NEWTON S PRINCIPIA. 331 



the density. The most simple assumption is to suppose 

 that it varies as the density. Supposing this to be the 

 truth it is not difficult to investigate the density of the 

 strata. But it is an assumption, and must stand or fall 

 according as its results agree with, or differ from, those of 

 observation. Fortunately it enables us to integrate the 

 equation connecting the density and ellipticity of any 

 stratum, and thus the ellipticity of the external stratum 

 furnishes us with a test of the truth of the law. 



Taking for granted the truth of the law, a very simple 

 calculation will give us the corresponding law of density. 

 Let us consider the earth as a perfect sphere, and let us 

 neglect the effect of the centrifugal force. The strata 

 of equal density will then all be spheres. Let p be the 

 density of that stratum whose radius is x. Let p be the 

 pressure at that stratum referred to a unit of area. 



We have first to find the attraction on a particle situated 

 in the stratum whose radius is r. The attractions of all 

 the external strata is manifestly nothing. To find the 

 attractions of the internal ones, we have merely to suppose 

 them concentrated into their common centre and attracting 

 according to the usual law. This will manifestly give 



where p is the force of attraction of a unit of mass at a 

 unit of distance. The law of fluid equilibrium will then 

 give 



-(1.) 



Also we have by our assumption 



dp = xp.dp - - - - (2.) 



whence we get 



