituted as follows : — A central globe, 3559 miles in radius, at the 

 initial temperature of the earth, which as yet has not sensibly 

 cooled, surrounded by 400 concentric spherical shells, each one mile 

 in thickness, the rate of cooling in each shell being uniform through- 

 out, and equal to its value at the outer surface at that shell. 



The results of the calculation are shown by the curve in the figure 

 accompanying the paper, and the following conclusions are deduced, 

 taking the time since consolidation provisionally at 174,240,000 

 years : — 



1. Folding by lateral pressure takes place only to a certain depth 

 (about five , miles) below the earth's surface, and below this depth 

 changes to stretching by lateral tension. 



2. Stretching by lateral tension, inappreciable below a depth of 

 400 miles, increases from that depth towards the surface ; it is 

 greatest at a depth of 72 miles (i.e., just below the depth at which the 

 rate of cooling is greatest) ; after this it decreases, and vanishes at a 

 depth of about five miles. 



3. Folding by lateral pressure commences at a depth of about 

 five miles, and gradually increases, being greatest near the surface of 

 the earth. 



No great importance is attributed to the numerical results. The 



