4-30 Mr. Ivoi'y on the Theory of the Figure of the Playlets 



centi'e to the surface, and likewise to the case when it consists 

 of a Solid liucleus covered with one or more fluids. 



The labours of Maclaurin and Clairaut are here classed to- 

 gether, because they are ccmfined to investigating certain pro- 

 perties of elliptical spheroids. But why is it necessary to sup- 

 pose a particular figure ? Ought not the figure of a planet, if 

 we suppose it fluid, to be deduced from the general laws of 

 equilibrium ? What is it that makes the investigation suc- 

 ceed when an elliptical spheroid is supposed, thus seeming to 

 exclude every other figure ? Is there but one figure of equi- 

 librium ? And, if this be the case, how is the demonstration 

 to be made out ? 



The second division of our knowledge respecting the figure 

 of the planets is likewise the development of the general view 

 of the problem proposed by Newton. A fluid body, whether 

 homogeneous or composed of strata of variable density, if it 

 be at rest and subjected only to the attraction of its particles, 

 cannot be in equilibrio^ unless it have the figure of a sphere. 

 If the sphere begin to revolve about an axis, the fluid will 

 subside at the poles, and become protuberant at the equator ; 

 and the question is to investigate the nature of the new figure 

 according to the laws of equilibrium. There are two points 

 that must be previously discussed, in order to ensure success 

 in this research. We must be able to estimate the variation 

 of the attractive force produced by the change of figure; or, 

 which is the same thing, a method is required for computing 

 the attraction of a body that is little different in its figure from 

 a sphere. We must likewise know the conditions necessary 

 to the equilibrium of a fluid body, the particles of which are 

 acted upon by any forces. In the Memoirs of the Academy of 

 Sciences for IVSi, Legendre first gave a method for the attrac- 

 tion of spheroids little different from spheres in a form fit to 

 be employed in the investigation of their figures of equilibrium. 

 He deduced from his analysis that the earth, supposing it ho- 

 mogeneous and to be a figure of revolution, must have its me- 

 ridians elliptical in order to fulfill the conditions of equili- 

 brium ; which is in fact to solve the problem as conceived by 

 Newton without making any gratuitous assumption. Laplace 

 generalized and improved the method of Legendre, adapting 

 it to the supposition of a variable density, and to the case of a 

 fluid covering a solid nucleus. In his hands it has become 

 an extensive branch of analysis, and is made the foundation of 

 the theory of the figui'e of the planets contained in the third 

 book of the Mecanique Celeste. 



With regard to the equilibrium of a fluid body, Newton 

 employed the principle of the balancing of the columns reach- 

 ing 



