Plane and solid analytic geometry; an elementary textbook . NL = VON- NA = V(2 a - x)x. From the similarity of the tri-angles OMP and ONL, OM: ON:: MP: NL, or x: : 2 a — x : : y : V(2 a — x): Fig. 101. Hence y1 = - za — x which is the rectangular equation of the cissoid. It isevidently symmetrical with respect to the X-axis, andhas the line x = 2a as an asymptote. 104. The conchoid. — Let A be a fixed point at a dis-tance a from a fixed line OX. Draw the line APthrough A cutting OX at B, and on this line lay off aconstant distance BP(=b) both ways from B. Thelocus of P is called the conchoid o
Plane and solid analytic geometry; an elementary textbook . NL = VON- NA = V(2 a - x)x. From the similarity of the tri-angles OMP and ONL, OM: ON:: MP: NL, or x: : 2 a — x : : y : V(2 a — x): Fig. 101. Hence y1 = - za — x which is the rectangular equation of the cissoid. It isevidently symmetrical with respect to the X-axis, andhas the line x = 2a as an asymptote. 104. The conchoid. — Let A be a fixed point at a dis-tance a from a fixed line OX. Draw the line APthrough A cutting OX at B, and on this line lay off aconstant distance BP(=b) both ways from B. Thelocus of P is called the conchoid of Nicomedes. To find its rectangular equation, take the fixed line Ch. XV, §104] HIGHER PLANE CURVES 199 OX as the X-axis, and OA as the F-axis. Drop a per-pendicular from P on the X-axis and continue it to meetAK, drawn parallel to the same axis. Then But Hence or AP = V*2 + (# + a)2, and #P = £. AP=KPBP MP g2-K,y + fl)2=Q/ + a)262 y2 Y. The fixed point ^1 is called the pole, and the fixed line OX the directrix of the conchoid. If a 5, the lower branch of the curve cuts the F-axis in a single point above A. Note.—Among the most noted problems of the ancient mathema-ticians were the Trisection of an Angle and the Duplication of the Cubeby the aid of ruler and compass alone. It has lately been shown thatthe solution of these problems in this way is impossible. Both problemsinvolve the solution of a cubic equation, and both may be made to dependupon the construction of two mean proportionals between two straightlines. This has been accomplished in various ways by aid of higherplane curves, and it was for this purpose that both the Conchoid andCissoid were invented. 200 ANALYTIC GEOMETRY [Ch. XV, § 10/ 105. The cycloid.—The path described by a point onthe circumference of a circle which rolls on a
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