396 Rate of Mountain-building on the Surface of a Planet. 



4. Sir W. Thomson's well-known solution in his " Secular 

 Cooling of the Earth " applies without sensible error to the 

 earth, and even to smaller bodies, for many millions of years 

 from the time of consolidation. This being the case, the rate 

 of cooling at a given depth and time is for a considerable 

 period independent of the radius of the planet. 



Let z and £ be any pair of depths below the surface of a 

 sphere for which the values of S0 are numerically equal, z 1 

 being the greater. Then, r being the radius of the sphere, 



r — z 



J — 



; =1+ ^> 



r — z' r — z n 



which increases as r decreases, since z 1 is greater than z and 

 less than r. 



Hence, the smaller the radius of the sphere the greater are 

 the ratios of r n to r , of r n -i to r l , and so on; and therefore 

 the deeper is the surface of zero-strain at any time below the 

 surface of the sphere. 



5. The amount of folding of any thin shell of radius r n and 

 thickness a is 87rar n d' x ~, where 



rf=^ +1 -H+i+ ■■■■+rJ.Sd n ), 



' n 



X being a constant, and r y the radius of the surface of zero- 

 strain. 



Now, in any given time, with the assumed law of cooling, 

 the values of od n , etc., are the same whatever be the radius of 

 the sphere. If, then, the radius of the sphere be large com- 

 pared with the depth of the surface of zero-strain (which is 

 the case in the early periods of a planet's history), the above 

 expression shows that d varies very nearly as r w , the radius of 

 the shell. 



Suppose, for a moment, that at any time since consolida- 

 tion the depth of the surface of zero-strain is independent of 

 the radius of the planet, and let the crust between its surface 

 and the surface of zero-strain be always divided into the same 

 number of shells, so that at a given time the thickness (a) of 

 each shell is the same. 



In this case, then, ^irard varies as r 2 very nearly, and 

 therefore, if the depth of the surface of zero-strain were the 

 same in all planets after the same period of cooling, the total 

 amount of rock-folding in a given time in a planet would 

 vary as the area of its surface. 



But, as shown above, the depth of the surface of zero-strain 



* Ibid. p. 236. 



