[ 491 ] 



XXIII. On the Equilibrium of a Mass of Homogeneous Fluid at liberty. By James 

 Ivory, K.H. M.A. F.R.S., Instit. Reg. So. Paris. Coi^esp., et Reg. Sc. Gottin. 

 Corresp. 



Received April 30, — Read May 29, 1834. 



Of the questions to which the publication of the Principia gave rise, none has been 

 attended with greater difficulty than that which relates to the figure of the planets. 

 In this research it is required to determine the figure of equilibrium of a mass of fluid 

 consisting of particles that mutually attract one another at the same time that they 

 are urged by a centrifugal force caused by a rotation about an axis. Geometers have 

 long ago adopted a theory of the equilibrium of fluids which is said to be perfect, and 

 to leave only mathematical difficulties to be surmounted in every problem : but it 

 must be admitted that the utility of this theory amounts to very little ; for it has 

 failed in solving the fundamental problem for determining the figure of equilibrium 

 of a homogeneous planet in a fluid state. This is the more remarkable, because Mac- 

 LAURiN, soon after the origin of such inquiries, demonstrated with accuracy and ele- 

 gance, that a planet supposed fluid would be in equilibrium if it had the figure of an 

 oblate elliptical spheroid. To every one that reflects, the question, not easily an- 

 swered, must occur. Why has it been found impossible to deduce the discovery of 

 Maclaurin from the analytical theory ? If we suppose that the theory is physically 

 correct, and that mathematical difficulties alone oppose its successful application, 

 there is great probability that these would have yielded, as in other instances, to the 

 repeated attempts of geometers. 



But if Clairaut's theory of the equilibrium of fluids be examined attentively and 

 without prejudice, other difficulties of greater moment will present themselves. In a 

 homogeneous fluid at liberty, if the forces in action be such as to make the problem 

 possible, the equilibrium, according to the theory, requires only one condition, namely, 

 that the forces urging every particle in the surface be directed perpendicularly towards 

 that surface. The solution is thus made to depend entirely upon the differential equa- 

 tion of the surface, and seems to demand that this equation be determinate, and ex- 

 plicitly given : for if the equation be indeterminate, or not explicitly given, how can it 

 be said that the problem is solved ? If the forces which urge the particles of the fluid 

 are explicit functions of the coordinates of the point on which they act, so that when 

 the values of the coordinates are assigned, the algebraic expressions are completely 

 ascertained, there is no doubt that the equation of the fluid's surface will be known, 

 and the figure of equilibrium will be determined. With respect to such problems, the 



