256 The Rev. Samuel Haughton on the 



or if e, e', denote the ellipticities of the principal sections, passing through the 

 greatest and least diameter, and mean and least, respectively ; since X^ = 2e, 

 \'^ — 2e', we obtain finally for the ellipticities of the principal sections 



from which it appears that the ellipticity of the section passing through the 

 greatest and least diameters is four times greater than the ellipticity of the sec- 

 tion passing through the mean and least diameters. 



If the planet be supposed to revolve on its axis with an angular rotation 

 different from that of its revolution round the central body, the equality <^ = » 

 will no longer subsist, and we should therefore use equations (1) to determine 

 the ellipticities of the principal sections. The result is 



6=^(30 + .), e' = if«; (4) 



(p and s being the quantities already defined, and depending on the central 

 body and rotation of the planet respectively. If the central body be supposed 

 ,•^0 remote as to produce no effect on the figure of the planet, then = 0, which 

 renders the ellipticities equal, and corresponds to the figure of revolution as- 

 sumed by the planet, if acted on only by its own attraction, and the centrifugal 

 force caused by its rotation.* If, therefore, we suppose the spheroid of revolution, 

 whose ellipticity is e = ^ a, described, having the axis of rotation for its least 

 diameter, the effect prodvxced by the attraction of the central body will be 

 measured by the shape and magnitude of the couche included between this 

 spheroid of rotation and the ellipsoid which forms the actual surface of the 

 planet. The friction between this couche and the interior spheroid, which 

 would constitute the surface of the planet, if the central body ceased to exist, 

 will tend to render the motions of rotation and revolution of the planet equal 

 to each other, and when the difference of these motions has fallen within the 

 narrow limits indicated by analysis, will destroy the libration produced by 

 the action of the central body in rendering those motions exactly equal. It 

 may be proved by simple geometrical considerations, that if the planet separates 

 from the central body, as a nodular or annular mass, without much friction, that 



• Vid. PoissoN, "Traite de Mecanique," Tom. ii. p. 544. 



