232 
MR. C. DAYISON ON T1IE DISTRIBUTION OF STRAIN IN THE 
where “ k denotes the conductivity of the solid, measured in terms of the thermal 
capacity of the unit of bulk ; Y, half the difference of the two initial temperatures ; 
v 0 , their arithmetical mean ; t, the time; x, the distance of any point from the middle 
plane; v, the temperature of the point x at time £.”* 
(3) Differentiating v with respect to t, we obtain 
_ _ v * . -««/4 
dt 2\/ ( 7 tk) t* 
the rate of cooling at the point x at time t. Differentiating this expression with 
respect to x, we find 
dH _ Y 1 fx 2 
dx dt t*\2id 
Hence d~vjdxdt is equal to zero when x — v /(2k^). In other words, at any given 
epoch, the rate of cooling is a maximum at the depth for which x is equal to x /(2kI).^ 
Hence the rate at which any shell parts with its heat increases to a certain depth 
below the Earth’s surface, where it is a maximum, after which it decreases towards the 
centre, and the depth of the surface of greatest rate of cooling is continually 
increasing, and varies as the square root of the time that has elapsed since the 
consolidation of the globe. 
(4) Consider any two consecutive spherical shells below the surface of greatest rate 
of cooling. Since the upper shell cools the more rapidly, its inner surface would, if 
free, contract more than the outer surface of the shell below ; but, being forced to 
remain of the same radius as the latter after its contraction, it follows that the upper 
shell must be stretched or rent in order to rest upon the lower. Owing to the great 
pressure at that depth, and also to the slow rate of cooling, there can be little doubt 
but that the upper shell will be stretched and not rent. 
Consider, again, two consecutive shells above the surface of greatest rate of cooling. 
In this case the lower shell cools the more rapidly; the inner surface of the upper 
shell, if free, would not therefore contract so much as the outer surface of the shell 
below. The upper shell must then either be stretched less than the lower, or must 
be crushed and folded in order to rest upon it. It will be shown afterwards that, 
above the surface of greatest rate of cooling, the amount by which each shell is being 
stretched gradually diminishes towards the surface of the Earth, until at a certain 
depth it is zero (the shell at this depth being now unstrained through cooling), and 
that, outside this depth, every shell is being crushed or folded. 
* ‘Edinb., Roy. Soc. Trans.,’ vol. 23, 1864, pp. 161-162; Thomson and Tait’s ‘Natural Philosophy,’ 
A PP- D -, §§ G). (0). 
t This paragraph is the substance of a letter by Professor G. H. Darwin on “ The Formation of 
Mountains and the Secular Cooling of the Earth,” published in ‘Nature,’ Feb. 6, 1879 (rol. 19, p. 313). 
If t be taken as 98,000,000 years, the depth at which the rate of cooling is greatest is about 53 miles. 
