356 Royal Society, 



partial waves which pass through the plate, and with those which 

 pass beyond the diiFracting edge with undiminished velocity, modify 

 the usual system of fringes in the manner described by the author 

 in the present paper. 



" 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 Centrifugal Force caused by re- 

 volving about one of the Axes, made perpendicular to the Surface." 

 By James Ivory, K.H., M.A., F.R.S. L. and Ed., Inst. Reg. Sc, 

 Paris, Corresp. et Reg. Sc. Gotting. Corresp. 



Lagrange, who has considered the problem of the attractions of 

 homogeneous ellipsoids in all its generality, and has given the true 

 equations from wliich its solution must be derived, inferred from 

 them that a homogeneous planet cannot be in equilibrium unless it 

 has a figure of revolution. But M. Jacobi has proved that an equi- 

 librium is possible in some ellipsoids of which the three axes are un- 

 equal and have a certain relation to one another. His transcend- 

 ental equations, however, although adapted to numerical computa- 

 tion on particular suppositions, still leave the most interesting points 

 of the problem unexplored. 



The author of the present paper points out the following property 

 as being characteristic of all spheroids with which an equilibrium 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 pass- 

 ing 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 it will not. For the re- 

 sultant of the centrifugal force and the attraction of the mass must 

 be a force perpendicular to the surface of the ellipsoid, which re- 

 quires that the directions of the three forces shall be contained in 

 one plane. This determination obviously comprehends all spheroids 

 of revolution ; but, on account of the complicated nature of the at- 

 tractive force, it is difficult to deduce from it whether an equilibrium 

 be possible or not in spheroids of three unequal axes, a problem which 

 is unconnected with the physical conditions of equiUbrium, and which 

 is a purely geometrical question respecting a property of certain 

 ellipsoids. 



The author then enters into an analytical investigation, from 

 which he deduces the fundamental equation. 



«-mr« = ^-r:rv5 <'•> 



the three axes of the ellipsoid being 



and A, B, C, constants, afterwards expressed by certain definite in- 

 tegrals. He then remarks that every ellipsoid which verifies this 

 formula is capable of an equilibrium when it is made to revolve with 

 a proper angular velocity about the least axis ; for, in this case the 



