330 NEWTON S PRINCIPIA. 



pendent of the nature of the internal structure. There is 

 a certain general connection between the form of the 

 surface and the variation of gravity which he establishes 

 on the above suppositions. This general connection has 

 been lately demonstrated without making any hypothesis 

 respecting the distribution of matter in the interior of the 

 earth, but merely assuming the law of universal gravita 

 tion. 



The investigations of Laplace lead to the same result 

 as those of Clairaut, but there is this difference between 

 their modes of reasoning. Clairaut assumes that the 

 forms of the strata are spheroidal, and then shows that 

 the whole will be in equilibrium. Laplace merely assumes 

 that they differ but slightly from spheres, and then de 

 duces from the condition of equilibrium that their forms 

 are spheroidal. The solution of the question would in 

 fact lead to a functional equation : we cannot write down 

 the condition of equilibrium without knowing the attrac 

 tion, and we cannot find the attraction without knowing 

 the form of equilibrium. The analysis by which Laplace 

 was enabled to prove the strata spheroidal is entirely his 

 own invention. Its power is very great, and in many 

 other investigations it has proved a useful engine of dis 

 covery. 



6. We have seen that the earth cannot be homogeneous; 

 we have also learnt from the investigations of Clairaut 

 and Laplace that it consists of strata of different densities 

 increasing from the circumference to the centre : it becomes 

 an interesting question to determine this law of density. 

 Legendre was the first who ascertained what is very 

 probably the true law. But it is the more finished results 

 of Laplace that we shall now consider. They are entirely 

 built upon one assumption. We know that in gases and 

 fluids the ratio of the change of pressure to the change in 

 density is constant. But in solids and semi-fluid bodies 

 it is more natural to suppose that this ratio increases with 



