248 Mr. Ivory on the Theory of the Figure of the Earth. 



reference to what is peculiar to the initial figure of the fluid. 

 The sphere at rest possesses all the properties of a homo- 

 geneous mass in equilibria ; that is, the gravitation is perpen- 

 dicular to the surface, and every particle is equally attracted 

 in opposite directions by a hollow shell comprised between 

 any two concentric surfaces. When a change is induced upon 

 the sphere by a centrifugal force, the laws of equilibrium can 

 be exactly fulfilled only when the new figure is an elliptical 

 spheroid. But when ihe derangement of the spherical surface 

 is very minute, it is obvious that both the conditions of equili- 

 brium may be considered as fulfilled, if we preserve the per- 

 pendicularity of gravity to the sinface. For the attractive pro- 

 perty of the level strata being exact at first, the small varia- 

 tion of their figure induced by the centrifugal force, can- 

 not be supposetl to effect suddenly any perceptible deviation 

 from the figure of ecjuilibrium. Legendre first proved that a 

 homogeneous fluid nearly spherical, revolving about an axis, 

 and having the gravitation perpendicular to the surface, must 

 be very nearly an elliptical spheroid. In the particular cir- 

 cumstances supposed, the true figure is thus found, Jiot by a 

 rigorous method, but approximately, although one of the con- 

 ditions of equilibrium is left out. This is indeed true appa- 

 rently ; but in reality, as we have shown, both the conditions 

 of equilibrium are vciy nearly fulfilled, and the more nearly 

 the less is the deviation from the spherical figure. We may 

 also object to the process of Legendre, that it substitutes a 

 mathematical projjerty which is true only in particular circum- 

 stances, and to a limited extent, in place of the physical prin- 

 ciples that are universally true in all cases whatever. 



The transition of the fluid sphere to an oblate figure nearly 

 an elliptical spheroid, by the effect of a small centrifugal force, 

 is not confined to the hypothesis of a mutual attraction of the 

 j)articles in the inverse proportion of the square of tlie distance. 

 It is likewise true if we suppose that every j^article is attracted 

 to the common centre by a force which is either the same at 

 all distances from the centre, or varies as any power of the 

 distance. In one particular law, namely, when the cential 

 force varies directly as the distance, the oblate figures are ex- 

 actly elliptical spheroids, as Herman first proved in his Pho- 

 ronomia. 



The discoveries of Legendre on the attraction of spheroids, 

 were greatly extended by Laplace. In the third book of the 

 Mt'canique Celeste, this subject is treated with the utmost ge- 

 nerality, and with all the resources and all the elegance which 

 the improved state of analysis is able to furnish. But it must 

 be observed that, in ap[)lying this mathematical theory to the 



figure 



