IN THE EARTH’S CRUST RESULTING FROM SECULAR COOLING. 
243 
Earth’s surface, and that this layer, instead of that of greatest cooling, must be taken 
to represent Favke’s elastic membrane. 
It appears that the mathematical discussion of the problem in his paper is un¬ 
necessarily laborious,'" and he has not made various important deductions as to the 
integral results of distortion and as to the magnitude of the effects to be expected. 
I, therefore, offer the following note with the intention of rendering more complete an 
important chapter in the mechanics of geology. 
When a spherical shell expands wdth rise of temperature, it may be said to stretch, 
in one sense of the word. If the shell were one of the layers of the Earth, such a 
stretching would have no geological effect, for it would merely involve a change of 
density. The term “stretching” then requires an explanation in connection with 
Mr. Davison’s paper. The stretching which we have to consider is, in fact, simply the 
excess of the actual stretching above that due to rise of temperature. The negative 
of such stretching is a contraction, and it would actually be shown by a crumpling 
of strata. 
If p be the density of a body, and e its modulus of linear expansion for temperature, 
then it is obvious that when the temperature is raised 6 degrees the density becomes 
p( 1 — Se6). Now suppose that a spherical shell of radius r expands so that its 
radius becomes r(l + a), and suppose that at the same time its temperature is raised 
6 degrees. Then, if k be the modulus of stretching, so that unit length is stretched 
by a length k, we have, in consequence of the above explanation of stretching, 
a 
— k -f- e6. 
( 1 ) 
Now let us consider the geometry of changes in a sphere such that a shell of 
internal radius r, thickness Sr, and density p, expands until its internal radius r 
becomes r (1 + a) and its density p becomes p (1 — 3ed). 
The external radius r + Sr clearly then becomes 
r (1 + a) -f- Si 
Thus the mass of the shell 47 rr 2 p Sr becomes 
47 rr 2 p Sr 
d 
1 + dr 
d 
1 + 2a + — Ira) — SeO 
dr 
Then, since the mass remains unchanged, we must have 
2a ~b — (ra) — 3 eO = 0. 
dr v ' 
This “ equation of continuity ” may clearly be written 
— (r 3 a) = Sedr 3 . 
dr x ' 
* Cf. foot-note, p. 234. 
2 I 2 
