IN THE EARTH’S CRUST RESULTING FROM SECULAR COOLING. 
245 
Integrating, with regard to time, from the time t to zero, we shall get the total 
amount of stretching in the layer x from the epoch of consolidation down to time t. 
Thus 
K=e \ ( i 
dx. 
Now with Thomson’s notation 
’ dv 
Since 
^dx = Y — v. 
J x CvOC 
dv 
y 
£2/4* t 
dx ( 7 r/ct) 
4 
dv 7 V r tva t 7 2 V Kt ... dv 
tC — dx = 7 -- ate -x/iKt _ -__ g __ 2 ^^ 
J X 
dx 
Hence (5) becomes 
( 7 Ttct) 
K = 
( 7 r/cty 
TT 6 Kt dv 
N - v -^- c b 
dx 
* 
(5) 
( 6 ) 
This expression gives us the total amount of contraction since consolidation in 
terms of the temperature and the temperature gradient. I shall return to this 
expression later. 
Differentiating (6) with respect to the time, we have 
dK 
dt 
= e 
But 
Hence 
or 
dv 
dt 
x dv 
2 1 dx ’ 
dK 
dv 
dt 2t C dx 
dK 
dv 
dt 21 C dx 
dv 
dt 
and 
x 
c 
~x 
6 /c dv 
c dx 
(jid d?v 
c dx dt 
d 2 v 1 / x 3 
dx dt 
2 1 \2fct 
— 1 
dv 
dx 
12 Kt 
Sx 2 
6*4 / x 2 
\ 2*4 
6*4 
c 2 
(?) 
When x exceeds more than a small fraction of the Earth’s radius dv/dx is very 
small; hence in (7) we may regard x/c as small, and write the equation 
dK 
dv 
dt 21 C dx 
6 Kt 
(8) 
* If ® had not been treated as small, we should have got 
e(V-») 
c (c 2 4- 6*4) 2*4 pc 3 — (c — *) 3 
2*4 Pc 3 
{c —x f L 
+ 4 Kt 
dv 
; dx 
(c — x) 3 (c — x)° L, « 
and it is easy to see that this leads to the same result as the above for all practical purposes. 
