Original and Actual Fluidity of the Earth and Planets. 265 



This equation is identical with that derived from the supposition that the 

 Earth is completely fluid, and is therefore independent of the law of density and 

 ellipticity of the solid parts of the Earth ; it determines the relation which neces- 

 sarily exists between the law of density and ellipticity of the fluid portions of the 

 Earth. If the law of density of the fluid parts be given, the integral of this dif- 

 ferential equation -w^^ give the law of ellipticity, involving two constants, one 

 of which is determined by the condition that the density does not become infi- 

 nite at the centre, and the other constant may be expressed in terms of the 

 ellipticity of the surface which bounds the fluid. If we suppose that there is 

 a fluid nucleus inside the Earth, whose radius is ai, and ellipticity e,, equation 

 (12) will give for the bounding surface of the nucleus the following, 



Elf*' o 1 (^^ d.a^e a^r* de wiajr* , 



^]r'--^^}f -d^--5]iTa=^Ar' ^ ^ 



If, also, we assume, as we may in the case of the Earth, that the external 

 surface is perpendicular to gravity, equation (12) may be applied to this sur- 

 face, although not fluid. Hence we obtain, 



Equations (14) and (15) assert, respectively, that the inner and outer sur- 

 faces of the solid shell are perpendicular to gravity. 



In the case of the Earth, the integral at the right-hand side of these equa- 

 tions is known, because the mean density of the Earth is known. The integral 

 at the left-hand side of equation ( 15) is also known ; since it may be expressed in 

 terms of the difierence of the moments of inertia with respect to the polar and 

 equatorial axes, which is given by the inequalities of the Moon's motion pro- 

 duced by the structure of the Earth, or by the phenomena of precession and 

 nutation, which are produced by the same cause. In fact, if C, A denote the 

 moments of inertia with respect to the polar and equatorial axis respectively, 



C-A 



S^rf* d.a'e ,, ,, 



Also the first and second integrals, on the left-hand side of equation (14) are 

 known from the differential equation (13), if we assume the law of density 

 of the fluid parts to be known. 



