LIMITING STRENGTH OF ROCKS UNDER STRESS 135 
numerous solutions. One solution of the problem was obtained 
by Sir G. Darwin' by assuming that the stress is connected with 
a displacement by the same equations as hold in the ordinary theory 
of an elastic zmcompressible solid, ....” 
The theoretical developments turn on the calculation of stress- 
differences corresponding to surface inequalities representing con- 
tinental blocks and mountains. Expressions for the hydrostatic 
pressure at any depth are not explicitly worked out although it 
would be possible to calculate from the analysis given the pressure 
at any depth due to those harmonic inequalities which Love has 
shown can be made to represent the existing distribution of land 
and water and the .general form of continental blocks. For 
most purposes it is sufficient to calculate the pressure at any depth 
as that due to the weight of the corresponding column of rock: 
this formula gives a value of greater relative accuracy the greater 
the depth at which the pressure is calculated. 
§9. DEPTH AT WHICH CAVITIES CAN EXIST 
(t) Collapse Due to Stress-Differences in the Earth's Crust 
Love? shows that on the isostatic theory the maximum stress- 
difference due to a harmonic inequality of the third order repre- 
senting a continental block of maximum elevation 2 kms. is approxi- 
mately the weight of a column of rock equal to 0.0208 of this 
greatest height, i.e., about o.0114 of a metric tonne per sq. cm. 
(162 pounds per sq. in.). This value is greater than that brought 
about by inequalities of the first and second order. The stress- 
difference is also worked out for a parallel series of mountain ranges 
400 kilometers apart and having a height of 4 kms. from crest to 
valley-bottom; it is shown that the greatest value of the stress- 
difference is about .26 metric tonnes per sq. cm. (3,700 pounds per 
sq. in.). In both cases the stress-difference is very much less than 
2 § so that no state of stress-difference due to weights of continents 
or mountains is intense enough to cause the material of the 
earth’s crust to rupture in the neighborhood of a small cavity. 
Darwin, ‘“‘On the Stresses Caused in the Interior of the Earth by the Weight of 
Continents and Mountains,” Phil. Trans. Roy. Soc., CLX XIII (1882). 
2'Love, op. cit., 36; also 47. 
