Page:Encyclopædia Britannica, Ninth Edition, v. 6.djvu/311

Rh CONIC SECTIONS 283 Piior. IT. If a cone be cut by a plane which does not pass through the vertex, and which is neither parallel to the base ncr to the plane of a subcontrary section, the common section of the plane and the surface of the cone will be an ellipse, a parabola, or an hyperbola, according as the plane passing through the vertex parallel to the cutting plane falls without the cone, touches it, or falls within it. Let ADBV (figs. 45, 46, 47) be any cone, and let ONP be the V or it will meet VA, produced 47. common section of a plane passing through its vertex and the plane of the base, which will either fall without the base, or touch it, or fall within it. Let FKM be a section of the cone parallel to VPO ; through C the centre of the base draw CN perpendicu lar to OP, meetingthe circumference of the base in A and B ; let a plane pass through V, A, and B, meeting the plane OVP in the line XV, the surface of the cone in VA, VB, and the plane of the section FKM in LK ; then, because the planes OVP,MKF are parallel, KL will be parallel to VN, and will meet VB one side of the cone in K ; it will either meet VA the other side in H, as in fig. 45, within the cone ; or it will be parallel to VA, as in fig. 46 beyond the vertex, in H, as in fi&amp;lt; Let EFGM be a sec tion of the cone parallel to the base, meeting the plane VAB in EG, and the plane FKM in FM, and let L be the inter section of EG and FM ; then EG will be parallel to NB, and FM will be parallel to PO, and therefore will make the same angle with LK, wherever the lines FM, LK cut each other; and since BN is perpendicu lar to PO, EG is per pendicular to FM. Now the section EFGM is a circle of which EG is the diameter (Prop, ii.), therefore FM is bi sected at L, and FL 2 = EL.LG. CASE 1. Let the line PNO be with out the base of the cone. Through K and H (fig. 45) draw KR and HQparallel to AB. The triangles KLG&amp;gt; KHQ are similar, as also HLE, HKR ; therefore and KL:LG = KH:HQ, HL:LE=KH:KK; therefore KL. HL : LG. LE (or LF 2 ) : : KH 2 : HQ. KR. Now the ratio of KH 2 to HQ. KR is the same wherever the sections HFKM, EFGM intersect each other ; therefore KL. HL has a con stant ratio to LF 2, consequently (Prop. xii. on the ellipse) the sec tion HFKM is an ellipse, of which HK is a diameter and MF an ordinate. CASE 2. Next, suppose the line ONP to touch the circumfer ence of the base in A. Let DIS (fig. 46) be the common section of the base and the plane FKM ; the line DIS is evidently parallel to FLM, and perpendicular to AB, therefore DI 2 = AI IB hence DI 2 : FL 2 = AI. IB :EL. LG. But since EG is parallel to AB, and IK parallel to AV, AI is equal to EL, and IB:LG = KI:KL, therefore DP : FL 2 = KI : KL. Hence it follows from Prop. xi. on the parabola that the section DFKMS is a parabola, of which KLI is a diameter, and DIS, FLM ordinates to that diameter. CASE 3. Lastly, let the line PNO fall within the base ; draw VT (fig. 47) through the vertex parallel to EG. The triangles HVT, HEL are similar, as also the triangles KVT, KGL, therefore HT:TV=HL:LE, and KT:TV = KL:LG; therefore HT. KT : TV 2 = HL. LK : LE. LG or LF 2. Hence it appears k that HL.LK has to LF 2 a constant ratio, there fore the section DFKMS is an hyperbola, of which KH is a trans verse diameter and FM an ordinate to that diameter (Prop. xii. on the hyperbola). From the four preceding propositions it appears that the only lines which can be formed by the common section of a plane and a cone are these five : 1. Two straight lines intersecting each other in the vertex of the cone ; 2. A circle ; 3. An ellipse ; 4. A para bola ; 5. An hyperbola. The first two of these, however, viz., the pair of straight lines and circle, may be referred to the hyperbola and the ellipse ; for if the axes of an hyperbola be supposed to re tain a constant ratio to each other, and, at the same time, to dimi nish continually, till at last the vertices coincide, the hyperbola will evidently become two straight lines intersecting each other in a point ; and a circle may be considered as an ellipse, whose axes are equal, or whose foci coincide ; so that the only three sections which require to be separately considered are the ellipse, the para bola, and the hyperbola,. PART V.-OF CURVATURE. DEFINITIONS. If a circle touch a curve at any point P and pass through another point Q on the curve, then if Q move up to P the limiting position of the circle, when Q coincides with P, is called the circle of curvature of the curve at P. The centre of this circle is called the centre of curvature of the curve at the point P. PROPOSITION I. The common chords of any conic and any intersecting circle are equally inclined to the axis or axes of the conic. Let P, Q, R, S be the points of intersection of a conic and a circle. Let PR, QS intersect in 0. Then because P, Q, R, S lie on a circle PO. OR = QO. OS (Eucl. iii. 35) ; and because POR, QOS are two chords in a conic, the ratio PO. OR : QO. OS (in the parabola) = parameter of PR : parameter of QS. (in the ellipse and hyperbola) = square on the semi-diameter paral lel to PR : square on the semi-diameter parallel to QS. Now parameters of chords in the parabola, and semi-diameters parallel to chords in the ellipse and hyperbola, are equal only when the chords are equally inclined to the axes. The same proof applies to the pairs of chords PQ, RS and PS, QR. COROLLARY L If a circle touch a conic at P and cut it in Q and R, the chords PQ, PR are equally inclined to the axis, and the chord QR and the tangent at P are also equally inclined to the axis. This is seen by considering the case of S moving up to and coincid ing with P. COROLLARY 2. If the circle of curvature of a conic at a point P intersect the conic again in Q, then the chord PQ and the tangent to the conic at P are equally inclined to the axis of the conic. This is seen by considering the case of R and S, both moving u&amp;gt; to and coinciding with P.