THE ELEMENTS OF THE CONIC SECTIONS WITH Qre sertiono OF THE CONOIDS. BY JAMES DEVEREUX HUSTLER, B.D., F.R.S., LATE FELLOW AND TUTOR OF TRINITY COLLEGE. FOURTH EDITION. CAMBRIDGE: PRINTED AT THE UNIVERSITY PRESS: FOR J. & J. J. DEIGHTON, AND T. STEVENSON, CAMBRIDGE; AND WHITTAKER & CO., AVE-MARIA LANE, LONDON. M.DCCC.XLV. ADVERTISEMENT. — 4 — IT has been the Editor's object in these Elements to provide short and easy proofs of all the Propositions in Conics which are required for NEWTON'S PRINCIPIA. The method of deriving the chief properties of the Ellipse and Hyperbola directly from the Cone was communicated to the Royal Society in 1843 by SIR FREDERICK POLLOCK, F.R.S., and is here introduced with his permission. THE ELEMENTS OF THE CONIC SECTIONS. On te Iaraboala. DEFINITIONS. 1. THE solid generated by the revolution of a right-angled triangle BOC about either of its sides BO as an axis, is a Right Cone. SCHOLIUM. The other side CO describes a circle COD, whose center O is in the axis BO, and to whose plane the axis is perpendicular. So also any other 1 2 THE PARABOLA. line NE parallel to OC in the revolving triangle describes a circle having its center N in the axis, and its plane perpendicular to the axis. Hence conversely, every section of the Cone perpendicular to the axis is a circle, having its center in the axis. 2. If a right cone be cut by a plane AGK which is parallel to a plane touching the cone along the slant side BC, the section AGK is called a Parabola. B w A K 3. If BCD be that position of the revolving triangle which is perpendicular to the cutting plane AGK,I their common section AH, which is parallel to BC (Eucl. 16. xI.) is called the Axis of the Parabola, and the point A, the Vertex. 4. If AW be drawn parallel to CD, and a point S be taken in the axis AH, so that 4AS may be a third proportional to BW and A/W, the point S is called the Focus. THE PARABOLA. 5. If the axis be produced to L, so that AL may be equal to AS, the perpendicular LM to the axis at the point L is called the Directrix. T 6. Any line PNR perpendicular to the axis and terminated both ways by the curve, is called an Ordinate to the axis; the part AN of the axis between the vertex and ordinate, the Abscissa; and the ordinate BC through the focus, the Latus Rectum. 7. Any line MPV parallel to the axis, is called a Diameter; a line QVQ' parallel to the tangent at any point P, is called an Ordinate to the point P or diameter PV; and the part PV of the diameter, the Abscissa. Also that ordinate, which passes through the focus, is called the Parameter. 4 THE PARABOLA. 8. A straight line PG perpendicular to the tangent at any point P and terminated by the axis, is called a Normal; and the part NG of the axis, the Subnormal. 9. The part of any diameter between one of its own ordinates and the intersection of a tangent at the extremity of the ordinate, is called the Subtangent. Thus NT is the subtangent to the axis. THE PARABOLA. PROP. I. The rectangle contained by 4AS and the abscissa AN is equal to the square of the semi-ordinate NP. B C X _D Let ENPF be any circular section, which being perpendicular to the axis is perpendicular to the plane BCD passing through the axis (Eucl. 18. xi.): also the cutting plane GAK is perpendicular to BCD. Hence the common section PNIR is perpendicular to AHi (Eucl. 19. xi.), and likewise to EF the diameter of the circle, which therefore bisects PR in F. (Eucl. S. III.) But AN: NF:: BW: XA W:: A W(EN): 44AS (Def. 4.).4A. 4S x AN= EN x NF= NP2, by the property of the circle. COR. 1. The axis bisects all its ordinates. COR. 2. AN oc NP~. 2 G 6 ~~~THE PARABOLA. PR~OP. II. Th'1e distance SP of any point P in the parabola from the focuts is equal to the perpendicular distance PM of- 1ic samn point from the directrix. T, For 4AS x AN + SNI2 - ATL2 (End. 8. il.) Or N./P 2 ~ SN2 (SP2) = PM 2. Therefore SP= PiL Con 1. The latus rectum BSC 'is equal to 4/IS. For draw BKC perpendicular to the directrix. Then SB=BS AL+AS=2 AS (Def. 5.) CoR. 2. The rectangle by the latus rectum -BC and the abscissa.AN is equal to the square of the -semiordinate NP. THE PARABOLA. 7 PROP. III. If a straight line QP cut the parabola in P, Q, and the directrix in H, then HSD drawn through the focus makes equal angles with the focal distances SP, SQ. F M [t QI Draw PM, QF, perpendicular to the directrix, and PE parallel to SQ; then the triangles HPE, HQS, and the triangles HMP, HQF, are similar. Hence PE: QS:: HP: HQ:: PM3: QF; But QS= QF, and.-. PE = PI= SP. Hence z PSE = PES= L QSD. (Eucl. 29. I.) COR 1. If any straight line HP, not parallel to the axis, cuts a parabola in one point P, it will cut it again. For with center P and radius PS or PJI, which is necessarily less than PH, suppose a circle described cutting HS in E. In HP, produced if necessary, take HQ a fourth proportional to HE, HS, HP; and draw PM, QF, perpendicular to the directrix. Then Q is a point in the curve, for SQ = QF, by similar triangles, as in the proposition. 8 THE PARABOLA. Con. 2. If PQ move parallel to itself until the NJ I-X____./ points P and Q coincide, and HQ touches the parabola, the z PSQ vanishes, and the angles made by SP, SQ with SH are right angles. Conversely, if SH be drawn perpendicular to any focal distance SP, cutting the directrix in H, then HP being joined touches the curve at P. COn. 3. Let PS meet the curve again in R; then HR being joined touches at B. The tangents therefore at the extremities of any parameter meet in the directrix. In the case of the latus rectum, the tangents meet in the intersection of the axis with the directrix, and are at right angles to each other. PROP. IV. A tangent PH bisects the angle SPM*. For z HSP is a right angle (Prop. 3. Cor. 2.); and HP is common to the two right-angled triangles * The reference is always to the Figure immediately preceding. THE PARABOLA. 9 HSP, HMI P; also SP=-PIM; therefore the triangles are equal, and z SPH =- MPH. Con. A tangent at the vertex is perpendicular to the axis, and parallel to all the ordinates of the axis. PROP. V. The focal distance SP is equal to i. The abscissa of the axis AN + AS. II. The distance ST of the tangent's intersection with the axis. ilI. The distance SG of the normal's intersection with the axis. 1'0 1~~~0 ~,HE PARABOLA. Fori. K1P~=_P]J=NL=AN~A4L=AN+.AS. II. zLSPT= L MPT-altern.LzSTP;:..SP-=ST, ill. The right angle GPT = L PG T +,z L GTP LPGS + L SPT;, take away LSPT, and L SP G= zLSGP;:.-SP =SG. PROP. VI. Thle s-ubnormal NG = 2. AS. For SG =SP= AN~+AS (Prop. 5.) 1=SN + 2AS; Take away SN, and NG= 2AS. PRO0P. VII. The subtangent NT of the axis is douible of the abscissa AN. For ST= SP= AN+ AS (Prop. 5.); Take away AS,. -and A T=AM. CoR,. If L. be the latus rectum, L x AN= NP (~Prop. 2. Cor. 2): hence L: NP:: NP: AN:: 2 NP:2 AN(NT). THE PARABOLA. 11 PROP. VIII. A perpendicular SY from the focus upon any tangent PT intersects PT in the tangent at the vertex A. Draw (Prop. 4. Then (Eucl. 2. Hence / to PT. the tangent A Y, which is parallel to PN Cor.), and let it meet PT in Y: join SY. AN= AT (Prop. 7.), and.-. PY= TY vi.) Also SP=ST, and SY is common: SYP = z SYT, and SY is perpendicular CoR. The triangles SA Y, SPY being similar, SP: SY:: SY: SA, and SP: SA:: SP2 SY. Also SY2= SP x SA, and oc SP. 12 1Q ~~~THE PARABOLA. PROP. IX. The rectangle by.,the latus 7ectuinz (L) and the part QF of any diameter cut off by an ordinate to the axis, is equal to the rectangle'by the segments, PF, FR, of'the ordinate. T, Draw QEY perpendicular to the axis: Then PP x FR = NVP2 _ NP2 (Eucl. 5.iji.) - NP2- QE2 = L xAN-4-L xAE (Prop. 2. Cor. 2.-) =LxEN=LxQF. THE PAIRABOL.A. 1 I fl- 3 PROP. X. -if fron either extremzity of an ordinate QQ' to any diameter a _perpendicular QD be let fall upon the diameter, the square of QD is equal to the rectangle by the latus rectum and the abscissa PV. The As QVD, 1PNT,' SPY are m'anifestly similar; Hence QD: DV: NP: NT L:PR (Prop. 7. Cor.) L xDV= PR xQD=,PR xPF. Also L x DP L x QF = RF x PF (Prop. 9.) Hence L x PV= PF (Endl. ~3. ii.) =QD2. In the. same mnanner if Q'D' be the perpendicular from the other extremity Q', it may be shewn that L x PJ7V Q'D'2.? COR. QD = Q'D, and the similar triangles Q VD, Q'VD' are therefore equal; hence QV= Q'V; i. e. a diameter bisects all its own ordinates. PROP. XI. The rectan~gle by four times the focal distance SP, and the abscissa PV is equal to the square of the wemi-ordinate QV. For QV2: QD9-::SC:Y.Or QV21: L x PV:.: SP SA (Prop. 8..Cor.):4 SP: L; 4SPxPV=QV2?. 14 THE PARABOLA. COR. 1. Let EF be the parameter. Then SP= ST (Prop. 5.) = P; also EV2= 4SPx PV= 4SP2. T F Hence EV= 2SP, and the parameter = 4SP. COR. 2. The rectangle by the parameter and abscissa of any diameter = the square of the corresponding semi-ordinate. COR. 3. The abscissa varies as the square of the semi-ordinate. Con. 4. If HI be drawn parallel to the axis V from any point H in a tangent, then HKoc PH2. For, if the parallelogram PHKVF be completed, HKE = the abscissa PV, and PH = the semi-ordinate KIV. THE PARABOLA. 1. 15 PR~OP XII. The rectangle. by the parameter (P) of any diameter PY and the part IM of any other diaimeter eut off by an ordinate QQ' of thefirst, is equal to the rectangle by the segments QM, 1JQ' of the ordinate. K iDraw LE parallel to QQ'. Then QZJx MQ'-= QV2-_ V1J2 (Eucl. 5. Ii.) = QV2-LE2=PxPV-PxPE (Prop. 11. Cor. 2.) = P x E = EP x LM. COR. 1. Let any other ordinate UKH; whose parameter is 1?', intersect QQ' in 1!: then QM x IQ': IHDx MIK:: Px LIi: P'x LM P:P'. COR. 2. If the lines QQ', 11K move parallel to themselves and intersect 'each other, either within or without the parabola, the rectangles of their segments are always in the sam-e constant ratio. 16 16 ~~~THE PARABOLA. P~ROP. XIII.0 if an ordinate PQ and a tangent PR be drawn fromn the same point P, any diameter DEF terminated by the, ordinate and tangent is. divided, by the curve in- the samet Iprop~ortion in which itse( f divides the ordinate. Draw QR parallel to the Then DE: QR::PlY:: PE2 axis;:PR2 (Prop. 11. Cor. 44.):P Q2-, And QR:DE:: PQ:PF;:.'DE:DE:: PE:PQ, Di~do. DE,:. EF:: PE: EQ. COR. 1. If DEEY bisects PQ, it~ is then - the dia.. meter of which PQ is, an- ordinate, and DEE is the'. THE PARABOLA. '7 '17 subtangent (Def. 6.): but then DE = EF, i. e. the 1) subtangent is double of the abscissa in all cases. COR. 2. If a tangent be drawn at Q, it must meet the diameter DEE in the same point D. COR. 3. The tangent at E is parallel to the ordinate PQ; hence PL = LD. PROP. XIV. If twio parabolas, PAN, QAN, wihose latera recta are L, L', respective1y, have a common axis and vertex, the areas PAN, QAN, ctt off by a common ordinate QPN, are in th/e subduplicate ratio of the latera recta. 18 THE PARABOLA. Draw OM parallel to QN and PE, QF perpendicular to OM, Then (Eucl. 1. vi.) rectangle PM: rectangle QM:: PN::: /L x AN: QN VL' x AN which constant ratio holds for all the corresponding rectangles thus inscribed in the areas PAN, QAAN, and componendo for the sums of them in each, into how many parts soever the abscissa AM is divided. Let the number of the parts MN be increased, and the magnitude of each diminished, indefinitely. Then in the limit the sums of the inscribed rectangles are respectively equal to the areas PAN, QAN. (NEWTON. Lem. 4.) The areas therefore are in the constant proportion of V/L to ViI. COR. Area ASP: area ASQ:: v/L: V'. THE PARABOLA. 19 PROP. XV. The parabolic area, contained by the curve, the abscissa, and the semi-ordinate of any diameter, is twothirds of the parallelogram completed of the abscissa and semi-ordinate. Q, q, are points in the curve indefinitely near; PR, QT, tangents at P, Q; QR, qr, parallel to PV; VY, RZ perpendicular to QT. Then the triangles TVY, RQZ being similar, and TV double of PV or QR, (Prop. 13. Cor. 1.) VY is also double of RZ. But area QVvq: area RQqr:: VY: IRZ; that is, area QVvq is double of RQqr. Hence the whole area PQV is double of the whole area PQR, and is therefore two thirds of the parallelogram VR. THE PARABOLA. PROP. XVI. To determine the diameter of the circle of curvature at any point of a parabola, and the chord wzhich passes throughc the focus. DEF. If a curve PQ and a circle PqL touch the P LA: same straight line PR at the same point P, the circle is said to be a Circle of Curvature to the curve, when their deflections QR, qR from the common tangent PR are ultimately equal, or, which is the same thing, when indefinitely small arcs PQ, Pq, of the curve and circle, being equally deflected from the common tangent, are coincident. COR. If any chord PL be drawn, and the subtense qQR parallel to PL, then the triangles Pq L, PqR are equiangular (Eucl. 29. I. and 32. iII.) Hence PL = chord. Pql] But in the evanescent state of the figure Pq R, the arcs PQ, Pq are coincident and equal, and the subtenses QR, qr: also -the chord Pq is then equal to the are Pq (NEWT. Lem. 7.) THE PARABOLA. 2 21 Hence PL == y Q PQ and the evanescent state. Let now PLKC be the circle QJ? being taken in of curvature at any point P of the parabola, PL the chord through the focus S, PKC the diameter. QR parallel to PL, and QV parallel to PT cutting SP in X. Then QR==PX=PV, because SP=ST(Prop. 5.) Also QV is ultimately equal to the arc PQ. (NEWT. Lem'. 7.) Hene P =PQ2 QV2 _ 4SP XPV HencQ1L QR Q =4SP. By sim. A' SPY, SKCL, the diameter PKIC PL>8 P 4SJ?' 4_____ 4 SP9 = =y or =S --— A(Prop.8 )Szr 4 SP" xSA_ _ LxSP' 0SYxSPxSA- SYJ 3 ON D EFINITIONS DEFINITIONS. 1. IF a right cone GAD be cut by a plane A4PI through both slant sides, the section is called an Ellipse. 2. If GAD be that position of the revolving triangle which is perpendicular to the cutting plane APM, their common section AM is called the Axis Miajor; and the points A, M, are called its Vertices: also the point C which bisects 4AM is called the Center. THE ELLIPSE. 23 3. A line BCL drawn through the center perpendicular to the axis major and terminated both ways by the curve, is called the Axis Minor. 4. If with the extremity B of the axis minor as center, and radius the semi-axis major, a circle be Q L Q Ail S e: described, cutting the axis major in the two points S and H, those two points are called the Foci. 5. A perpendicular PNR to the axis major, terminated by the curve, is called an Ordinate to the axis, and the segments AN, NM, into which it divides the axis, the Abscisse. Also the ordinate through either focus is called the Latus Rectum. 6. Any line PCG through the center is called a Diameter; and a diameter DCK drawn parallel to the tangent at the extremity P of PCG is called the Conjugate Diameter to PCG. 7. A line QQ' drawn parallel to the tangent at any point P is called an Ordinate to the point P or diameter PG; and the segments PV, VG, of the diameter, the Abscissce. 24 THE ELLIPSE. 8. A perpendicular PO to the tangent at any point P, terminated by either axis, is called a Normal; and the part ON of the axis, the Subnormal. 9. If in the axis major produced a point X be taken, so that CX is a third proportional to CS and CA, a perpendicular to the axis major through that point X is called the Directrix. — ___ --- THE ELLIPSE. PROP. I. The rectangle by the abscissa? AN, NM, of the axis major is to the square of their semi-ordinate NP as the square of the axis major to the square of the axi-s minor. A. 0 D As in the Parabola, Prop. 1, PNJ? is perpendicular to AM and to EF, and is bisected by EF: the axis therefore bisects all its ordinates. Draw MQK parallel to AD. Then AN:EN::AM:MK::AC:MQ And NM:NF::AM:AD::AC:AO AN x ATM: EN x NE (NP2):A C2: A 0x MQ. When NP coincides with BC, ACx CM: BC2:: AC2: AO xMQ B C2=AOxMQ; and ANxNXM: NP:AC: B C2. COR. 1. NP'o AN xNM oc L1 _ A T21N. COR. 2.NP': CA21 --- CN2:: B C2: A C'. 4 26i THE ELLIPSE. Con. 3. If AQMi be the circle on the axis major, and PN be produced to meet the circumference in Q; then, since CA' - CN2= NQ', PN: QN:: BC: AC. C E CoR. 4. Let CQ cut the circle upon the axis minor in q; join P q, and produce it to the axis minor in n. Then because P.N: QN -: Cq: CQ (Cor. 3.), Pqn is parallel to CN and perpendicular to BC. Hence qn': CNM:: Cn (PM2): QM 2 by sim. As Or Bn x nE: Pn2:: BC2: A4C2. CoR. 5. BnxnEocy-Pn. CoR. 6. Pn:qn:: AC: BC. Con. 7. As in the Parabola, Prop. xiv, the Ni S C N area ANP of the ellipse: the area ANQ of the THE ELLIPSE. 27 circle:: NP: NQ, i. e.:: BC: AC. The whole areas therefore of the ellipse and circle are in the same proportion. Hence the area of the ellipse = area of BC the circle x ABC which for different ellipses oc AC x ACoc AC x BCoc the rectangle by the axes. COR. 8. The areas of circles being as the squares of their diameters, the area of an ellipse = the area of that circle whose diameter is a mean proportional between the axes. COR. 9. Area ASP: area ASQ\ BC AC. and area ACP: area A CQ PROP. II. The distance SP of any point P in the ellipse from the focus is to the perpendicular distance PX of the same point from the directrix in the constant ratio of SC to AC. L B p Il * or ///T/\X\ L. _- = ^w<\\v _ C T A S N C H M T Let SL, perpendicular to AMi, meet the circle on the axis major in L; also let LT a tangent at L, 28 28 ~~~~THE ELLIPSE. meet the axis in. T, and NVP produced in K. Join. CK, CL.. The perpendicular TX to the axis through T, is the directrix (Def. 9): also SL = BC, 'from Def. 4. Then NP2: CiA 9 - CNV: B C': -AC' (Prop. 1.- Cor. 2. And SN':2 KL': ST2: TL' SL2 (B C?): CL2 (AC') by sim. A%.~NP + SN (P)KL'+GCA'-'NBC2: AC2 B ut KL2 + CA' - C'N2= CKC2- CN'=KiV~ Hence SP KN:BC:AC And KY: TN (PX): SC: SL (BC) by sim. S Therefore SP: PX: SC:CA. Pu.op. III. Thze svumof the two focal distances SP, HP, is equal to the axis major. Take M~T'= AT; draw the directrix TX'; and produce XP to meet T'X' in XI. and lIP: PX:SC: A C(Prop. 2.):: A C: CT SP+ HP: PX+ PX' (XX'):: AC:a CT But XX'= TT`= 2CT;.-. SP+ HP = 2AC. COR.1 HP=2AC-SP. COR. 2.ASX SJI= SL'=BC'. COR. S3. Let SV be the semi latus rectum: then SV: SL (BC):: BC: AC (Prop. 1. Cor. 3.) that is, the latus rectum is a third proportional to the axis major and axis minor. THE ELLIPSE. Prop. IV. The focal distance SP =AC SC 2 PSN' radius being unity. (2AC-SP)2== HP' (Prop. 3. Cor. 1.) = =SP2+SI[2- 2 Sflx SN (Eucl. 13. ii.) Or4AC2-4ACx SP+SP2=SP2+4SC'-4SC.SN Hence A C x SP_-SC x SN= A C-2 SC2 = BC2. But SN= SP x cos. PSN, rad =1, AC x SP-_SC x SP xcos. PSN=BC2, And SP= BC2 AC- SC x cos. PSN' PROP. V. if one q/ the focal distances SP be produced, the line PT which bisects the exterior angle HPL, touches the curve at P. In SP produced take PL = PH, and in PT take any point T; join ST meeting the ellipse in Q: join also LT, HT, HQ. Then LT=HT (Eucl. 4. i.); also ST+ HT= ST + TL, and is therefore greater than SL, or than 30 THE ELLIPSE. SP+ HP or than SQ + HQ. Hence the point Q lies between S and T (Eucl. 21. I.), or 7' is without the ellipse; and since every point of PT, except P, is without the ellipse, PT touches at P. COR. 1. SP, HP make equal angles with the tangent PT. COR. 2. A tangent at the extremity of either axi is perpendicular to that axis. COR. 3. Complete the parallelogram SPII G. Y R= S Jr3 Ia" z Then SG+ GH=SP+ PH (Eucl. 34. i.), and G is a point in the ellipse. Join CP, CG; then because the diagonals of parallelograms bisect each other, and that SC = CH, PCG is a straight line and a diameter; i. e. the center bisects all diameters. COR. 4. Draw the tangents YT, RZ, at P, G: Then SPY= z HPT (Cor. 1.) and..- = suppt. of SPH= - supplement of SGH= z HGZ; and z SPG=alternate L PGH;.Y. YPG = L PGZ, and YT is parallel to RZ. THE ELLIPSE. 31 PROP. VI. The perpendiculars from the foci upon any tangent intersect the tangent in the circumfejrence 9f a circle whose diameter is the axis major. Produce HP to W, making PW= SP; let SW cut the tangent in Y, and join CY. w M 0 The as SPY, WPY are equal in all respects (Eucl. 4. I.), whence z SYP = z T YP, and SY is perpendicular to PY. Also SY= YW, and CS= CH, therefore CY is parallel to HW, and by sim. as, CY= I HW= g (SP+ HP)= CI; therefore Y is a point in the circumference of the circle whose radius is CA. In like manner, Z is a point in the circumference of the same circle. 32 THE ELLIPSE. COR. Let a conjugate diameter CD cut either focal distance HP in E. Then by the proposition, CY is parallel to HP, and PECY is a parallelogram. Hence PE =CY= C4A. PROP. VII. The rectangle by the perpendiculars frSom the foci upon the tangent is equal to the square of half the minor axis. Produce ZH to the circumference in 0; join CO. Then z OZY being a right angle is in a semicircle, and 0, Y, are the extremities of a diameter: OCY is therefore a straight line and a diameter; also the as OHC, YSC are similar and equal, and SY= OH;. SYx HZ= OH x HZ= Al x HM (Eucl. 35. II.) =BC2 (Prop. 3. Cor. 2.) COR. Since z SPY= z HPZ (Prop. 5, Cor. 1.), the AS SPY, HPZ are similar. SP SP Hence SY= HZ x HP and SY2= BC2 x SP SP Also SY~ c~ HP or oc2 4 C- SP PROP. VIII. The semi-axis major is a mean proportional between the distance CN of any ordinate from the center and the distance CT of the intersection of the tangent,at P with the axis. THE ELLIPSE. 33 y S C N 1-I M T Produce NP to Q; and join TQ, CQ Then TYS: SYt T And TZ': HZJ *.. TYx TZ(MTx TA): SYx HZ(BC2):: TN2: PN And BC2:A C:: PN2:Q. Hence MTx TA: CA2:: TN2 QN2 Compdo. CT2: CA2 (CQ")::QT2 QN2. The triangles CQT, TQN have therefore a common angle CTQ, the sides about two other angles proportional, and the third angle TNQ in one a right angle; they are therefore similar (Euc. 7. vi.) Hence z CQT is a right angle, and TQ touches at Q; i.e. the tangents at P and Q intersect the axis in the same point T'. Also CN: CQ(CA):: CQ(CA): CT. COR. Produce the tangent TP to meet the axis minor in t; and join qt. 5 34 THE ELLIPSE. T Then Ct: Cn:: CT: NT:: CT2: QT2:: Cq2 or BC2: Cn2 Hence Ct: BC: BC: Cn, as for the axis major. ] PROP. IX. The rectangle by the normal PO to either axis and the perpendicular POF upon the conjugate diameter is equal to the square of half the other axis. tp B \ A S C \/6 N T 0 Let PN produced meet the conjugate in R. Then the angles N and F being right angles, a circle THE ELLIPSE. 35 would circumscribe the quadrilateral FONR; hence PO x PF = PN x PR (Eucl. 36. III.)= Cn x Ct = BC2 (Prop. 8. Cor.) Similarly Po x PF = CN x CT= C2. Con. 1. The triangles PON, Pon, being similar, ON: Pn (CN):: PO: Po::BC2: AC2':L 2AC (Prop. 3. Cor. 3.) Hence the subnormal ON= 2C x CN, L being 2AC the latus rectum. Similarly on =-L x Cn. CoR. 2. From 0 draw OL perpendicular to SP; then from the quadrilateral EFOL, whose opposite PF x PO angles F and L are right angles, PL = BC2 AC = half the latus rectum. PROP. X. If one diameter DCK be conjugate to another PCG, conversely PCG is conjugate to DCK. B X AN R S /N He T Draw PN, DR, ordinates to the axis, and the tangents PT, DX. 36 36 ~~~~THE ELLIPSE. Then CN x CT= CA' (Prop. 8.) Take away (CN', and CNx NT= CA' - (CN' (Euci. ~3. ii.) =ANx Nit!. (CNx NT: CRxRX:: PN2: DR' (Prop. 1. Cor. 1.) NT: Ci?', by sim. ~s* Hence CN: RX:NT: CRI:: PN: DR?; &, (' PN, DRX are sim ilar (Euci. 6. vi.) and PC is parallel to DX. PROP. XI. The distance of one of the ordinates PN, DR from the center is a mean proportional between the distances o~f the other ordinate from the center and firom the intersection of its tangent with the axis. CN x CT== CR x CX, for each = CA' (Prop. 8.) Hence CN: CR:: CX: CT:CD: PT bysim.As CPT, CDX CR: NT by sim. A CDRI, PNT. So CR: CN:: CN: RX. COR. 1. CNx NT (C.R 2>=CA4'-CN' (Prop. 10.) Hence CA' ('N' + CR2. By a similar proof, if Pn, Dr, be perpendicular to the axis minor, Cn' ~ Cr', =BC2. Wherefore AC' + BC = CN + Cn' + CR' + Cr' = CP' + CD'. COR.2. SP'~IIP2'2SPX HP=4AC2. But SP' +HP' 2SC2+ 2 CP9' (Eucl. 12 and 13. ii.) SP x -HP= 2A4C2-SC'-_CP' =AC + BC'- CP' =CD2, by the preceding Corollary. COR.. NP': ANx NM (C.R):: B C2: A C' NP:CR:: BC:AC So DR: CN:: BC: A C. THE ELLIPSE. 37 COR. 4. NP: CR:: DR: CN. The as CPN, CDR, are therefore equal. (Eucl. 15. vi.) PROP. XII. All parallelograms, circumscribing an ellipse at the extremities of conjugate diameters, are equal. Ift,- Et be If tangents be drawn at the extremities of two conjugate diameters, they form a parallelogram circumscribing the ellipse (Prop. 5. Cor. 4. and Prop. 10.) of which CPaK is a fourth part. Draw POF perpendicular to DCK. Then by sim. As PON, CKR CK(CD): PO:: CR: PN:: AC: BC (Prop. 11. Cor. 3.).. CDx PF: POx PF(BC2): ACxBC: BC2 (Prop. 9.) Hence CD x PF= AC x BC, or the parallelogram PK= the rectangle by the semi-axes. COR. The whole area of the ellipse, which varies as AC x BC (Prop. 1. Cor. 6.), varies also as CDxPF, or as the circumscribing parallelogram. 38 38 ~~~~THE ELLIPSE. PIZop. Xlii. The rectangle by the abscissw PV, VG, of any diameter is to the square o~f their semi-ordlinate QV as the squtare of the semii-diameter CP to the square of tie sYemi-conjugate CD. Let the ordinate QQ' meet the axis in 0: draw RPNM, VY, QL, EDE, perpendicular to the axis: let YV meet CR in X, and OX meet LQ in K. Then since KCL: QL:: XY: VY:: RAT: PN, by similar triangles, K is in the circumference of the circle (Prop. 1. Cor. -3.) But CD is parallel to PT;:TN: CF:: PN: DF::RN: EF; whence the A" TNR, C~FEY, are similar, and EC is parallel to R T and perpendicular to GE. In like manner KO is perpendicular to CR. Now CP2 C V2: CR2: C'X2 CP2 [_ 2 C CR?2-X2 (KX):: CR2: CR2 -And KX-2: Q V2: CE2:- CD' CP'2- CV' (PVx VG): Q VI:CP2 CD'. THE ELLIPSE. 3 39 COR. 1. P1" x VG Q'V':: CP2 CD', by the same proof; Q QV= Q'V; i. e. a diameter bisects all its own ordinates. COR. 2. PVx VG or Cl?'_ CV'oc, QV'. cop.. 3. Since CP'- CV2: Cl?':: Q V2: CD', Divo CV2: C~P2:: CD' - Q VI: CD2, Whence CD' - Q PI oc CV', Pu~or. XIV. If. any line AB intersect a diameter QCR in F, and CD1 be parallel to AB, then the rectangle AF, FB:the rectangle QF, FR:0: CD)':CQ'. Draw Cl? conjugate to CD and therefore bisecting A4B in V: also draw QI parallel to AB. Then CD'- QI': CD'- A V:: CI': CV,'(Pr. 13. Cor. 3.) Divdo.A P-,_Q.P: CI,-QP:: C!'- CV-': Cl'::QI'-FV-: QI2'by sim. A'. HenceA V'-FV2: CD'::.QI'-FV2: QI'(Eu. 12. v.) CQ - CF': CQ2 by sim. A. Or AFx F-B: CD'::QFxFR: CQ'. 40 THE ELLIPSE. COR. 1. If any two lines whatever AB, MN intersect in F, the rectangles AF x FB, ]MF x FN, are as the squares of the diameters CD, CO, drawn parallel to AB, MN, respectively. COR. 2 If AB, MfiN, move parallel to themselves, and intersect any where either within or without the ellipse in F, the rectangles AF x FB, MF x FN, have always the same constant ratio. COR. 3. If from any point K without the ellipse tangents KW, KZ be drawn, then are KW and IKZ in the same proportion as the diameters to which they are respectively parallel. For take any other point, and draw, For take any other point F, and draw FMN, FQR parallel to KW, KZ. Then if FMN move parallel to itself until it coincides with KTW, the rectangle FM, FN, becomes the square of KW at the point of contact: and so the rectangle FQ, FR becomes the square of KZ. But FM x FN to FQ x FR, and therefore KW2 to KZ2, is as the squares of the diameters to which they are parallel. THE ELLIPSE. 41 PROP. XV. If a tangent QT drawn at the extremity Q of any ordinate QV meet the diameter CP to which the ordinate belongs in T; then CP is a mean proportional between CV and CT. At the extremities, P, G of the diameter draw PL, GMi touching the ellipse, and meeting the tangent QT in L and M. Then MfQ: MG:: LQ LP(Prop. 14. Cor. 3.) And MQ: LQ:: MG: LP VWhence VG PV:: TG TP by sim. triangles.'. VG-PV: G+PV:: TG TTP: TG+ TP Or 2CV: 2CP:: 2CP: 2CT. COR. 1. A tangent at the other extremity of the ordinate would meet the diameter CP produced at the same point 7. COR. 2. PVx VG= CP- CV2= CVx CT-CV2 = CVx V. 42 THE ELLIPSE. PROP. XVI. The diameters which bisect the lines joining the extremities of the axes are equal and conjugate..3 A C M Let AB, B1~ be bisected in E, F: then the As B CE, BCF are manifestly equal and similar, and z ECB = z FCB; therefore CP and CD making equal angles with CB are similarly drawn in the two elliptic quadrants and consequently are equal. Also AB is bisected by CP and is therefore an ordinate belonging to the diameter CP (Prop. 13. Cor. 1.) But 4AC= CI, and BF= FM; hence CFD is parallel to AB or is conjugate to CP. PROP. XVII. To determine the diameter of curvature and the chords through the center and focus of an ellipse. I. The chord PI through the center. Draw QR parallel to PI and Q V parallel to the tangent; then wen n PV is evanescent, PQ= Q V. Now PVx VG: QV2:: CP: CD2 (Prop. 13.); QVhe Ce PQQ hence p = VG x; and ultimately PQ 2CP X CD2 2 CD' CP2 CP- P THE ELLIPSE. 43 ii. The diameter PK. In the quadrilateral CFKI, the opposite angles F and I are right angles, and therefore a circle would circumscribe it. Hence PK x PF= CP x PI= 2 CD, 2CD2 by the preceding case, and PK= —P-. iII. The chord PL through the focus. Let it cut the conjugate in E; then PE=the semi-axis major. Also, as in the last case, PL x PE 2 CDI~ =P PKx PF 2CD;.9. PL= PE= 2CDm ' pE2 x BC2 COR. The diameter PK pE = p= 73 9 J~r PPF3 2BC9 PE3 SPJ' (Prop. 12.) = PE x pF Lx ^-: by similar triangles. ON rwe VLrbo a. DEFINITIONS. 1. IF a right cone DEF, and another DE'F' which is equal and vertical to DEF, be cut by a plane RAMrp, each of the sections RAP, Mpr, is called an Hyperbola, and the two are called Opposite Hyperbolas. SCHOLIUM. Every section DGH of the cone passing through the vertex D is a triangle, because the slant side must coincide with the hypothenuse of the revolving triangle in some one position of it. THE HYPERBOLA. 45 2. The interval AM between the opposite hyperbolas is called the Axis Major, and the points A, M, are called Vertices: also the point C which bisects AM is called the Center. 3. If DHGgh be the triangular section through the vertex D whose plane is parallel to the cutting plane; and BC, perpendicular to AM at C, be taken a fourth proportional to DKi, KG, and AC; then BC is called the Axis Minor or Conjugate Axis; which is manifestly the same for both RAP and Mrp. 4. The two opposite hyperbolas BD, EK, whose axis major is BE and conjugate axis AM, are called the Conjugate Hyperbolas to AP, MP'. H M cnadS e hS 5. If with center C, and radius CS equal to the line joining the vertices A, B, of the major and minor 46 THE HYPERBOLA. axis, a circle be described cutting the axes produced in S, S, H, I', respectively, those four points are called the Foci. 6. If the axes A[I, BE be equal, the curve is called a Rectangular Hyperbola. In this case the conjugate hyperbolas AP, BD, are equal and similar. 7. Any line PCG drawn through the center and terminated by two opposite hyperbolas is called a Diameter, and the intersections of a diameter with the curves are called it's Vertices. 8. A perpendicular PNR to the axis major, terminated by the curve, is called an Ordinate to the axis; and the distances AN, NiiH of the ordinate from the vertices are called the Abscissw. 9. A line QQ' drawn parallel to the tangent at any point P and terminated by the curve, is called an Ordinate to the point P or diameter PCG; and the distances PV, VG of the ordinate from the vertices of the diameter are called the Abscisswe. 10. The Latus Rectum, Normal, Subnormal, Conjugate Diameter, and Directrix, are defined as in the Ellipse. 11. An Asymptote is a straight line which approaches nearer to meet a curve, the farther it is produced, but which being produced ever so far does never actually meet it. THE HYPERBOLA. 47 PROP. I. The rectangle by the abscissa AN, NM of the axis is to the square of their semi-ordinate NP as the square of the axis major to the square of the axis minor. k r G P Let DGH be the triangular section through D whose plane is parallel to RAP. Then as in the Ellipse, Prop. 1, PNR is perpendicular both to EF and AN, and is bisected by EF: the axis therefore bisects all its ordinates. Also GH is perpendicular to EF, which bisects it in K. Then AN: NF:: DK: FK And NM: EN:: DK: EK AN x NM: EN x NF NP(P)::D DKi2: EKx FK (KG2):: AC2: BC2 (Def. 3.) 48 THE HYPERBOLA. CoR. 1. PNM': CN2-CA2:: BC: AC2. COR. 2. PN2oc AN x NM o CN2- CA. COR. 3. In the rectangular hyperbola, the rectangle of the abscissa is equal to the square of the corresponding semi-ordinate. (Def. 6.) COR. 4. If AQ be a rectangular hyberbola, AP any B NI C A S other hyperbola having the same axis major, and QPN a common ordinate of the axis, then PN: QN:: BC: AC. Hence as in the Parabola, Prop. xiv, the areas APN, AQN, as also the areas ASP, ASQ, are in the same constant proportion of BC to AC. PROP. IIo The distance SP of any point P in the hyperbola from the focus is to the perpendicular distance PX of the same point from the directrix in the constant ratio of SC to CA. THE HYPERBOLA. 4 49 Let SL touch the circle upon the axis major at L; then SL =BC, from Def. 5; and XLT perpendicular to AMithrough L is the directrix. Produce LS to meet PN produced in K; and join UK. Then PN2: UN2- GA2:: BU2: A C2 (Prop. 1. Cor. 1.) Also SN2: KN':: LS' (B C2): CL2 (AU2) by sim. A s..PN2 +SN2 (SP2):KIN2~U N2 _UA42: B C2: A C2 But N2 + CM2 _ (CA'= CK112 _ CL'2 = KCL'. Hence SP: KL:BC: AC And KL: TN(PX):: SL: ST:: CS: SL (BC) by sim. As. Therefore SP: PX:: SC: AC, 6 THE HYPERBOLA. By the same proof, if LC be produced to L', and the tangent HL' be produced to meet PN in K'; also the directrix L'T'X', and PX' perpendicular to it, be drawn, and CK' joined, HP: PX':: CS: CA. hop. TIII. The diference of the focal distances SP, HP is equal to the axis major. For by Prop. 2, SP: PX SC: AC:: AC: CT. and HP:P: SA C Hence II-P- SP: PX'-PX(XX'):: AC: CT But XX ' T== = 2 CT; therefore HP - SP=2AC. COR.1. HP =2AC+SP. COR. 2. ASx SM= SL2 (Euel. 36. iii.)=BC2. COR. 3. If SV be the semi-latus rectum, AC': BC2:: ASx SM(BC2): SV2 (Prop. 1.) The latus rectum is therefore a third proportional to the axis major and minor. PROP. IV. The focal distance SP B CB' AC - SC x cos. PSN' radius beingo unity. For since (2AC~+ SP)2 = HP', (Prop. 3. Cor. I.) 4AC'2 +4AC x SP+ SP'= SP2 + SH2 + 2-SH x SNT Whence A1Cx S.P- SCx SN=SC_-AC2=BC' Or, since SN= SP x cos. PSN, A C x SP - SC x SP x cos. PSN= BC2, BC2 And SP = AC-SC Cos.PSN' THE HYPERBOLA. 51 PROP. V. The line PT, which bisects the angle SPH, touches the hyperbola at P. Ht~ t A S From EPH cut off PL= SP, and in PT take any point whatever T. Join HT, LT, AST; then LT- ST (Eucl. 4..) In ST take any point Q nearer to S than T is, and join HQ. Then HT is less than HQ + Q ' (Eucl. 20. i.) and taking away ST, HT-ST is less than HQ-SQ. The nearer therefore to S the point Q is taken, the greater is the difference HQ-SQ, which in the limit = HS, when Q coincides with S. But since HT is less than HL + LT, taking away LT, HT-LT or HT- ST is less than HL or HP-SP. It follows, that there is some point Q between 7' and S, where HQ- SQ= HP-SP, and 6-2 THE HYPERBOLA. where consequently the curve cuts ST. The point T is therefore without the hyperbola, and the same being true of every point in PT, except P, PT touches the curve at P. COR. 1. SP and HP make equal angles with every tangent PT. COR. 2. A tangent at the vertex of either axis is perpendicular to that axis. PROP. VI. The perpendiculars from the foci on any tangent intersect the tangent in the circumference of a circle whose diameter is the axis major. In PH take PW=SP, and then HW=A4M; let SW cut the tangent in Y, and join CY. THE HYPERBOLA. Then the triangles SYP, WYP are equal in all respects;.-. z SYP = WYP, and SY is perpendicular to PY. Also SY = YW, and SC = CH;.'. CY is parallel to HW, and = HW= CA; i. e. Y is a point in the circumference. In like manner, if HZ be perpendicular to the tangent, Z is also in the circumference. COR. Join CZ which is parallel to SP, as in the proposition; and let SP produced meet the conjugate diameter in E. Then PECZ is a parallelogram, and PE= CZ= AC. PROP. VII. The rectangle by the perpendiculars from the foci on any tangent is equal to the square of half the minor axis. Let HZ meet the circle again in 0, and join CO; then the right angle YZO is in a semicircle; hence YCO is a diameter and a straight line; also the triangles CSY, CHO are equal in all respects; hence HO=SY, and SYx HZ=HO x HZ= HA x HM (Eucl. 36. III.)=BC2 (Prop. 3. Cor. 2.) SP COR. SY-'BC2x HiP' as in the Ellipse, and SP SP SY H+ SP HP A C + SP' 54 THE HYPERBOLA, PROP. VIII. The semi-axis major is a mean proportional between the distance CN of any ordinate to the axis from the center and the distance CT of the intersection of the axis with the tangent at the extremity of the ordinate. j?' Draw TQ to the circumference perpendicular to AM; and join NQ, CQ. Then TY: SYt P And TZ: HZ "TN PN.T. TYx TZ: SYx HZ(BC2):: TN2: PNT And BC2 C: AC P: PN2 CN- CA But TYx TZ= AT x TIM= QT2 Hence QT2: TN2:: CA2: CN2-CA Compdo. QT2 QN2:: CA2 (CQ2 ): CN2. The triangles QTN, CQN are therefore similar (Eucl. 7. vI.), and z CQN is a right angle; and NQ touches at Q. The tangents TP, NQ have therefore the same subtangent TN. Also CN: CQ(CA):: CQ(CA): CT. THE HYPERBOLA. COR. 1. Draw Pn perpendicular to the axis minor, and produce the tangent PT to meet the axis minor in t. Then Ct: PN(Cn):: CT: NT:: CQ2: NQ2 (CN-2 CQ2):: BC2: PN2 (Prop. 1. Cor. 1.) Whence Ct: BC:: BC: Cn, as for the axis major. COR. 2. Let DL, touching the conjugate hyperbola at D, intersect the axes in L, t; draw DR, Dn, perpendicular to the axes. THE HYPERBOLA. Then Cu X nt= Cu2- Cu X Ct (Euc. 2. ii.) =Cu2-BC2. But CL.x.CL x:Dn (CB):: Ct: t CR:Dn2:CnuX Ct::BC2:A C2 CuXut:Cn2-BC2:Du2(Prop. I.Cor. 1.) Hence CL xCR =A C2 Coin.3 CR x RL: CL x CR? (A C2):RL: CL::DR (Cu):Ct DR 2 Cu x Ct (B C2). Hence CR xRL: DR':: A C2: BC2. Piop. IX. The rectangle by the normal PO to either axis and the perpendicular PF upon the conjugate diameter is equal to the square of half the other axis. THE HYPERBOLA. 57 n B,R _B.~~~' L N P 0 C T A S N Let PN produced meet the conjugate in R. as in the Ellipse, Prop. 9. PO x PF = PN x PR= Cn x Ct=BC2. Similarly Po x PF= CN x CT= CA2. Then CoR. 1. As in the Ellipse, the subnormal ON L 2Al C -=.C x CN and on = L x Cn, L being the latus rectum. COR. 2. As in the Ellipse, if OL be drawn perpendicular to the focal distance SP, PL = half the latus rectum. THE HYPERBOLA. PROP. X. If tangents be drawn at the vertices of the axes, the diagonals of the rectangle so formed are asymptotes to the four curves. r II/ -PR - - 7 \A TV e / p Let NP meet the diagonal CE produced in Q. Then NQ: CN:: AE2 (BC2): AC2 o:: p2p ~ CN2_ CA2 (Pr. 1. Cor. 1.) Now as CN increases, the ratio of CN2 to CN1V- CA2 continually approaches to equality; but CN2- CA2 is never actually equal to CN2, if CN be ever so much increased. Hence NP is always less than NVQ, but approaches continually nearer to equality with it. By the same proof, CQ is an asymptote to the conjugate hyperbola BP'. COR. 1. The two asymptotes make equal angles with the axis major, and with the axis minor. THE HYPERBOLA. CoR. 2. The line AB joining the vertices of the conjugate axes is bisected by one asymptote and parallel to the other. It is bisected by CQ, because the diagonals of the rectangle BCA E bisect each other; and it is parallel to Ce, because EO= OC and EA=Ae (Eucl. 2. vi.) COR. 3. All lines perpendicular to either axis and terminated by the asymptotes are bisected by the axis. COR. 4. In the rectangular hyperbola, the asymptotes are at right angles to each other. PROP. XI. If any line Qq perpendicular to either axis be terminated by the asymptotes, the rectangle of the segments, into which the curve divides it, is equal to the square of half the conjugate axis. For NQW: CN2 NP'": 2N- CAe (Prop. 10.).. NQ- NP2: CA2::NP2: C CA2 BC2: CA2 (Prop. 1. Cor. 1.) Hence NQ2- NP2 or QP x Pq=BC2, and is therefore the same wherever in the curve the point P is taken. 60 THE HYPERBOLA. PROP. XII. If any line whatever Rr, making a given angle with either asymptote, cut the curve in P, the rectangle by the segments RP, Pr, is invariable. x Through any other point W of the curve draw Zz parallel to Rr, and through P, W, draw Qq, Xx perpendicular to the axis. Then, by similar triangles, RP: QP:: ZW: XW Pr: Pq:: Wl: Wx,.. RPxPr: QPxPq:: ZWx Wx: XW x Wx But QPxPq=XWx Wx (Prop. 11.). RPxPr=ZWx W%. COR. 1. Let Er move parallel to itself until the points P, p, coincide, as at E. Then LEK touches at E, and RPx Pr= LE x EK. Also by the same proof Rp xpr =LE x EK. THE HYPERBOLA. 6 1. CoR. 2. RP x Pr =1?- p xpr, or RP x Pp + RPxpr = RPxpr ~ Pp xpr (Euci. 1. ii.): whence RP=pr. COR. LE= EK, and RP x Pr= LE'. COR. 4. Let the diameter CE cut 1?r in V: then VJ?== Vr, by similar triangles, and RP=pr; whence PV= VP; i. e. a diameter bisects all it's own ordinates. PROP. XIII. If from any point P in the curve straight lines PD, PH, be drawn parallel to the asymptotes, their rectangle is invariable. Draw the tangent LPI( and Qq perpendicular to the axis. Then, by similar triangles, PI: PQ:: AO:A4E PD: Vq:: OE: AE P,. D x PH: PQ x Pq:: A02: AE2 But PQ x Pq=AE 2(Prop. 11.).. PD x PH =A 02 = (AC'+ BC') a constant quantity. 62 THE HYPERBOLA. CoR. 1. Draw DF perpendicular to CH; then the triangle DCF is always similar to itself. Hence CD x CH and DF x CII are in a constant ratio; and therefore DFx CH or the parallelogram DH is a constant quantity. COR. 2. The triangle CLK is a constant quantity, being double of the parallelogram DI-. CoR. 3 PHOC. PROP. XIV. If a straight line PLD be drawn parallel to one asymptote, and terminated by the conjugate hyperbolas, it is bisected by the other asymptote. ~-~a/ I D - For CL x LP= A O (Prop. S13) = BO2= L x LD;.e. PL = D. CoR. 1. If ab touch at P, it is bisected in P (Prop. 12. Cor. 3.); therefore Ca=2 CL (Eucl. 2. vi). For this reason a tangent dD at D must meet Ca THE HYPERBOLA. 63 in the same point a. Also since PL- LD, Cb =Cd, by similar triangles; and Cb or Cd=2LP =PD. Hence CD is parallel to ab, and CP to ad (Eucl. 33. I.) COR. 2. Draw b4Ic touching at I, and from c draw cGd touching at G. Then cb, cd being bisected at IC, G, GK is parallel to bd and bisected in I. Hence, as in the last CorY, Cd= Cb, and ad, cd, meet Cd in the same point d. Also IK = 1 Cb LD, whence CI= CL. (Prop. 13. Cor. 3.) The triangles CIK, CLD are therefore equal in all respects, and in like manner the triangles CLP, CIG. Hence PCG, DCK are straight lines and conjugate diameters; they are likewise bisected by the center C. COR. 3. The tangent ad is equal and parallel to the diameter PG, and ab to the conjugate DK. CoR. 4. The figure abed is a parallelogram, of which CPaD is a fourth part. COR. 5. In the rectangular hyperbola, every diameter PCG is equal to it's conjugate DCK. PROP. XV. The parallelograms formed by tangents at the vertices of any pair of conjugate diameters have all the same area. For the parallelogram CPaD = 2 A CPa = A Cab (Eucl. 38. i.) a constant quantity (Prop. 13. Cor. 2.) 64 THE HYPERBOLA. COR. Draw PF perpendicular to CD; then the parallelogram CPaD = CD x PF. Now when the tangents are drawn at A and B, CD coincides with CB, and PFwith AC. Hence CDxPF=ACxBC. PROP. XVI. The diference of the squares of any two conjugate diameters is equal to the difference of the squares of the axes. xB L X --- — / -^ / C A Draw CZX perpendicular to AB and PD, Then CP2 - CD2 = PX- DX2 = 4PL x LX(Eucl. 8. II.) But LX: OZ:: CL: CO, by sim. AS.:AO:PL (Prop. 13. Cor. 3.) 4PL x LX(CP'- CD)) = 4A0 x OZ =. C2 BC2. COR. As in the Ellipse, Prop. 11. Cor. 2, the rectangle by the focal distances is equal to the square of the semi-conjugate diameter. PROP. XVII. The rectangle by the abscissee PV, VG, of any diameter PG is to the square of their semi-ordinate QV as the square of CP to the square of the semiconjugate CD. THE HYPERBOLA. 6 655 Let QQ' meet the asymptotes in 1?, r. Then /~~~~~ RQ x Qr or RV2 - QV2= PL' (Prop. 12. Cor. 3.) =CD' (Prop. 14. Cor. 3.).. Q72=RV2-PL2; But UP2 UP2: V: R2 PL2 Divdo. CV2- UCP2: RV2- PL2 UP2 PL 2 Or PVx VG: QV2 UP2 CD2. CoR. 1. QV2oc PV x VG oc CV2 -CP2. COR. 2. Let QOX, parallel to PG, cut CD in 0 and the opposite hyperbola in X. Then V2_- CP2 CP2 QV2: UJY Compd'. CV2: CD2 '+ QV2 Or Q0:U CD2+ C0I And so X02: CD2+ C02I Hence QO=OX; i. e. every line terminated by two opposite hyperbolas is bisected by that diameter to whose conjugate it is parallel. COR. 3. C V2 oc CD2 + Q V; and Q O'o CD2'+ C02. 7 66 66 ~~~THE HYPERBOLA. PROP. XVIII. If any line AB intersect a diameter RQ produced in F, and CD be the semi-diameter. parallel to AB, then AF xFB: QF xFR: CD':C' z R N V Let CPV be the diameter bisecting AB, and draw QI parallel to AB; then (Prop. 17. Cor.:3.) CD2+A4V2: CD2+ Q12::CV2: Cli Divd -AV2QI2: CD'+1:C2CI2: C12 FJ72-Q12:QI~by sim-A~ Hence AV2- FV2: CD2::FV - Q12I: Q12 (E. I19. v.) CF2-CQ2: CQ~bysim.tAs.Or AFxFB:CD':QF xFR:CQ2. COR. 1. If any other line MN cut AB in F, and. CH be the semi-diameter parallel to MN, then AFxFB: AfFxFN:.: CD 2:CH 2. THE HYPERBOLA. 67 COR. 2. Draw FKL parallel to VG, and cutting the opposite hyperbolas; also draw AZ parallel to VG: then (Prop. 17. Cors.) AZ" (O"): KO:: CD+ CZ2: CD + CO2 DivdO. FO_ -KO2 CZ _ C0: I KO: CD2 + CO Or KFxFL: AFxFB:: CP2: CD2.K. KFx MFL: MFx FN:: CP2: CH2. CoR. 3. If AB, KL, MN, move parallel to themselves, and intersect in F either within or without the hyperbola, the rectangles AF x FB, KF x FL, MF x FN, have always the same ratio to each other as the squares of the diameters to which the lines are respectively parallel. COR. 4. If from any point R tangents RWI, RZ be drawn to either of the opposite hyperbolas, these are to each other as the diameters to which they are respectively parallel. As in the Ellipse, Prop. 14. Cor. 3, the rectangles FK x FL and FM x FN, and therefore the squares of RW and RZ, are as the squares of the parallel diameters. 7-2 THE HYPERBOLA. PROP. XIX. If a tangent QT at the extremity Q of any ordinate QV meet the diameter CP to which the ordinate belongs in T; then CP is a mean proportional between CV and CT. At the extremities P, G of the diameter let the tangents PL, GM meet the tangent QT in L, M. Then MQ: MG:: LQ: LP (Prop. 18. Cor. 4.) And therefore MQ LQ:: MG: LP Hence VG: PV:: TG: TPbysim. A'..'. VG + P:VG-PV:: TG+ TP: TG-TP Or CV: 2CP:: 2CP: CT. COR. 1. A tangent at the other extremity of the ordinate would meet the diameter CP in the same point T. COR. 2. PVx VG= CV - CP2= CV2- CVx CT =CVx VT. THE HYPERBOLA. 69 PROP. XX. To determine the diameter of curvature and the chords through the center and focus of an hyperbola. As in I. II. III. the Ellipse, Prop. 17. 2 CD2 The chord PI through the center =2 C-. 2 CD2 SP' The diameter PK = PF or = L xS. 2 CD2 The chord PL through either focus= A C 70 THE HYPERBOLA. PROP. XXI. If BDE be a cone, from which an hyperbola RAP is cut; DGEH a circular section perpendicular to the axis of the cone; BGH a triangular section through the vertex B parallel to RAP; BTG, BTH planes touching the cone along the slant sides BG, BH; then CO, CQ, the common sections of BTG, BTH, with the cutting plane RAP, are the asymptotes of the hyperbola. M B L Draw BL parallel to DE, meeting AM in L. Then the axes of the hyperbola being in the proportion of BF to FH, the z GBH or the equal z OCQ (Eucl. 16. xi.) is the angle between the asymptotes. Now by similar triangles ALB, BFE and CLB, BFT, AL: CL:: TF: FE and therefore AC: CL:: TE: FE. In like manner by similar triangles THE HYPERBOLA. 71 MLB, BFD, and CLB, BFT, ML: CL:: TF: DF and therefore CM: CL:: TD: DF. But by the property of the circle*, TE: FE:: TD: DF. Therefore CA= CM (Eucl. 9. v.). Hence C is the center of the hyperbola, and CO, CQ are the asymptotes. * If DLE be a circle; T any point in the diameter produced; TL a tangent at L; LF perpendicular to the diameter; P any other point in the circumference; then TP: PF is in the constant ratio of TL: LF. P T D F C E Join CL, CP. Then CT: CL (CP):: CL (CP): CF (Eucl. 8. vi.), and the As TCP, FCP, are similar, (Eucl. 5. vi.) Hence TP: PF:: CP (CL): CF:: TL: LF. Therefore TD: DF and TE: EF are in the same ratio. ON THE SECTIONS OF ~tw tonoti+ DEFINITIONS. 1. THE solids generated by the revolution of conic sections about their axes are called Conoids. 2. If a Parabola so revolve, the solid is called a Paraboloid. 3. If an Ellipse revolve about it's axis major, the solid is called a Prolate Spheroid; and if it revolve about the axis minor, an Oblate Spheroid. 4. The solid generated by the revolution of an Hyperbola about it's axis major is called an Hyperboloid. SCHOLIUM. Every ordinate to the axis of revolution describes a circle whose center is in the axis; therefore all sections, whose planes are perpendicular to the axis, are circles. Also the revolving plane, since it always passes through the axis, is perpendicular in every position to the planes of all the circular sections. THE CONOIDS. 73 PROP. I. If a paraboloid be cut by a plane parallel to the generating plane, the section is the same as the parabola which revolves. Let FAE be that position of the generating parabola which is perpendicular to the cutting plane RAP, and FPER any circular section. Then both the planes FPE, RAP are perpendicular to FAE, and therefore PNR is perpendicular to FAE and to the lines AN, EF. Also AN is parallel to the axis of the solid. If L be the latus rectum of the revolving parabola, L x AN = EN x NF (Parabola, Prop. 9.) = NP2. Wherefore the curve RAP is the parabola whose latus rectum is L. 74 THE CONOIDS. PROP. II. Any other section of a paraboloid is an ellipse. L M er Let the generating plane AiMBF be perpendicular to the cutting plane AMP, AM their common section, which cutting the parabola in one point A, must cut it again, as at M (Parabola, Prop. 3. Cor. 1.). Also let BV be the diameter to which AM belongs, and draw QyR an ordinate of the axis. Then (Parabola, Prop. 12. Cor. 2.) ANx NM: ENx NF:AV: A M QVx VR Or ANx MNx: NP::N Ay: L x BV. The curve is therefore an ellipse, whose major axis is AM, and whose semi-axis-minor is a mean proportional between L and BV. PROP. III. All sections of a spheroid, except those which are perpendicular to the axis, are ellipses. THE CONOIDS. 75 Let ABD represent a prolate spheroid, CB the V MZL A semi-axis minor of the revolving ellipse, CD a semidiameter parallel to AM: then (Ellipse, Pr. 14. Cor. 1.) ANx NM: ENx NF (NP2):: CD: CB2. The curve is therefore an ellipse, whose axis major is AIi, and whose axis minor is to A1M as CB to CD. If ABD represent an oblate spheroid, the circular section EPF must be made perpendicular to BC; and by the same demonstration APM is an ellipse, whose axis minor is AM. COR. 1. All sections whose planes are parallel to each other, are similar ellipses, because CB and CD remain unaltered. COR. 2. If ABD be an oblate spheroid, and KL touch the generating ellipse at the extremity K of the axis major, all sections made by planes passing through KL are similar to the generating ellipse. For KL is parallel to CB and therefore the plane of every such section is parallel to the generating plane in some one position of it. 76 THE CONOIDS. PROP. IV. If an hyperbola and its asymptote revolve about the axis major, the sections of the hyperboloid and cone so generated, made by any plane, are similar figures. /L D B X R Through. draw XY parallel to the axis minor CB. Then QN:GN:: QA: XA And NR: NH:: AR: AY. QN x NR: GNx NH:: QAxAR: XA x AY Or QN x NR: NO:: CD2: CB' (HYP. P. 11 & 12.):: AN x NM: NP2 (P. 18. Cors.) i. e. the rectangle by the abscissae has to the square of the semi-ordinate the same ratio in both cases. But that ratio in the section of the cone is a constant ratio, and it is therefore the same constant ratio in the section of the hyperboloid. Also the conclusion is the same whether AM cut the hyperbola AFM only, as in the figure, or the two opposite hyperbolas. Hence, THE CONOIDS. 77 i. If QAR cuts both asymptotes, the section APM is an ellipse. ii. If QR is parallel to one asymptote, APIi is a parabola. III. If QR cuts one asymptote and also the other produced backwards, APM is an hyperbola; and if QR be parallel to the axis major, APM is similar to the generating hyperbola. COR. If the cone be cut by a plane which touches the hyperboloid at any point, the section is an ellipse, whose axis minor is always equal to the axis minor of the generating hyperbola and whose axis major =2 CD.