Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/144



The velocity of a body revolving in any conic ection is to the velocity of a body revolving in a circle at the ditance of half the principal latus rectum of the ection, as that ditance to the perpendicular let fall from the focus on the tangent of the ection. This appears from cor. 5.

Wherefore ince (by cor. 6. prop. 4.) the velocity of a body revolving in this circle is to the velocity of another body revolving in any other circle, reciprocally in the ubduplicate ratio of the ditances; therefore ex æquo the velocity of a body revolving in a conic ection will be to the velocity of a body revolving in a circle at the lime ditance, as a mean proportional between that common ditance and half the principal latus rectum of the ection, to the perpendicular let fall from the common focus upon the tangent of the ection.

Suppoing the centripetal force to be reciprocally proportional to the quares of the diŧances of places from the centre, and that the abolute quantity of that force is known; it is required to determine the line, which a body will decribe that is let go from a given place with a given velocity in the direction of a given right line.

Let the centripetal force tending to the point S (Pl. 6. Fig. 3 .) be uch, as will make the body p revolve in any given orbit pq; and uppoe the velocity of this body in the place p is known. Then from the place P, uppoe the body P to be let go with a given velocity