of the Figure of the Earth. S45 



brium is not possible but when the fluid has the figure of an 

 elliptical spheroid. Laplace generalized and perfected the 

 analysis of Legendre, which is founded on the properties of a 



f (articular kind of functions. The same illustrious geometer 

 ikewise discovered an equation in partial fluxions relating to 

 the attractions of spheroids little different from spheres, which 

 takes place at their surfaces. Availing himself of all these 

 resources, Laplace was enabled to give a complete theory of 

 the figure of the planets, and of the variation of gravity at 

 their surfaces, which the reader will find explained at length 

 in the third book of the Mecanique Celeste. 



5. On reviewing all the researches relative to the figure of 

 the earth, it is remarkable that the discoveries of Maclaurin 

 stand apart by themselves, without much connection with the 

 rest. His method applies only to homogeneous spheroids ; 

 but of this case it furnishes an accurate and a general solu- 

 tion. All the other attempts to solve the problem are merely 

 approximations founded on the supposition that the spheroids 

 are not much different from spheres. As was observed by 

 Mr. Stirling, they do not accurately determine the figures of 

 equilibrium ; they only show that these figures will coincide 

 with elliptical spheroids when we neglect the squares and 

 higher powers of the ellipticities. 



If the conditions of equilibrium assigned by the hydro-' 

 statical theory were accurate and sufficient, we should expect 

 that the discoveries of Maclaurin would be deducible from 

 them. Yet this has been accomplished by no geometer. 

 Nay, when we push the approximation to the figure of equi- 

 librium beyond the quantities of the first order, the ellip- 

 tical spheroid seems to be excluded. It also appears unac- 

 countable that the solution of Legendre, supposing that it is 

 deduced from a sufficient theory, should bring out only by 

 approximation a figure which we know will accurately fulfill 

 all the conditions. 



These reflections, and others which it is not important to 

 mention, induced me to examine very narrowly the hydro- 

 statical theory of equilibrium. I ' distinguished two separate 

 cases ; one, when there is no attraction between particle and 

 particle ; and the other, when the particles are endowed with 

 mutual attractive powers. 



As an example of the first case, we may take Huyghens's 

 hypothesis respecting the figure of the earth ; in which every 

 particle of the fluid is acted upon by a centrifugal force, and 

 a constant attraction directed to the centre. The equilibrium 

 of the revolving mass requires that the resultant of the two 



Vol. 63. No. 313. May 1824. X x forces 



