736 JOSEPH BARRELL 



Darwin obtained his results by making the initial assumption 

 that the earth substance was incompressible but possessed elasticity 

 of form. The introduction of the factor of compressibility Darwin 

 showed to affect the result largely in the case of the second harmonic, 

 but for harmonics of infinitely high orders the resulting stresses are 

 independent of the modulus of compression. Consequently he 

 states, "it may be concluded that except for the lower harmonic 

 inequalities compressibility introduces but little change in our 

 results."^ 



Modifications imposed by long and large wave-lengths. — In Fig. i6, 

 curve A shows that for a wave-length of 400 km. the depth of maxi- 

 mum stress is 64 km. and at 122 km. the stress is 75 per cent of the 

 maximum. If this should be regarded as the beginning of the 

 asthenosphere it would mean that a large part of the stress would 

 be thrown on to the zone incapable of sustaining large stress- 

 differences. If the wave-length became 2,000 km. the lithosphere 

 would be subjected to but small stress-differences and the maxi- 

 mum strain would occur at a depth of 320 km., the middle of the 

 sphere of weakness. In order that Darwin's solution should hold 

 for these cases the height of the arches would have to be so small 

 that the resulting stress-differences would not exceed the elastic 

 limit of the asthenosphere. For greater loads disposed on the 

 surface in these large wave-lengths the stress relations would 

 approach those of a rigid crust overlying a fluid substratum. 

 Of this problem Darwin states, "The evaluation of .stresses in a 

 crust, with fluid beneath, would be tedious, but not more difficult 

 than the present investigation" (on the stresses caused by the 

 weight of continents and mountains).^ It is a different problem 

 from that solved by Love; the latter considering the stresses in 

 such a crust caused by a condition of isostasy, not by a lack of 

 isostasy. The limited mathematical training of the present writer 

 does not permit here the definite solution of this problem, but some 

 general observations can be made. 



For wave-lengths very large in comparison with the depth of the 

 lithosphere the stress-differences, if confined within the crust, 

 approach those existing in a continuous beam, each span being 



^ Scientific Papers, II, 500. 'Ibid., II, 502, footnote. 



