Analysis of the Mecan'iqile Ccleite b/M. La Place. 209 



t"oid above tbat of the raflius of the sphere, from which it 

 differs verv htlle ; and as the equation of equilil)riuii) is 

 hncar with respect to the radius of the spheroid, it will 

 still bo suisfied if we add any nun)ber of these excesses to 

 the constant part which enters into the expression of the 

 radius vector. The spheroid to which this radius belongs _ 

 is no longer of revolution ; it is forn)ed by the sphere from 

 which the spheroid differs but liiile, augmented bv any 

 number of layers similar to the excess of the primitive sphe- 

 roid of revolution above this sphere; these layers besides 

 being laid arbitrarily above each other. The author shows 

 that these resQlts may also be deduced from the reduction 

 into a series of the attractions of the spheroids ; which 

 proves that the results obtained by this method have all 

 possible ocnerallty, and that there is no fear of any figure 

 of equilibrium escaping them. This result confirms what 

 has been previously s.en, that the form given to the radius 

 of the spheroids is not arbitrary, and flows from the very 

 nature of their attractions. ■ 



The author afterwards takes up the general equation of 

 the equilibrium of spheroids differing little from spheres, 

 and covered with fluid layers of variable densities. He 

 deduces from it the equation for the figure of these layers. 

 Examining particularly the case in which the spheroid sup- 

 posed to be entirely fluid is not solicited by any foreign ac- 

 tion, he shows that it can then only be an ellipsoid of re- 

 volution whose ellipticities increase and densities diuunish 

 from the centre to the surface ; he obtains the ec^uation 

 which determines thj ratio of these quantities to each other, 

 and he deduces the limits of the oblatenessof the spheroid; 

 the former answering to the case of homoocneity, the other 

 to that in which gravity would be directed towards a single 

 point. Such must liave been the (igure of the earth, if sup- 

 posed to have been primitively fluid. In the case now in 

 question, the directions of gravity from the surface to the 

 centre no lonjrer form a right line, but a curve, the equa- 

 tion of which is determined by the author, and which is 

 the trajectory at right angles of all the ellipses, which 

 by their revolution form the layers of the level of the sphe- 

 roid. 



[To be continued.] 



Vol. 34. No. 137. Sept. 1809. O XXX. On 



