Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/142

 ratio of the principal latera recta directly. Pl. 6. Fig; 2.

From the focus S, draw SY perpendicular to the tangent PR, and the velocity of the body P will be reciprocally in the ubduplicate ratio of the quantity $$\textstyle \frac {SY^2}{L}$$. For that velocity is as the infinitely mall arc PQ decribed in a given moment of time, that is. (by lem. 7.) as the tangent PR; that is, (becaue of the proportionals PR to QT and SP to SY) as $$\textstyle \frac {SP \times QT}{SY}$$, or as SY reciprocally and $$\scriptstyle SP \times QT$$ directly; but SP x QT is as the area decribed in the given time, that is (by prop. 14.) in the ubduplicate ratio of the latus rectum, Q. E. D.

The princ[i]pal latera recta are in a ratio compounded of the duplicate ratio of the perpendiculars and the duplicate ratio of the velocities.

The velocities of bodies, in their greatet and leat ditances from the common focus, are in the ratio compounded of the ratio of the ditance inverely, and the ubduplicate ratio of the principal latera recta directly. For thoe perpendiculars are now the ditances.

And therefore the velocity in a conic ection, at its greatet or leat ditance from the focus. is to the velocity in a circle at the ame ditance from the centre, in the ubduplicate ratio of the principal latus rectum to the double of that ditance.

The velocities of the bodies revolving in ellipes, at their mean ditances from the common focus, are the ame as thoe of bodies revolving in circles, at the ame ditances; that is (by cor. 6. prop, 4-) reciprocally in the ubduplicate ratio of the ditaces.