732 JOSEPH BARRELL 



The results of this part suggest that in future investigations by 

 mathematicians upon the elastic competence of the earth, a prob- 

 able case would be to consider the isostatic compensation as uni- 

 formly tapering out through a depth twice as great as the depth 

 given for uniformly distributed compensation, that is, tapering out 

 through about 244-254 km. Further, it is not in accordance with 

 nature to assume that at this depth all shearing stresses disappear. 

 Such an assumption brings in artificialities nearly as great as those 

 involved in Darwin's assumption of no isostatic compensation. 

 Rather should the stress relations be solved as limited by some such 

 curve as is here shown, and determined by the nature of the depar- 

 tures from isostasy. It is possible that this may add still further 

 difhculties to the mathematical treatment of the subject, yet only 

 by closer recognition of the realities of nature can mathematical 

 analysis become of increasing value. 



SECTION A, PRESENTATION OF THEORY 

 RELATIONS OF LOADS AND STRESSES 



Stresses imposed by harmonic surface loads. — A harmonic series 

 gives a succession of sweeping curves such as are shown in Fig. 15. 

 The vertical scale may be made of any size and the curves may 

 be regarded as sections across a series of hills and valleys, or, on 

 progressively larger scales, anticlinoria and synclinoria, geanticlines 

 and geosynclines, continents and ocean basins. The parts of the 

 curves convex upward will then represent loads above the mean 

 surface and give rise to stresses acting downward. The broad 

 hollows give negative loads and the surface beneath is strained 

 upward by the pressures from surrounding regions. The inequali- 

 ties of the earth's surface may be taken as approximating to har- 

 monic undulations of simple or complex nature. By so taking them, 

 the stresses which they produce on the earth's interior may be 

 evaluated. 



G. H. Darwin treated this problem in his paper ''On the Stresses 

 Caused in the Interior of the Earth by the Weight of Continents 

 and Mountains."^ In this are investigated the stresses given 

 by positive and negative loads whose distribution follows a law 



^Phil. Trans. Royal Soc, CLXXIII (1882), 187-230. 



