SEC. Y.J OF NATURAL PHILOSOPHY. 139 to meet in d. For since Pr to Pt, PR to PT, pB to PB, pe to Pt, are all in the same ratio, pe and Pr will be always equal. After this manner the points of the trajectory are most readily found, unless you would rather describe the curve mechanically, as in the second construction. PROPOSITION XXIII. PROBLEM XV. To describe a trajectory that shall pass through four given points, and touch a right line given by position. CASE 1. Suppose that HB is the given tangent, B the point of contact, and C, 1., P, the three other given points. Jo n BC. and draw IS paral lel to BH, and PQ parallel to BC ; complete the parallelogram BSPQ. Draw BD cutting SP in T, and CD cutting PQ, in R. Lastly, draw any line tr parallel to TR, cutting off from PQ, PS, the segments Pr, Pt proportional to PR, PT respectively ; and draw Cr, Bt their point of concourse d will (by Lem. XX) always fall on the trajectory to be described. The same otherwise. 1 et tl e angle CBH of a given magnitude re volve about the pole B; as also the rectilinear rad:us 1C, both ways produced, about the pole C. Mark the points M, N, on which the leg BC of the angle cuts that radius when BH; the other leg thereof, meets the same radius in the points P and D. Then drawing the indefinite line MN, let that radius CP or CD and the leg BC of the angle perpetually meet in this Ikie; and the point of concourse of the other leg BH with the radius will delineate the trajectory required. For if in the constructions of the preceding Problem the point A comes to a coincidence with the point B, the lines CA and CB will coincide, and the line AB, in its last situation, will become the tangent BH ; and there fore the constructions there set down will become the same with the con structions here described. Wherefore the concourse of the leg BH with the radius will describe a conic section passing through the points C, D, P, and touching the line BH in the point B. Q.E.F. CASE 2. Suppose the four points B, C, D, P, given, being situated withont the tangent HI. Join each two by the lines BD, CP meeting in G, and cutting the tangent in H and I. Cut the tangent in A in such mannr: