Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/143

 For the perpendiculars are now the leer emi-axes, and thee are as mean proportionals between the ditances and the latera recta. Let this ratio inverely be compounded with the ubduplicate ratio of the latera recta directly, and we hall have the ubduplicate ratio of the ditances inverely.

In the ame figure, or even in different figures, whoe principal latera recta are equal, the velocity of a body is reciprocally as the perpendicular let fall from the focus on the tangent.

In a parabola, the velocity is reciprocally in the ubduplicate ratio of the ditance of the body from the focus of the figure; it is more variable in the ellipis, and les in the hyperbola, than according to this ratio. For (by cor. 2. lem. 14.) the perpendicular let fall from the focus on the tangent of a parabola is in the ubduplicate ratio of the ditance. In the hyperbola the perpendicular is les variable, in the ellipis more.

In a parabola, the velocity of a body at any ditance from the focus, is to the velocity of a body revolving in a circle at the ame ditance from the centre, in the ubduplicate ratio of the number 2 to 1; in the ellipis it is les, and in the hyperbola greater, than according to this ratio. For (by cor. 2. of this prop.) the velocity at the vertex of a parabola is in this ratio, and (by cor. 6. of this prop. and prop. 4.) the ame proportion hold in all ditances. And hence alo in a parabola, the velocity is every where equal to the velocity of a body revolving in a circle at half the ditance; in the ellipis it is les, and in the hyperbola greater.