246 
PROFESSOR G. H. DARWIN ON THE DISTRIBUTION OF STRAIN 
When x — 0, dK/dt is negative, and hence at the surface there is contraction or 
crumpling. The crumpling continues for some depth downwards, and vanishes when 
x 
c 
Taking, with Thomson, the foot and year as units, k appears to be about 400 ; if, 
therefore, i is t million years, Kt = 4 X 10 8 r. Now, c being 21 X 10° feet, 
kt/c 2 = t/10 6 approximately, and 
x 
24 x 10 8 
21 x 10 6 
r feet, 
= 114 r feet. 
If r be 100, x = 2 miles. 
Thus, if the time since consolidation be 100 million years, the present depth of the 
stratum of no strain is 2 miles, and the depth is proportional to the time since con¬ 
solidation. With a greater value of k the depth is greater. 
With regard to the value of dK/dt at greater depths, we observe that at a few 
miles below the stratum of zero strain 6/ct/c 2 becomes negligible compared with x/c. 
Hence dK/dt is approximately proportional to xdvjdx. Now, if we take the figure 
drawn in Thomson and Tait’s ‘Natural Philosophy,’ Appendix D, and augment or 
diminish each ordinate NP' proportionately to the corresponding abscissa, it is clear that 
we shall get just such a curve as that drawn by Mr. Davison. His curve might thus 
have been drawn without any computations at all. The function xdv/dx is pro¬ 
portional to dv/dt, and thus his dotted curve is of just the same kind as the other, 
excepting close to the surface. 
Now let us return to the expression for the integral stretching, viz. :— 
We have 
K = e 
(V-v) 
6 id dv 
c 2 C dx 
dv 
dx 
y 
(ir/ct')’ 1 
( 9 ) 
Hence, if x be small and t large (both of which conditions apply to the present 
time near the Earth’s surface), 
dv _ Y 
dx (7T/cty 
Hence 
K= e 
V -v- ^y 
C7T 5 
( 10 ) 
Now, as we have seen above, with such values as those with which we have to deal, 
Gid/c 2 is a small fraction, notwithstanding that it increases with the time. 
