244 
PROFESSOR G. H. DARAVIN ON THE DISTRIBUTION OF STRAIN 
Substituting for a in terms of tlie modulus of stretching as given by (1), we have 
Hence 
and therefore 
j- {kr* + dr*) = Zedi* . 
d / 7 o\ 
o d0 
€V dr ’ 
h=~-A 
' r 3 y dr , 
r 3 J 
dr 
( 2 ) 
where the integral is taken from r down to such a depth that there is no change of 
temperature. 
If, now, 6 represents the rise of temperature per unit time, and if we replace h by 
clK/dt, the rate of stretching per unit time, and if we make application of (2) to the 
case of the Earth, and write v for the temperature of the Earth at a depth x below the 
surface, and c for the Earth’s radius, we have 
dv 
dt 
and (2) becomes 
c — x = r ; 
dK e [ x , sr. dlv 
—— = — —-t (C — X ) 6 -—- 
dt (e — x)° J c ' ' dxdt 
(3) 
In inserting the limits to the integral, it is assumed that the temperature at the 
Earth’s centre is sensibly constant. 
The amount by which a great circle of radius (c — x) is being stretched per unit 
time is 
„ , ,dK 
2,7 
This expression, with the above value (3) for dK/dt, is Mr. Davison’s result. 
We know from Thomson’s solution for the cooling of the Earth that, when x 
exceeds a small fraction of the Earth’s radius the temperature gradient and its rate of 
variation in time are very small. Hence, when x is not very small, d'V/dxdt is very 
small; therefore we may with sufficient approximation replace (c — x) s under the 
integral sign by c 3 — 3 c~x. Outside of the integral we may simply neglect x. Also 
the limit c of the integral may be replaced by infinity, and this is desirable because 
Thomson’s solution is really applicable to an infinite slab and not to a sphere. With 
these approximations we get 
dIC 
dt 
d°~v 
dx dt 
dx. 
(4) 
