the Weight <>f Coiiii nmtx a ml Mountains. 257 



Professor Stokes. It may be that underneath each continent 

 there is a region of deficient density; then underneath this 

 region there would be no excess of pressure. 



For the present investigation it is to some extent a matter of 

 co as to which of these views is correct, for if it is 

 only the crust of the earth which possesses rigidity, or if Pro- 

 fessor Stokes's suggestion of the regions of deficient density be 

 correct, then the stresses in the crust or in the parts near the 

 surface must be greater than those here computed— enormously 

 greater if the crust be thin,* or if the region of deficient den- 

 sity be of no great thickness. 



With regard to the property of incompressibility, which is 

 here attributed to the elastic sphere, it appears from § 10 

 that even if we suppose the elastic solid to be very highly com- 

 pressible, yet the results with regard to the internal stresses 

 urt almost the same as though it were incompressible. I think 

 the hypothesis of great incompressibility is likely to be much 

 nearer to the truth than is that of great compressibility. I 

 shall, therefore, adhere to the supposition of infinite incom- 

 pressibility, bearing in mind that even great compressibility 

 would not much affect most of the results." 



I take then a homogeneous incompressible elastic sphere and 

 suppose it to have the power of gravitation and to be super- 

 ficially corrugated. In consequence of mathematical difficul- 

 ties the problem is here only solved for the particular class of 

 Burface inequalities called zonal harmonics, the nature of 

 which will be explained below. 



Before discussing the state of stress produced by these 

 inequalities, il will be convenient to explain the proper mode 

 of estimating the strength of an elastic solid under stress. 



At any point in the interior of a stressed elastic solid there 

 are three lines mutually at right angles, which are called the 

 principal stress-axes. Inside the solid at the point in question 

 imagine a small plane (say a square centimeter or inch), drawn 

 ess-axes; such a small plane will 

 he eaded an inter-face. f The matter on one side of the ideal 

 inter face might be removed without disturbing the equilib- 

 rium of the elastic solid, provided some proper force be 

 applied to the inter-faeo; in oiler words, the matter on one side 

 of an inter-face exerts a force on the matter on the other side. 

 Now a stress-axis has the property that this force is parallel to 

 the stress-axis to which the inter-face is perpendicular. Thus 

 along a stress axis the internal force is either purely a traction 



