12 GwVTHER, Conditions of Stresses in a Heavy Body. 

 where 



and 



cii 11 I bv 

 ^ +- + _ _, 



cr r r cd 



02/ Z^ I Cli 



2W = ^r- + - ~ ^^ 



cr r r cB 



and if we consider separately the terms 



«„y sin 6 and vjt) cos 6 in // and v respectively, 



then 



r-r-^ + n„-Vo = (i - 2(r)(Ar- + B) , 

 dr 



r^ + v„-u„= - 2(i - (T){Ar' - B) , 

 cr 



from the displacement equations ; and the conditions are 

 that 



?( ^;~ -"V,,-^ u„\= —2(1 + <T)rp , 



q\r^ + '^(«o - ^0) j = ( I - ^""yp , 



^|,.|^ +(.„-,,)} = (I -0|,(/^-% • . (8) 



The different conditions implied by these equations 

 cannot be satisfied, it being remembered also that p 

 vanishes both when r = a and r = d. 



The general conclusion at which I arrive is that the 

 stresses in a heavy body cannot reasonably be assumed to 

 be elastic stresses, that the conditions required in order 

 that the stresses should be elastic are very stringent, and 

 in the cases I have examined the stresses do not satisfy 

 these conditions. 



In the case of the Earth, which offers great interest, 

 the magnitudes of the stresses make it especially urgent 

 that the assumption that the stresses are elastic should 

 not be made without a very thorough investigation. 



