The tangent at P bisects the angle DPL, and is perpendicular to DL, which it bisects at a point I on the circumference of the circle whose diameter is AB, the major axis, the center being C, the middle point of D O. As P recedes from A, it is evident that the angles P D L, P L D, will increase, until D L assumes the position D T tangent to the given circle, when they will become right angles. P will therefore be infinitely remote, and the point I having then reached t, where D T touches the smaller circle, C t S will be an asymptote to the curve. This shows that the measurements from the convex arc, for the construction of A P, are made only from the portion F T of the given circumference. In the diagram the point Q is so chosen that D L produced passes through E, so that Q J, the tangent at Q, is parallel to P I. It will thus be seen that the measurements from the concave arc, for the construction of B Q, are confined to the portion G T of the given circumference. As D L E rises, the points P and Q recede from A and B, the points L and E approach each other, finally coinciding at T; at this instant I and J fall together at t, so that S S is the common asymptote to A P and B Q. In Fig. 2 the given point D lies within the circumference of the given circle. Bisect D F at A, and D G at B; about D describe an arc with any radius D P greater than D A, and about O another, with radius O P = O F--D P, these arcs intersect in P, and producing O P to cut the circumference in L, we have P D = P L. Similarly E D = E H, U D = U W, etc. And since P D + P O = L P + P O, D E + E O = H E + E O, and so on, the curve is obviously the ellipse of which the foci are D and O, and the major axis is A B = F O, the radius of the given circle. [Illustration: FIG 2.] If, as in Fig. 3, the given point be made to coincide with the center of the circle, the ellipse becomes a circle with diameter A B = F O. But if the point be placed upon the circumference, as in Fig. 4, the ellipse will reduce to the right line A B coinciding with F O. [Illustration: FIGS 3, 4, 5, 6.] In this case we may also apply the same process as in Fig. 1; D T becomes a tangent at D to the circumference, and the asymptotes coincide with the axis of the hyperbola, of which one branch reduces to the right line A P extending from A to infinity on the left, and the other reduces to the right line B G Q, extending from B to infinity on the right. If the circle be reduced to a point, as in Fig. 5, the resulting locus is a right line perpendicular to and bisecting D O. If on the other hand the diameter of the given circle be infinite, the circumference, as in Fig. 6, becomes a right line perpendicular to the axis at F, and the curve satisfies the familiar definition of the parabola, D E being equal to E H, D P equal to P L, and so on.