EARTH’S CRUST RESULTING FROM SECULAR COOLING. 
233 
Hence folding by lateral 'pressure takes place only to a certain depth below the 
Earth’s surface; at this depth it vanishes , and, passing through it downwards , folding 
gives place to stretching by lateral tension .* 
(5) In order to find the depth at which folding by lateral pressure vanishes, 
I propose to solve the following problem :— 
A globe, of radius r, is surrounded by a number of concentric spherical shells, 
called A 1} A 3 , A s . . ., of thicknesses a x , a 2 , a 3 . . ., respectively. The globe remaining 
at its initial temperature, the shell A i is cooled by t°, the shell A 3 by t 2 ° in the same 
time, and so on. The linear coefficient of expansion being e, and the same for all the 
shells, it is required to find the distribution of strain resulting from this method of 
cooling. 
[( 6 ) Consider the shell A^. If the globe were absent, the radius of its inner 
surface would become r(l — et x ). It is, however, obliged to remain of radius r, and 
it must, therefore, be stretched. Assuming the stretching to be uniform throughout 
the shell, and expressing the fact that the volume of the shell after cooling t° is 
equal to its original volume multiplied by 1 — 3 et 1} it will be found that the radius of 
the outer surface of the shell is 
where 
r x = r + a \- 
Proceeding in a similar manner with the other shells, and remembering that the 
radius of the inner surface of any shell after cooling must be the same as the radius 
of the outer surface of the shell below after its cooling, then the radius of the inner 
surface of the shell A„ + 1 becomes 
(rn 1 u —k) et n A h' n — i r n _ f) et u _y -f- ... -I- v ’) ct x 
^ n 9 
^n 
Now, let a 1 = a z = ... — a; also let t x — St 0 , t % — t Y ~ Sty, . . ., t n + 1 — t„ = St a . 
If the shell A, i + 1 had been allowed to contract, as if the globe and n interior shells 
were absent, the radius of its inner surface would have been (r + no) (1 — et /l + l ). 
Hence the amount by which a great circle of the inner surface of this shell is 
stretched is proportional to 
(r + ncif + ( r + n ~ 1 • a )' 3 • + . . . + r 3 . S^ 0 ], 
that is, if the shells be supposed infinitely thin and infinitely great in number, to 
e [ r+m „ dt , 
- -— z 6 , dz. 
(r + nay j r dz 
Whilst folding and crushing by lateral pressure must be accompanied by a development of heat, as 
pointed out by Mr. R. Mallet and others, stretching by lateral tension, on the other hand, must be 
accompanied by a cooling of the masses stretched. 
MBCCCLXXXVII.—A. 2 H 
