Mr. Ivory o« an Article in the Bulletin ties Sciences. 273 



ordinates, of which R" is the resultant. Now the equations 

 of Clairaut's theory comprehend only the two forces R and 

 R', and omit the third force R". For that theory computes 

 only the accelerating forces that act directly upon any particle 

 of the fluid ; whereas R" is a secondary force caused by the 

 attraction between two portions of the mass of fluid. The 

 least attention is sufficient to show that the same force cannot 

 possibly be included in any general theory applicable in all 

 cases ; because it is the effect of a particular hypothesis. 



What has just been said demonstrates the insufficiency of 

 Clairaut's theory for solving the problem of the equilibrium of 

 a planet in a fluid state. It is not a mathematical difficulty 

 that has obstructed the progress of this research since the time 

 of Newton ; it is the want of a clear conception of the physical 

 conditions of the problem, and the omitting of some of the 

 forces required for the equilibrium. Clairaut's theory, al- 

 though perfectly just when properly applied, has occasioned 

 geometers to waste their labour in attempting to solve analy- 

 tical equations which do not comprehend all that is necessary 

 to determine the question. 



It would serve no good purpose to reply to the objections 

 of the author in the Bulletin. His arguments, as well as those 

 formerly advanced by M. Poisson, are chargeable with incon- 

 sistency: for both these writers admit the attraction of C — A 

 upon the particles of A, and estimate its effects ; and although 

 that attraction is left out in Clairaut's theory, yet they uphold 

 the sufficiency of his equations for solving the problem*. 



If A be similar to C and similarly posited about the centre 

 of gravity of the planet, I have proved that A will be in equi- 

 libria separately, supposing the exterior matter is taken away 

 or annihilated. In this case, therefore, the force R will be per- 

 pendicular to the surface of A ; and as the three forces that 

 urge m must be in equilibria, it follows that the resultant of R' 

 and R" must be perpendicular to the same surface. And be- 

 cause the resultant of the two forces is perpendicular to the 

 surface of A, their joint action will produce the same inten- 

 sity of pressure upon every point of the surface, which is 

 therefore one of equable pressure. This has been proved 

 beibre, and it mostly removes the difficulty of solving the pro- 

 blem. For, these surfaces of etjuable pressure being all de- 



• In |)articular M. Poisson lias cRtimated tlic pressure upon m caused by 

 tlie attraction of (' — A upon a canal witiiin A. Now this pressure co- 

 exists witli the centrifugal force and the attraction of all the matter of the 

 planet upon tn : and, as the two latter forces alone are included in Clair- 

 aut's eijuations, it is evident that his theory is insufficient to solve the 

 problem. 



N. S. Vol. fi. No. 3i. Oc(. 1 829. 2 N rived 



