ON THE PHYSICAL STRUCTURE OF THE EARTH. 207 



wheu ia Rotation," p. 57,* I referred to a proof obtained by me of the 

 result alluded to, and I now may be allowed to submit this proof to 

 those interested in the question. 



(2) Let us suppose the earUi to consist of a solid spheroidal shell 

 composed of nearly similar spheroidal strata of equal density, and 

 having the ellipticities of the inner and outer surfaces small and 

 nearly equal. The shell is supposed to be full of liquid and to rotate 

 around its polar axis. Under these conditions the attraction of an 

 exterior body would tend to produce pressure between the fluid 

 nucleus and the inner surface of the shell. Whatever may be the 

 direction of this pressure, it can be resolved into a force normal to the 

 shell's surface and into forces iu its tangent plane. The normal force 

 might be efl'ective in causing a deformation of the shell, or, if the latter 

 were rigid, it would be destroyed by the shell's resistance. 



If friction existed between the materials of the shell and the fluid of 

 the nucleus, the resolved forces in the tangent ijlane would tend to 

 change the motion of the shell from the motion it would have if empty. 

 But if no friction and no adhesion existed between the particles of the 

 liquid and the shell's nearly spherical surface, and if the particles of the 

 liquid are free from viscidity and internal friction among themselves, 

 this purely tangential component could exercise no influence on the 

 motions of the shell. If the solid envelope containing fluid was bounded 

 by planes such as a prismatic vessel or box, it is manifest that unequal 

 normal pressures on the faces of such prism would tend to produce 

 couples, and thus possible rotations. Such a case has been considered 

 by Professor Stokes, and he has shown that a rectangular prism filled 

 with fluid will have the same motion as if the fluid was replaced by a 

 solid having the same mass, center of gravity, and principal axis, but 

 with much smaller moments of inertia corresponding to these axes. 

 But in a continuously curved and nearly spherical vessel the normal 

 pressure arising from the disturbance of the liquid could not produce 

 the same results. The tangential components of the forces acting at 

 the surface of the liquid could, in this case, be alone effective, and if no 

 friction or viscidity existed at this surface such tangential action would 

 totally disappear. The conclusion of Mr. Hopkins's first memoir is, 

 that if the ellipticity of the inner and outer surfaces of the solid shell 

 were the same, i^recessiou would be unaffected by the fluid, and any 

 small inequality of nutation would be totally inappreciable to observa- 

 tion (p. 423, Phil. Trans., 1839). This may be rendered more manifest 

 by recalling the general equations for the surface of a fluid obtained 

 by Poisson, Navier, Meyer, and other mathematicians when the internal 

 friction of the fluid is taken into account. It a, /i, y, be the angles 

 made by the normal to the curved surface of the fluid. A', Y, Z the com- 

 ponents parallel to the rectangular axes of a?, y., and z, it appears that 

 we shall have at the fluid surface, when nearly spherical, 



* proceedings of R. I, 4,, 2d series, vol. iii, Science. 



