VOL. XLII.] ) PHILOSOPHICAL TKANSACTIONS. 05] 



facility by the same method; which is also applied for determining the property 

 of the solid of least resistance, and serves for resolving the problem, when 

 limitations are added concerning the capacity of the solid, or the surface that 

 bounds it. 



The last chapter of the first book treats chiefly of gravitation towards 

 spheroids, of the figure of the planets, and of the tides. The author, 

 having occasion in those inquiries for several new properties of the ellipse, be- 

 gins this chapter by deriving its properties from those of the circle, by consider- 

 ing it as the oblique section of a cylinder, or as the projection of the circle by 

 parallel rays on a plane oblique to the circle. In this manner the properties are 

 briefly transferred from the one to the other, because by this projection the 

 centre of the circle gives the centre of the ellipse ; diameters perpendicular to 

 each other in the circle with their ordinates, and the circumscribed square, give 

 conjugate diameters of the ellipse with their ordinates, and the circumscribed 

 parallelogram ; parallel lines in the plane of this circle are projected by parallels 

 in the plane of the ellipse that are in the same ratio ; any area in the former is 

 projected by an area in the latter, which is in an invariable ratio to it; and con- 

 centric circles give similar concentric ellipses. It is likewise shown how pro- 

 perties of a certain kind are briefly transferred from the circle to any conic 

 section with the same facility. 



After demonstrating the properties of the ellipse, it is shown, that if the 

 gravity of any particle of a spheroid being resolved into two forces, one per- 

 pendicular to the axis of the solid, the other perpendicular to the plane of its 

 equator, then all particles, equally distant from the axis, must tend towards it 

 with equal forces ; and all particles at equal distances from the plane of the 

 equator, gravitate equally towards this plane ; but that the forces with which 

 particles at different distances from the axis tend towards it, are as the distances; 

 and that the same is to be said of the forces with which they tend towards the 

 plane of the equator. 



From this it is demonstrated, that when the particles of a fluid spheroid, of 

 a uniform density, gravitate towards each other, with forces that are inversely 

 as the squares of their distances, and at the same time any other powers act on 

 the particles, either in right lines perpendicular to the axis, that vary in the 

 same proportion as the distances from the axis, or in right lines perpendicular 

 to the plane of the equator, that vary as their distances from it, or when any 

 powers act on the particles of the spheroid, that niay be resolved into forces of 

 this kind ; then the fluid will be every where in equilibrio, if the whole force 

 that acts at the pole be to the whole force that acts at the circumference of the 

 equator; as the semidiameter of the equator to the semiaxis of the spheroid ; 



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