Page:Newton's Principia (1846).djvu/119

 

PROPOSITION VIII. PROBLEM III.

If a body moves in the semi-circumference PQA; it is proposed to find the law of the centripetal force tending to a point S, so remote, that all the lines PS, RS drawn thereto, may be taken for parallels.

From C, the centre of the semi-circle, let the semi-diameter CA be drawn, cutting the parallels at right angles in M and N, and join CP. Because of the similar triangles CPM, PZT, and RZQ, we shall have CP² to PM² as PR² to QT²; and, from the nature of the circle, PR² is equal to the rectangle $\scriptstyle QR\times {\overline {RN+QN}}$, or, the points P, Q, coinciding, to the rectangle $\scriptstyle QR\times 2PM$. Therefore CP² is to PM² as $\scriptstyle QR\times 2PM$ to QT²; and $\scriptstyle {\frac {QT^{2}}{QR}}={\frac {2PM^{3}}{CP^{2}}}$, and $\scriptstyle {\frac {QT^{2}\times SP^{2}}{QR}}={\frac {2PM^{3}\times SP^{2}}{CP^{2}}}$. And therefore (by Corol. 1 and 5; Prop. VI.), the centripetal force is reciprocally as $\scriptstyle {\frac {2PM^{3}\times SP^{2}}{CP^{2}}}$; that is (neglecting the given ratio $\scriptstyle {\frac {2SP^{2}}{CP^{2}}}$), reciprocally as PM³.   Q.E.I.

And the same thing is likewise easily inferred from the preceding Proposition.

 

SCHOLIUM.

And by a like reasoning, a body will be moved in an ellipsis, or even in an hyperbola, or parabola, by a centripetal force which is reciprocally as the cube of the ordinate directed to an infinitely remote centre of force.

 

PROPOSITION IX. PROBLEM IV.

If a body revolves in a spiral PQS, cutting all the radii SP, SQ, &c., in a given angle; it is proposed to find the law of the centripetal force tending to the centre of that spiral.

Suppose the indefinitely small angle PSQ to be given; because, then, all the angles are given, the figure SPRQT will be given in specie. Therefore the ratio $\scriptstyle {\frac {QT}{QR}}$ is also given, and $\scriptstyle {\frac {QT^{2}}{QR}}$ is as QT, that is (because the figure is given in specie), as SP. But if the angle PSQ is any way changed, the right line QR, subtending the angle of contact QPR