312 Mr. Ivory's Remarks on the Theory 



he proves that a homogeneous body of fluid revolving upon 

 an axis will be in equilibria when it has the figure of an oblate 

 spheroid very little different from a sphere, the ellipticity be- 

 ing \ of the proportion of the centrifugal force to the gravity 

 at the equator. In the hypothesis of Huvghens the ellipticity 

 of the earth would therefore be \ x 7 £g, or T j F , instead of 

 ff ig which it is in the theory of Newton. 



The centrifugal force remaining very small in proportion 

 to gravity, if we suppose that the attractive force placed in 

 the centre varies as some power, or even as some function, of 

 the distance, we shall still find the same ellipticity as when 

 the central attraction acts with the same intensity at all di- 

 stances. For, on account of the near approach of the figure 

 of equilibrium to a sphere, the variations of the central force 

 at the surface will introduce into the equation of the spheroid 

 no quantities but such as are of the second order, which are 

 to be neglected. 



In the Newtonian law of attraction, if we suppose a re- 

 volving fluid mass, which increases in density towards the 

 centre, to be in equilibrio, it is proved that the ellipticity will 

 be less than in the case of a homogeneous fluid ; and the 

 dense we suppose the matter near the centre, the more will 

 the ellipticity decrease. If the matter at the centre be infi- 

 nitely dense, we fall upon the hypothesis of Huvghens; 

 which is therefore one extreme case, the other extreme being 

 the supposition of a homogeneous fluid. The ellipticity of 

 Huyghens, or \ the proportion the centrifugal force to the 

 gravity at the equator, is therefore the least possible; and 

 that of Newton, or ^ of the same proportion, is the greatest. 

 It is extremely improbable that Nature will coincide with 

 either of the extreme cases ; and accordingly all the observa- 

 tions agree in giving the earth a mean figure between the two 

 limits. 



3. About iS years after the publication of the Principia, 

 Mr. Stirling communicated to the Royal Society of London 

 two elegant propositions, in which he proved the legitimacy 

 of Newton's investigation of the equilibrium of a homogeneous 

 fluid revolving about an axis. Two years afterwards Clairaut, 

 in two papers sent to the same learned body, treated the same 

 subject more fully, demonstrating the accuracy of the conclu- 

 sions obtained in the Principia, and extending his researches 

 to spheroids composed of strata of different densities. In 1 740, 

 or 53 years after the publication of Newton's work, Mac- 

 laurin's Dissertation on the Tides appeared, forming a re- 

 markable epoch in the history of this department of science. 

 He proved, by the most elegant and accurate geometry, that 



a homo- 



