C 57 ] 



III. Of such Ellipsoids consisting of homogeneous matter as are capable of having the 

 resultant of the attraction of the mass upon a particle in the surface, and a centri- 

 fugal force caused by revolving about one of the axes, made perpendicular to the 

 surface. By James Ivory, K.H. M.A. F.R.S. L. 8$ E. Instit. Reg. Sc. Paris. 

 Corresp. et Reg. Sc. Gottin. Corresp. 



Received June 20th, 1837. — Read 11th December, 1837. 



1 . J.N the Conn, des Temps for 1 837 it is announced that a homogeneous ellipsoid 

 with three unequal axes, and consisting of particles that attract one another accord- 

 ing to the law of nature, may be in equilibrium when it revolves with a proper velo- 

 city about the least axis. Lagrange has considered this problem in its utmost 

 generality. The illustrious Geometer found the true equations from which the solu- 

 tion must be derived : but he inferred from them that a homogeneous planet cannot 

 be in equilibrium unless it have a figure of revolution. Nevertheless M. Jacobi has 

 proved that an equilibrium is possible in some ellipsoids of which the three axes have 

 a certain relation to one another. The same thing is demonstrated by M. Liousville 

 in 23rd cahier of the Journal de TEcole Poly technique. M. de Pontecoulant has 

 also touched on the subject*. M. Jacobi has thus detected an inadvertence into 

 which those had fallen who preceded him in this research. He has shown that the 

 equations which, according to Lagrange, are capable of solution only in figures of 

 revolution, may be solved in a certain class of ellipsoids with three unequal axes. 

 But the transcendent equations of M. Jacobi, although fit for numerical computation 

 on particular suppositions, leave unexplored the points of the problem which it is 

 most interesting to know. 



It is easy to find a property characteristical of all spheroids with which an equili- 

 brium is possible on the supposition of a centrifugal force. From any point in the 

 surface of the ellipsoid draw a perpendicular to the least axis, and likewise a line at 

 right angles to the surface : if the plane passing through these two lines contain the 

 resultant of the attractions of all the particles of the spheroid upon the point in 

 the surface, the equilibrium will be possible ; otherwise not. This will be evident, if 

 it be considered that the resultant of the centrifugal force and the attraction of the 

 mass must be a force perpendicular to the surface of the ellipsoid, which requires 

 that the directions of the three forces shall be contained in one plane. This deter- 

 mination obviously comprehends all spheroids of revolution ; but, on account of the 



* Tom. iii. Th^or. Anal. 

 MDCCCXXXVIII. I 



