Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/177



The former of thee contructions (Fig. 5.) will become omething more imple by joining BP, and in that line, produced if need be, taking Bp to BP as PR is to PT; and through p drawing the indefinite right line be parallel to SPT; and in that line pe taking always pe equal to Pr; and drawing the right lines Be, Cr to meet in d. For ince Pr to Pt, PR to PT, pB to PB, pe to Pt, are all in the ame ratio, pe and Pr will be always equal. After this manner the points of the trajectory are mot readily found, unles you would rather decribe the curve mechanically as in the econd contruction.

To decribe a trajectory that hall pas through four given points, and touch a right line given by poition. Pl. 10. Fig. 1.

Suppoe that HB is the given tangent, B the point of contact, and C, D, P, the three other given points. Join BC, and drawing PS parallel to BH, and PQ parallel to BC, compleat the parallelogram BSPQ. Draw BD cutting SP in T and CD cutting PQ in R. Latly, drawing any line tr parallel to TR, cutting off from PQ, PS, the egments Pr, Pt proportional to PR, PT repectively; and drawing Cr, Bt, their point of concoure will (by lem. 20.) always fall on the trajectory to be decribed.