and Mathematics to Seismology. 177 



— i^-^JFJ^t.^*"} «> 



Here f is the distance between the element day where p is 

 applied, and the point x, y, z where u, v, w are measured. 

 The integration extends to all parts of the surface where p 

 differs from zero. 



The simplification in the formulae when n = '5, or the 

 material is incompressible, should be noticed. 



The slope — i.e., inclination to the plane of xy — introduced 

 into any horizontal plane depends only on the vertical dis- 

 placement iv. In particular, the slope of the surface depends 

 only on iv , the value of the vertical displacement when z = ; 

 and by (3) we obviously have 



w ={(l- V )/(2irn)}§(p/r)d a> . . . . (4) 



In (4) r is the distance between the element dco and the point 

 x , y on the surface to which w refers. 



Relation between Pressure and Gravitational Effects. 



§ 6. If we suppose t the thickness, p the density of the 

 material loading the surface, its gravitational forces are derived 

 from the potential 



V=y$(tp/r)d<o, (5) 



where 7 is the attraction between two unit masses at unit 

 distance. 



The pressure exerted by the load is 



where g is gravity at the surface. Here we may regard g 

 as the undisturbed value prior to the application of the load, 

 as the alteration in the vertical component is negligible for 

 our present object. Thus 



Comparing this with (4), we find for the surface value V of 

 V the simple relation 



Y = 27rnyiv /{g{l—r))} (6) 



