24"6 JNIr. I\ ory on the Theory 0/ the Figure of the Earth. 



O P Q must therefore be possessed of such a figure as to at- 

 tract all particles ia the inside with equal force in opposite 

 directions, because otherwise the fluid contained within it 

 could not be in cquilibrio with respect to the attraction of the 

 stratum. 



Since the taking away of the uppermost stratum will leave 

 the remaining fluid bounded by the surface O P Q in cquili- 

 brio, it is evident that we may apply to the stratum imme- 

 diately below that surface the same reasoning that has been 

 applied to the stratum below the surface R S T. Wherefore 

 the surface N L M will be a level surface, and the stratum 

 between the surfaces O P Q and N L M will attract all parti- 

 cles in the inside with equal force in opposite directions. The 

 same conclusion is equally true of all the successive strata. 

 And because what has been proved of the several strata, taken 

 separately, is manifestly true of the aggregate of any number 

 of them, we are to conclude, that all the level surfaces of a 

 homogeneous fluid in equilibria by the supposed forces are 

 similar to the outer surface, and that any stratum contained 

 between two level surfaces attracts particles in the inside with 

 equal force in opposite directions. 



These two propositions determine all the conditions neces- 

 sary to the equilibrium of a homogeneous fluid, the particles 

 of which are subjected to a centi'ifugal force, and to a mutual 

 attraction following the law observed in nature. They do not 

 indeed apply immediately to a body composed of fluids vary- 

 ing in their densities; but, as the defect of the received theory 

 of fluids is clearly pointed out, it becomes easy to deduce from 

 the same principles a satisfactory solution of every case. 



A problem in Natural Philosophy is not brought within the 

 domain of the Mathematics, until all the physical conditions 

 be previously investigated. It then becomes the province of 

 the geometer to tiace the necessary consequences of these 

 conditions, and to deduce from them the required solution. 

 When this natural order of research is not followed, the re- 

 sult is always attended witli something imperfect, something 

 not entirely satisfactory to the mind. The conditions of equi- 

 librium being now fully known, we are able to discover why 

 Maclaurin succeeded in proving that a homogeneous elliptical 

 spheroid is in equilihrio by the attraction of its particles and 

 a centrifugal force. The figure supposed, contains one of the 

 conditions of equilibrium, and is the only figure compatible 

 •with that condition. Newton proved that a shell of matter, 

 comprised between two similar, and similarly situated elliptical 

 surfaces, or between two concentric spherical surfaces, will at- 

 tract all particles in the inside with equal force in opposite di- 

 rections : 



