NEWTON S PRINCIPIA. 329 



spheroids, having their minor axes in the same direction, 

 but not necessarily of the same ellipticity. He then shows 

 that when there is a certain relation between the density of 

 any stratum and its ellipticity the fluid will be in equili 

 brium. Assuming that the density increases with the depth, 

 it follows that the ellipticities must decrease from the surface 

 to the centre, so that the strata are more and more nearly 

 spherical the nearer they are to the centre of the earth. 

 Suppose the earth to consist of a spheroidal nucleus 

 formed of spheroidal strata of different densities, sur 

 rounded by a very thin layer of fluid (the sea), and sup 

 pose the laws of the density and the ellipticity of the 

 strata to be any whatever, except that the ellipticity of 

 the outermost stratum is the same as that of the thin 

 layer of fluid upon it, then if G be the equatorial gravity, 

 g the gravity at latitude A, s the ellipticity of the outer 

 most stratum, m the ratio of the centrifugal force at the 

 equator to the equatorial gravity, 



g = G (1 + n sin. 2 A) 



5 



n = 2 m ~~ * 



This is a very remarkable proposition : the law of gravity 

 along the surface of the earth is then quite independent 

 of the law of density in the interior. It also furnishes us 

 with a method of determining the ellipticity by observa 

 tions on the force of gravity in different parts of the earth. 

 It is usually called &quot; Clairaut s Theorem.&quot; 



If the earth was not originally fluid the strata of equal 

 density may not have been spheroidal. But merely 

 assuming that they differ but little from spheres, and 

 that the surface is covered by a fluid in equilibrium, 

 Laplace has shown that the changes in the force of gravity 

 at the surface, and for all external points, is quite inde- 



