Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/146

 the line PK, and is thence given. But if the body goes from its place P with a yet greater velocity, the length PH is to be taken on the other ide the tangent; and o the tangent paing between the foci, the figure will be an hyperbola having its principal axe equal to the difference of the lines SP and PH, and thence is given. For if the body, in thee caes, revolves in a conic ection o found, it is demontrated in prop. 11, 12, and 13, that the centripetal force will be reciprocally as the quare of the ditance of the body from the centre of force S; and therefore we have rightly determined the line PQ, which a body let go from a given place P with a given velocity, and in the direction of the right line PR given by poition, would decribe with uch a force. Q. E. F.

Hence in every conic ection, from the principal vertex D, the latus rectum L, and the focus S given, the other focus H is given. by taking DH to DS as the latus rectum to the difference between the latus rectum and 4DS. For the proportion, SP + PH to PH as 2PS + 2KP to L, becomes, in the cae of this corollary, DS + DH to DH as 4DS to L, and by diviion DS to DH as 4DS - L to L.

Whence if the velocity of a body in the principal vertex D is given, the orbit may be readily found; to wit, by taking its latus rectum to twice the ditance DS, in the duplicate ratio of this given velocity to the velocity of a body revolving in a circle at the ditance DS (by cor. 3. prop. 16.) and then taking DH to DS as the latus rectum to the difference between the latus rectum and 4DS.

Hence alo if a body move in any conic ection, and is forced out of its orbit by an impule; you may dicover the orbit in