336 NEWTON S PRINCIPIA. 



the shell and the contained fluid is that the general effect is 

 the same as if the whole were solid. This method there 

 fore fails to tell us anything of the thickness of the crust. 

 But when we proceed to consider the case in which both 

 the solid shell and the inclosed fluid are of variable density, 

 we arrive at a different result. The disturbing forces will 

 of course depend on the law of density : taking the law 

 which we have already investigated, we can calculate these 

 forces and compare them with those obtained in the case 

 of a homogeneous shell. The required alterations can then 

 be made in differential equations of the motion of the pole. 

 We can thus find the precession. In order that the result 

 thus found may agree with that found by observation, there 

 must be a certain relation between the ellipticity of the 

 internal surface of the solid part and the mean thickness of 

 the crust. This internal surface is a surface of equal soli 

 dity. If we knew, then, what function the solidity of a 

 body is of the temperature and pressure, we should be able 

 to express the ellipticity of a surface of equal solidity as 

 a function of the depth. By equating the two values thus 

 found, we should have an equation to determine the thick 

 ness of the crust. But we do not know the law of the 

 ellipticities of the surfaces of equal solidity. If heat did 

 not affect solidification, they would be the same as 

 the surfaces of equal density. If density did not affect 

 solidification, they would be the same as the surfaces 

 of equal temperature. But both are acting causes. 

 Hence the ellipticity of a surface of equal solidity passing 

 through any point must lie between those of the surfaces 

 of equal density and equal temperature passing through the 

 same point. The ellipticity of the former decrease, those 

 of the latter increase, from the surface to the centre. 

 Hence, in order that the numerical value of precession may 

 be accounted for, it is sufficient that the thickness of the 

 crust shall not be less than a certain value, found to be 

 about one-fourth to one-fifth the radius. 



This reasoning fails therefore to give more than an 



