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MR. IVORY ON THE EQUILIBRIUM OF A MASS OF 



What has been said is well elucidated by the investigation that has been given of 

 the exact figure of equilibrium, when all the forces in action are taken into account. 

 Assuming that the problem is possible, it has been found that the supposition is veri- 

 fied, and all the conditions of equilibrium fulfilled, when that body is an oblate ellip- 

 tical spheroid, and only when it has that figure. If the body H K I, whether its 

 dimensions be finite or infinitely small, have the figure mentioned, and if the centri- 

 fugal and attractive forces be so adjusted that their resultant is, at every point, per- 

 pendicular to the surface of the spheroid, the procedure of Clairaut will generate a 

 series of figures all similar to one another, and all in equilibrium ; but, as this propo- 

 sition is exclusive, if we substitute for H K I a body of a different form, supposed 

 infinitely small, none of the successive figures will be in equilibrium, although in the 

 long run, when they have acquired finite dimensions, they will fulfill the condition of 

 the perpendicularity of the forces to the upper surface. 



The discussion in which we have been engaged is of importance, because it shows 

 the insufficiency of the methods usually employed for determining the equilibrium of 

 a homogeneous fluid consisting of attracting particles. In this problem an equilibrium 

 is not sufficiently established by making the upper surface perpendicular to the re- 

 sultant of the forces acting upon the particles contained within it, nor by proving 

 that all the narrow canals diverging from an interior particle, and terminating in 

 the surface, press with equal intensity ; nor can the problem be solved by attending 

 solely to the forces that act upon the particles individually*. 



On the Method of Investigation followed in the Paper published in the Philosophical 



Transactions for 1824. 



The equilibrium of a homogeneous planet may likewise be investigated by the 

 method employed in my first paper on this subject, published in the Philosophical 

 Transactions for 1824. As this method admits of being treated in few words, and 

 will contribute to illustrate the principles on which the solution of the problem de- 

 pends by placing them in a new light, I am induced to add a short explanation of it, 

 more especially as it will give me an opportunity of stating clearly what is really 

 liable to objection in that paper. 



* In a Memoir published in 1784, Legendre has arrived at this conclusion, that the elliptical spheroid is 

 exclusively the figure of equilibriuna of a homogeneous planet. To the mathematical processes employed by 

 that eminent geometer, no objections can be made. But, on examination, it will appear that the grounds on 

 which his investigation really rests, are these two : first, the equation of the upper surface of the fluid, which is 

 a necessary condition of equilibrium ; secondly, an expression for the radius of the spheroid assumed arbitra- 

 rily and without reference to an equilibrium. Such a procedure can never be admitted as a complete and an 

 a priori solution of the problem, unless it were first proved that every figure that can possibly fulfill the con- 

 ditions of equilibrium is necessarily included in the expression assumed for the radius of the spheroid. No 

 particular spheroid can be deduced from the equation of the upper surface alone, without first making a sup- 

 position respecting the expression of the radius : and this is an evident proof, that the equation is indetermi- 

 nate and comprehends many different figures. 



