as deduced from Experiments made with the Pendulum. 5 



actually take place in a homogeneous fluid in equilibrio, and 

 it is sufficient to determine the figure which it will assume. 



In the particular case of a homogeneous fluid in equilibrio 

 by a centrifugal force, and the attraction of its particles in the 

 inverse proportion of the square of the distance, the same con- 

 clusion may be obtained in a different way. It is easy to 

 prove, from the nature of the forces in action, that in such a 

 body all the level surfaces are similar to one another ; and this 

 property, while it leads immediately to the law already laid 

 down, is of itself sufficient to determine the figure which is 

 necessary to the equilibrium of the fluid. 



The theory we have been explaining has been opposed, and 

 has been rejected superciliously without examination. But it 

 is founded in truth, and will ultimately be adopted. No other 

 way but by investigating the physical properties of equilibrium, 

 can be successful in simplifying a very difficult subject, and in 

 rendering it completely satisfactory. 



But the fact is, that, supposing an attraction according to 

 the law of nature and a centrifugal force, there is no other 

 figure known which is competent to the equilibrium of a ho- 

 mogeneous fluid, except the elliptical spheroid of revolution. 

 Such therefore must be the figure of the central body, or 

 nucleus, of a mass entirely fluid, but consisting of strata of 

 variable density. When the body is nearly spherical Clairaut 

 has already shown that the equilibrium requires it to be an 

 oblate spheroid ; for it is manifest the method of solution fol- 

 lowed by that geometer coincides with the view of the problem 

 taken here. Clairaut has carried his calculations no further 

 than the first power of the ellipticity ; and in pushing the ap- 

 proximation so as to include the square of the same quan- 

 tity, I have obtained the following equations, taking care to ex- 

 hibit them in as simple a form as possible. The symbols g, e, 

 denote the density, and the double of the ellipticity, both 

 being functions of a, the axis of revolution of a stratum of 

 equal density, varying from the centre to the outer surface of 

 the fluid: A is an unknown function of the same variable 

 quantity. 



= j -^ ~ ?^ ~ i " 1 { F Side] 



e_ fzd(a=e) 5f fjd.a^ J_ f z d{a s e"-) 8 



"*" 7 ' a? 21 : ' " a? ~5 ' a 1 ~ QA^ ' 



2 * o5 



