Original and Actual Fluidity of the Earth and Planets. 263 



in assuming the original fluidity of all the planets, when there exists so re- 

 markable an exception in the case of the planet Mars.* 



III. — On the Structure of the Earthy supposed partly Fluid and partly Solid. 



In the following investigation I shall suppose the Earth composed of ellip- 

 tical couches of small ellipticit}', the density of each couche being constant and 

 a function of its distance from the centre. The surfaces bounding the couches 

 must be perpendicular to the resultant of the forces acting upon the particles 

 composing them, in the parts of the Earth which are supposed fluid, and also 

 at the boundary between the solid and fluid parts, since the friction of the fluid 

 would render the bounding surface perpendicular to the resultant, if not so ori- 

 ginally. The only external forces supposed to act upon the particles are the 

 centrifugal forces arising from the earth's rotation. 



The condition that any surface bounding one of the couches of equal density 

 shoidd be perpendicular to gravity is contained in the following equation : 



const = F-f A^, (7) 



in which Y is the potential of the earth, and 



A^ = ^«,V - ^»V^ ; (8) 



r denoting tlie radius of the surface, w the angular velocity, and » = cos-e — I, 

 6 being the angle contained between the radius vector and the axis of rotation.f 

 The potential contained in (7) is composed of two parts, one relating to the 

 couches inside the surface considered, and the other to the couches outside the 

 same surface. The value of the potential of a body constituted as we have 

 supposed the earth, on an external point, is, 



* It has been remarked by Laplace (Mec. Cel. Tom. ii. p. 370, and Tom. v. p. 287), that the 

 ellipticities of the principal sections of the Jloon, deduced from the moments of inertia obtained by 

 the observations of Tobias Mater and Nicollet, are nearly Tivs ^^^ TtVj > *°<^ t'^^' both these 

 ellipticities are greater than those of the figure of the Moon, if supposed fluid and homogeneous, 

 wliich would give the maximum ellipticity. We have, therefore, in the Moon a case similar to 

 Mars, viz., the actual ellipticity is greater than the major limit of the fluid hypothesis; but it is 

 easier to admit that gravity is not perpendicular to the surface in the case of the Moon. 



f Mec. Celeste, Tom. ii. p. 66. 



