Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/249

 Rh area's VDba, VIC are always equal; and the nacent particles DcxE. XCY of the area's VDca, VCX are always equal; therefore the generated area VDba will be equal to the generated area VIC, and therefore proportional to the time; and the generated area VDca is equal to the generated ector VCX. If therefore any time be given during which the body has been moving from V, there will be alo given the area proportional to it VDba; and thence will be given the altitude of the body CD or CI; and the area VDca, and the ector VCX equal thereto, together with its angle VCI. But the angle VCI, and the altitude CI being given, there is alo given the place in which the body will be found at the end of that time. Q. E. I.

Hence the greatet and leat altitudes of the bodies, that is the apides of the trajectories, may be found very readily. For the apides are thoe points in which a right line IC drawn thro' the centre falls perpendicularly upon the trajectory VIK; which comes to pas when the right lines IK and NK become equal; that is, when the area ABFD is equal to ZZ.

So alo, the angle KIN in which the trajectory at any place cuts the line IC, may be readily found by the given altitude IC of the body: to wit, by making the ine of that angle to radius as KN to IK; that is as Z to the quare root of the area ABFD.

If to the centre C (Pl. 17. Fig. 5.) and the principal vertex V there be decribed a conic ection VRS; and from any point thereof as R, there be drawn the tangent RT meeting the axe CV indefinitely produced, in the point T;