234 
MR. C. DAVISON ON THE DISTRIBUTION OF STRAIN IN THE 
If this expression for any shell be negative, then that shell will be folded and not 
stretched.] '* 
(7) Comparing the above with the notation used by Sir W. Thomson, it will be 
seen that t is proportional to his dv/dt; his x corresponds to c — z, where c is the 
radius of the Earth, and z is the distance of any point from the centre of the Earth. 
Hence the dtjdz of the preceding paragraph is proportional to — (d/dx) (dv/dt) in Sir 
W. Thomson’s notation. 
Now 
d fdv 
dx \dt 
V 1 / x* 
2 ( 777 c) P \2 Kt 
e -x*/4 Kt 
Substituting in the integral in § (6), we have 
S« + i — 
P 
P (r + na ) 2 
(c — x ) 3 ^ — ly e x2/iKt dx, 
where g is a constant. 
No practical use can be made of this expression, and we are, therefore, obliged to 
return to the last expression in § (6), in order to find the radius of the shell which 
experiences no strain through cooling. 
(8) If, in the expression for dv/dt in § (3), we put x = 2 v /(kZ) 2 , we find that, at 
any given time, the rate of cooling is proportional to 2 e -£2 . At the depth where the 
rate of cooling is greatest (for which 2 = 1/^/2) the value of ze~ z ~ is ‘42888. 
Again, at the depth for which 2 = 4‘00, the value of this expression is ‘00000045014. 
Hence, at any time, the rate of cooling at the latter depth is about one-millionth of 
its value where it is greatest. For our present purpose I shall, therefore, assume that 
below the depth for which 2 = 4‘00 the rate of cooling is practically insensible, and 
it will be seen afterwards that the quantities neglected do not appreciably affect the 
results. 
If the temperature of melting rock be 7000° F., the consolidation of the globe may 
have taken place about 98,000,000 years ago ; but, as Sir W. Thomson remarks, “we 
are very ignorant as to the effects of high temperatures in altering the conductivities 
and specific heats of rocks, and as to their latent heat of fusion. We must, therefore, 
allow very wide limits in such an estimate as I have attempted to make ; but I think 
we may with much probability say that the consolidation cannot have taken place less 
than 20,000,000 years ago, or we should have more underground heat than we actually 
have, nor more than 400,000,000 years ago, or we should not have so much as the 
least observed underground increment of temperature. ”t 
* The paragraphs witliin square brackets (added June 21, 1887) are abridged from the proof 
contained in the original paper communicated to the Society, partly because the proof there given was 
unnecessarily long, and partly because another and more elegant proof is given, in the note following 
this paper, by Professor G. H. Darwin. 
f ‘ Edinb. Roy. Soc. Ti'ans.,’ vol. 23, pp. 160-161; ‘Natural Philosophy,’ App. D., §§ (J), (Z). 
