BARTH’S CRUST RESULTING FROM SECULAR COOLING. 
237 
(13) Let us suppose, for a moment, that the rate of cooling is always inappreciable 
at the depth for which z — LOO, i.e., for which x = 2 fft). 4*00. This depth is 
continually increasing, and varies as the square root of the time that has elapsed 
since the consolidation of the globe. Dividing the crust above this depth always 
into the same number of spherical shells, the thicknesses of these shells therefore 
increase in proportion to the square root of the time. Now, the rates of cooling of 
any two particular shells have at any time the same proportion to one another, and 
therefore the proportional values of St 0 , 8 t x , 8 t 2 , . . . , in § (6) are always the same. 
But the cubes of the radii of the shells above the surface of greatest rate of cooling 
diminish in a less ratio, as the time increases, than the cubes of those below that 
surface; so that the depth of the unstrained surface would increase in a proportion 
rather greater than the square root of the time. On the other hand, the rate of 
cooling of any particular shell varies inversely as the time and, although at any 
time, the rate of cooling at the depth for which z = 4'00 is always about one-millionth 
of its rate at the depth where it is greatest at that time, yet in early periods the rate 
of cooling might be sensible at a depth greater than that for which z = -TOO. This 
would have the effect of slightly diminishing the proportion alluded to above. Hence, 
within certain limits, it is true that the depth of the unstrained surface increases as 
the square root of the time that has elapsed since the consolidation of the globe. 
From the value given in § ( 11 ) we can therefore determine approximately the depth 
of the unstrained surface at any other time. 
If we assume, as is generally done in the theory of the Conduction of Heat, that k, 
the rate of conductivity, is independent of the temperature, then the rate of cooling of 
any particular shell varies only when the time changes, and therefore the depth of the 
unstrained surface is, at any time, independent of the initial temperature of the 
globe. 
(14) Making use of the conclusion of the last paragraph, we can also determine 
approximately the law according to which the amount of rock stretched or folded in a 
given time changes. Considering any shell above the unstrained surface, the amount 
of it folded in a given time has been shown, in § (10), to be Sna'r' 8 . Now, approxi¬ 
mately, r may be considered constant, 8 to vary inversely as the time, and a directly 
as the square root of the time; so that 87 tolv 8 varies nearly inversely as the square 
root of the time. This is the case with every shell above the unstrained surface, and 
therefore with the total folding of all such shells. Hence folding by lateral pressure 
ivas effected most rapidly in the early epochs of the Earth’s history, and, since then, the 
toted amount of rock folded in any given time decreases nearly in proportion as the 
square root of the time increases. 
(15) The same law being approximately true of the total amount of rock stretched 
by lateral tension, it follows that the ratio of the amount of rock stretched to the 
* For dv/dt = — V/ N . 2 e~~ 2 . 1 jt, wliicli varies as 1/t, since z is a constant for any particular shell. 
