242 Mr, Ivory on the Figjirc of the Planets, 



the level surface*. Again, because the expression of A.V is 

 independent of the coordinates of all the points in, or within, 

 the level surface, it follows, from the nature of that function, 

 that the attraction of the stratum upon all the interior particles, 

 will be equal to zero, or will be equal in all opposite directions. 

 If a stratum laid upon an}' level surface possess either of the 

 two properties; namely, equality of pressure or nullity of at- 

 traction upon all the interior particles, it will necessarily be 

 possessed of the other. 



We are nov.' prepared to estimate the force of M. Poisson's 

 objections. In the article written by him, and inserted in the 

 Atmales de Chimie et de Physiqtic, tom. xxvii. p. 225, he sets 

 out with giving a succinct and clear exposition of the princi- 

 ples of the usual theory. In the case of a homogeneous fluid 

 these principles are only two^ which relate, the one to the al- 

 gebraic expressions of the accelerating forces, and the other 

 to the perpendicularity of gravity to the outer surface. The 

 necessity and the sufficiency of these two conditions in all cases 

 whatever in which a homogeneous fluid is considered, are 

 conveyed in these imperative words, Ilfaut et il siijfit. There 

 is no objection to this theory when the accelerating forces are 

 explicitly given. But when the particles of the fluid attract 

 one another, the accelerating forces at any point in the in- 

 terior vary according to the situation of the point with regard 

 to the attracting matter; on this account the equation of the 

 level surfaces is extremely complicated ; and it by no means 

 appears that, in this case as in the more simple one when the 

 accelerating forces are explicitly given, we can infer the ex- 

 istence of the interior level surfaces, which is necessary to the 

 equilibrium, from the equation of the outer surface alone. I 

 have therefore departed from the usual theory, and have 

 sought to deduce the conditions of the equilibrium of a homo- 

 geneous planet entirely fluid, by a difft^rent procedure. The 

 conclusions I have obtained rest on their own evidence. All 

 attempts to solve completely the equation of the outer sur- 

 face of a homogeneous mass of fluid in equilibria, have hitherto 

 failed, and it is not known what figures come under it. If 

 it comprehend ellipsoids exclusively of all other figures, the 

 usual theory will agree with my investigation ; if it extends to 

 other figures, it is certain, from what I have proved, that these 

 are not figures of equilibrium. 



* In a short paragraph, p. 16/ of this Journal, for last month, V(r)— V'(r), 

 or A.V, is said to be the pressure in the interior of the fluid. It is not, 

 however, the whole pressure, but only that part of it produced by the at- 

 traction of the exterior stratum. The whole paragraph is not much to the 

 purpose, and had better have been left out. 



M. Poisson's 



