Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/120



Hence if the given point S, to which the centripetal force always tends, is placed in the circumference of the circle, as at V; the force will be reciprocally as the quadrato-cube (or fifth power) of the altitude SP.

The force by which the body P in the circle APTV (Pl. 3. Fig. 4.) revolves about the centre of force S is to the force by which the ame body P may revolve in the ame circle and in the ame periodic time about any other centre of force R, as $$\scriptstyle RP^2 \times SP$$ to the cube of the right line SG, which from the firt centre of force S, is drawn parallel to the ditance PR of the body from the econd centre of force R, meeting the tangent PG of the orbit in G. For by the contruction of this propoition, the former force is to the latter as $$\scriptstyle RP^2 \times PT^3SP$$ to $$\scriptstyle SP^2 \times PV^3$$; that is, as $$\scriptstyle SP \times RP^2$$ to $$\textstyle \frac {SP^3 \times PV^3}{PT^3}$$ or, (becaue of the imilar triangles PSG, TPV) to $$\scriptstyle SG^3$$.

The force by which the body P in any orbit revolves about the centre of force S, is to the force by which the ame body may revolve in the ame orbit, and in the ame periodic time about any other centre of force R, as the olid $$\scriptstyle SP \times RP^2$$, contained under the ditance of the body from the firt centre of force S, and the quare of its ditance from the econd centre of force R, to the cube of the right line SG, drawn from the firt centre of force S, parallel to the ditance RP of the body from the econd centre of force R, meeting the tangent PG of the orbit in G. For the force in this orbit at any point P is the ame; as in a circle of the ame curvature