Royal Society, 67 



are necessarily the same ; so that the expression which he obtains 

 for gravity at any point on the earth's surface, besides being a func- 

 tion of the latitude of that point, and of the radii and ellipticities of 

 the shell's inner and outer surfaces, contains functions depending on 

 the constitution of the shell and nucleus. He states that this ex- 

 pression for gravity is not merely speculative, but that it will be 

 found to assist in explaining certain apparent anomalies detected by 

 observation in the variation of gravity at the earth's surface, as well 

 as in pointing out the limits assigned by observation to the thickness 

 of the crust. 



III. The Laws of Density of the Shell and Nucleus. 



According to the author's views in a subsequent section, it ap- 

 pears that the solidification of the earth could not proceed simulta- 

 neously from the centre towards the surface, and from the surface 

 towards the centre. He therefore, in determining the laws of den- 

 sity of the shell and the nucleus, restricts his investigations to the 

 latter case, in which the solidification proceeds from the surface to- 

 wards the centre. 



IV. The Forms of the Strata of the Shell. 



The author conceives a surface to exist which may be called the 

 effective surface of separation of the perfect fluid of the nucleus and 

 the imperfectly fluid portion adhering to the shell, the form of which 

 surface will depend on the pressures which the fluid exerts. As it 

 may be shown that the pressure of the perfect fluid will not be con- 

 stant, the surface of separation will tend to assume a form different 

 from that of the inner surface of the shell. If we admit that the 

 matter composing the nucleus becomes denser in assuming the solid 

 state, the author concludes that the inner surface of each stratum 

 added to the shell will be more oblate than its outer surface ; and 

 that thus the tendency will always be to render the inner surface of 

 the shell more and more oblate. He then deduces an expression 

 for the ellipticity of the fluid surface. 



V. The principal Moments of Inertia of the Earth. 



From his investigations the author concludes that, as the thick- 

 ness of the shell increases, the difference between the greatest and 

 the least moment of inertia of the earth also increases ; which con- 

 clusion is independent of any knowledge of the absolute laws of 

 density of the earth's interior. 



VI. On the existence of a Solid Nucleus within the Earth. 



The conclusion arrived at here is, that if a solid nucleus existed, 

 as the pressure on it would be continually diminishing, while its 

 temperature would remain nearly constant, this nucleus, instead of 

 increasing in magnitude, would tend to return to its original fluid 

 state. 



F2 



