and Rate of Mountain-building on its Surface. 395 



expression be positive, the corresponding shell is stretched ; 

 if it be negative, the shell is folded. The following reasoning 

 shows that there exists a surface of zero-strain within the 

 planet*, below which its crust is stretched, and above which 

 it is crumpled. 



2. At the surface of greatest rate of cooling, S0 is zero. 

 Let r x be the radius of this surface, and r n the radius of the 

 planet. Then, if cooling has. not yet sensibly begun at the 

 centre, we know that 



800 + 80,+ .... +S0_ 1 = Sg r+1 + +S0 M _ 1 + 80„, 



numerically, the terms on the left side being positive, and on 

 the right negative. 



On either side of the surface of greatest rate of cooling, let 

 there be the same number of shells, and their thicknesses such 

 that, numerically, S0 =86 n , $di = S6 n -i, and so on. 



Then, in the series 



r„ 3 .80 o + J - 1 3 .S0 1 + .... +r„ 3 .80 n , . . (1) 

 we have 



S<9 n =-S0 o , and r n >r , 

 . " . r n 3 . S0 n + r 3 . S6 is negative. 



This is the case with every pair of terms equidistant from the 

 beginning and end of the series. 



Therefore, the sum of the series is negative. 



But the terms from r 3 . B6 to rj^ . 80 a -i are all positive. 

 Starting, then, from the first term, it follows that up to and 

 including a certain term r y z . h6 y , y being greater than x and 

 less than n, the sum is zero. 



Hence, there exists a surface of zero -strain within the 

 sphere. 



If cooling have begun at the centre, this simply cuts off 

 some of the terms from the beginning of the series, and its 

 effect is to still further deepen the surface of zero-strain. 



3. If the radius of the sphere be infinitely great, the ratios 

 of r n to r , of r n _i to i\ , &c, are unity, and the sum of the 

 series (1) is zero ; i. e. the surface of zero-strain coincides 

 with the surface of the sphere. In other words, on a globe 

 of very large radius, provided its surface be initially smooth 

 and spherical, no mountain- ranges can be formed by contrac- 

 tion from secular cooling, during a very long time from the 

 commencement of its history. And, in any case, the course 

 of geological change on such a body will probably be very 

 different from what we know it to have been upon the earth. 



* The same proof holds of course for any sphere, however small, 

 cooling from a uniform temperature. 



2D2 



