AEBF; it may be proved as before, that either the single attraction of one towards the other, or the mu tual attraction of both towards each other, will be as the distance of the centres pS.
If spheres are dissimilar and inequable in proceed. ing directly from the centre to the circumference ; but are every where similar at every given distance in a circumference around ; and the attractive force of every point is as the distance of the attracted body : the whole force, with which two spheres of this kind attract each other, is proportional to the distance between the centres of the spheres.
This is demonstrated from the preceding proposi tion, in the same manner as the Proposition in Chap. V. p. 691. col. 1. of Physical ASTROSONIY was demon strated.
Cor. Those things which are demonstrated of the motion of bodies round the centres of conic sections, take place, when all the attractions are made by the force of spherical bodies of the quality already de scribed, and the attracted bodies are spheres of the same kind.
If any circle A EB is described with the centre S; and two circles EF, of are described with the centre P, cutting the former in E, c, and the line PS in Ff; and ed be let fall perpendicular to PS; then, if the distance of the arcs El?, ef is supposed to be continually diminished, the limit of the ratios of the variable line Dd to the variable line Ff is the same as the ratio of the line PE to the line PS.
For, if the line Pe cuts the arc EF in q; and the right fine Ee, which approaches nearer than by any assignable difference to the arc Ee, he produced, and meet the right line PS in T ; and SG be let fall from S, perpendicular to PE : because of the similar tri angles DTE, dTe, DES, Dd will be to Ee, as DT to TE, or DE to ES : and, because of the similar triangles Eeq, ESG, Ea will be to eq or Ff, as ES to SG ; and, ex cepo, D d to Ff, as DE to SG ; that is, because of the similar triangles PDE, PGS, as PE to PS.
If EFfc, considered as a surface, by reason of its breadth being indefinitely diminished, describes a sphe rical concavo•convex solid by its revolution round the axis PS, to the several equal particles of which there tend equal centripetal forces ; the force, with which that solid attracts a particle placed in P, is in a ratio compounded of the ratio of the solid DE' x Ff, and the ratio of the force, with which a given particle in the place Ff would attract the same particle in P.
For, if we first consider the force of the spherical surface FE, which is generated by the revolution of the arc FE, and is any where cut in r by the line de; the annular part of the surface, generated by the revolu tion of the arc rE, will be as the small line Dd, the radius of the sphere PE remaining the same ; as Ar. chimedes has demonstrated in his book concerning the sphere and cylinder. And the force of this, ex erted in the direction of the lines PE or Pr, placed around in a conical surface, is as this annular surface itself ; that is, as the line Dd; or, which is the same, as the rectangle under the given radius PE of :he sphere, and that line Dr/ : but that force, acting in the direction of the line PS tending to the centre S, is less, in the ratio of PD to PE, and therefore as PD X Dd. Let the line DF be now supposed to be divided into innumerable equal particles, each of ' which may be called Dd; and the surface FE will be divided into as many equal annuli, whose forces will be as the sum of all the rectangles PD x Dd; that is, as and therefore as DE'. Let the surface FE be now multiplied into the altitude Ff; and the force of the solid EFfe, exerted upon the particle P, will be as x Ff; supposing that the force is given, which any given particle Ff exerts up on the particle P at the distance PF. But, if that force is not given, the force of the solid EFfe be as the solid DE' x Ff, and that force not given, jointly.
If equal centripetal forces tend to the several equal parts of any sphere ABE, described about the centre S ; and, from the several points D, perpendiculars DE are erected to the axis of the sphere AB, in which any particle P is placed, meeting the sphere in E ; and in those perpendiculars the lengths DN are ta DE" ken, which are as the quantity , and the force, which a particle of the sphere, placed in the axis at the distance PE, exerts upon the particle P, jointly ; I say, that the whole force, with which the particle P is attracted towards the sphere, is as the area ANB, contained between AB the axis of the sphere, and the curve line ANB, which the point N continually touches.