/« reply to M. Poisson. 323 



ferent from a sphere. If the centrifugal force be only a very 

 small part of the attraction, which is true with respect to all 

 planets, the problem is simplified extremely, as we are per- 

 mitted to reject the square and other powers of the fraction 

 expressmg the proportion of the two forces. By takino- ad- 

 vantage ot this simplification Legendre proved, in the Mim. 

 deVAcad. des Sciences 1784, that the elliptical spheroid is the 

 only figure deducible from the equation of the outer surface 

 ol a planet entirely fluid. The process of Legendre has been 

 practically executed only as far as the first power of the cen- 

 trihigal force : but it is shown that the calculation may be con- 

 tmued to any required degree of exactness ; that there is only 

 one series expressing the radius of the spheroid; and, conse- 

 quently only one figure which will answer the conditions of 

 the problem, although the nature of that figure is not precisely 

 ascertained. M. Poisson has lately remarked that the figure 

 m question must be an elliptical spheroid, the radius bting 

 expanded in a series; because, when the centrifugal force is 

 very small it is known that the conditions of equilibrium can 

 be fulfilled by an elliptical spheroid, and by one such figure 

 only, which must therefore coincide with the single figure 

 found by the other process. But there is some reason to de- 

 mur with respect to M. Poisson's argument, inasmuch as the 

 equilibrium is not equally sure in the two cases. In the el- 

 iiptica spheroid, the equilibrium is fully ascertained, as all 

 will allow; but it is not so certain in the other case, since it 

 is inferred from the single condition of the perpendicularity 

 of gravity to the surface of the planet. 



In order to throw some light on Legendre's method of 

 investigation, it is requisite to examine it more particularly. 

 Ihe attractive force at the surface of a sphere, of which the 

 radius and the density are both unit, being if, we shall put 

 -3- X a for the centrifugal force at the distance I from the 

 axis of rotation, and « will be the proportion of the latter force 

 to the former. Let r be a radius of the spheroid drawn from 

 the centre of gravity to a point in the surface of the fluid; 

 e the angle vyhich r makes with the axis of rotation; and V 

 the sum of all the molecules of the s,)lieroid divided by their 

 respective distances from the same point of the surface : then, 

 the equation of the fluid's surface, which expresses the per- 

 pendicularity of gravity, will be, 



V + -J-r'sin^U = C (1) 



C being a constant quantity. If we suppose « = 0; that is, if 



