Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/229

 and uppoe the latus rectum of the figure RPB to be L. From cor. 9. prop. 16. it is manifet that the velocity of a body, moving in the line RPB about the centre S, in any place P, is to the velocity of a body decribing a circle about the ame centre, at the ditance SP, in the ubduplicate ratio of the rectangle $$\scriptstyle \frac 12 \times SP$$ to $$\scriptstyle SY^2$$.. For by the properties of the conic ections ACB is to $$\scriptstyle CP^2$$ as 2AO to L and therefore $$\textstyle \frac {2CP^2 \times AO}{AP}$$ is equal to L. Therefore thoe velocities are to each other in the ubduplicate $$\textstyle \frac {CP^2 \times AO \times SP}{ACB}$$ to $$\scriptstyle SY^2$$. Moreover by the properties of the conic sections, CO is to BO as BO to TO, and (by compoition or diviion) as CB to BT. Whence (by diviion or compoition) as CB to BT. Whence (by diviion or compoition) BO - or + CO will be to BO as CT to BT, that is AC will be to AO as CP to BQ; and therefore $$\textstyle \frac {CP^2 \times AO \times SP}{ACB}$$ is equal to $$\textstyle \frac {BQ^2 \times AC \times SP}{AO \times BC}$$. Now uppose CP, the breadth of the figure RPB, to be diminihed in infinitum, o as the point P may come to coincide with the point C, and the point S with the point B, and the line SP with the line BC, and the line ST with the line BQ; and the velocity of the body now decending perpendicularly in the line CB will be to the velocity of a body decribing a circle about the centre B at the ditance BC, in the ubduplicate ratio of $$\textstyle \frac {BQ^2 \times AC \times SP}{AO \times BC}$$ to $$\scriptstyle SY^2$$, that is (neglecting the ratio's of equality of