Rigidity of the Earth. 485 



The above is most easily seen by considering that rj is the 

 rotation of the radius vector r, and 



i{ 



T 'dr r'de) 



is the rotation relative to the radius vector. 



The two rotations in planes perpendicular to the plane 

 of o) are clearly zero. 



All the conditions of equilibrium are satisfied by assuming 

 that a shell of uniform density which is spherical in the 

 unstrained state will be strained into a spheroidal shell. 

 This will be evident in the course of the subsequent work. 



On account of the difficulties of the analysis I am 

 assuming that the sphere is incompressible. 



Since r denotes the unstrained distance of an element 

 from the centre of the sphere both the preceding 

 assumptions are involved in the equation 



e = re(cos 2 — jj) (6) 



Both the ellipticity e and the density p must be regarded 

 as functions of r. We may regard r itself as the parameter 

 of a shell of uniform density. 



The assumption of incompressibility gives also the 

 equation 



= S=^|-(f 2 e)-f-4^| 5 (i ?S in^). ... (7) 



v z $r K y sm 6 0U K J v ' 



Equations (6) and (7) enable us to express rj in terms of e. 

 Thus, using equation (6"), 



hhw-i 00 "- 1 *)?*™ (8) 



Equations (7) and (8) now give 



71 sin e= '?i ^*) ( ( cos2<? - J ) sin e M +/w 



l 4 (A) {i cos ^~i cosfl } +/w 



7 



2 cos sin 2 0-(r 3 *) +/M, . (9) 



where /(r) is an arbitrary function of r which arises from 

 integrating with respect to 0. But since f] is zero when 6 is 

 zero it follows that f(f) is also zero. 



Phil. Mag. S. 6. Vol. 22. No. 130. Oct. 1911. 2 K 



