CONIC SECTIONS
A CONIC section is the curve in which a plane cuts a cone, which is defined in Euclid’s Elements as “a solid figure described by the revolution of a right-angled triangle about one of the sides containing the right angle, which side remains fixed.” Though the properties of conic sections can be investigated from this point of view, we consider it more advantageous to start from the following definition, which is derived from one of the properties which all conic sections possess in common.
Definition.— If a point move in such a way in a plane that its distance from a fixed point in the plane always bears a fixed ratio to its distance from a fixed straight line in the plane, the point will trace out a conic section.
The curve is called an ellipse if the distance from the fixed point is less than, a parabola if it is equal to, and a hyperbola if it is greater than, the distance from the fixed straight line.
The fixed point is called a focus, and the fixed straight line a directrix of the curve.
The fixed ratio of the distance from the focus to the distance from the directrix is called the eccentricity of the curve.
The discovery of the conic sections seems to have originated in the school of Plato. It is probable that the followers of that philosopher were led to the discovery of these curves, and to the investigation of many of their properties, in seeking to resolve the two famous problems of the duplication of the cube and the trisection of an angle, for which the artifices of the ordinary or plane geometry were insufficient. Two solutions of the former problem, by the help of the conic sections, are preserved by Eutocius, and are attributed by him to Menaechmus, the scholar of Eudoxus, who lived a little after the time of Plato.
The writings of Archimedes that have reached us show that the geometers before his time had advanced a great length in investigating the properties of the conic sections. This author expressly mentions numerous demonstrations of preceding writers, and often refers to properties as known to mathematicians. His own discoveries are worthy of the most profound and inventive genius of antiquity. In the quadrature of the parabola he gave the first and the most remarkable instance that has yet been discovered of the exact equality of a curvilinear to a rectilinear space. He determined the proportion of the elliptic spaces to the circle; and he invented many propositions respecting the mensuration of the solids formed by the revolution of the conic sections about their axes.
It is chiefly from the writings of Apollonius of Perga that we know how far the ancient mathematicians carried their speculations concerning these curves. (See Apollonius.) His work on the conic sections, written in eight books, was held in such high estimation by the ancients as to procure for him the name of the Great Geometer. The first four books of this treatise only have come down to us in the original Greek; in these the author claims no further merit than that of having collected, amplified, and arranged the discoveries of preceding mathematicians. One improvement he introduced deserves particular notice. The geometers who preceded him derived each curve from a right cone, which they conceived to be cut by a plane perpendicular to its slant side; and Apollonius was the first to show that all the curves are produced from any sort of cone, whether right or oblique, according to the different inclinations of the cutting plane. An Arabic MS. discovered in 1658, and two others brought from the East a few years later, contain the first seven books of the treatise of Apollonius; the eighth book appears to be irrecoverably lost. Dr Halley, who in 1710 put forth a correct edition of the Conics of Apollonius, guided by the account of the different books preserved by Pappus, has given a very able restoration of the eighth book. The last four books of the Conics of Apollonius, containing the higher or more recondite parts of the science, are generally supposed to be the fruit of the author’s own researches, and do much honour to his geometrical skill and invention. Even in our times the whole treatise must be regarded as a very extensive work on the conic sections,—modern mathematicians having made few discoveries of which there are not some traces to be found in the work of Apollonius.
The geometers who followed Apollonius seem to have contented themselves with commenting on his treatise. It was only about the middle of the 16th century that the study of this branch of mathematical science was revived; since that time no part of mathematics has been more cultivated, or has fc been illustrated by a greater variety of ingenious writings. The applications of the properties of these curves in natural philosophy have, in modern times, given them a degree of importance which they did not formerly possess; and a knowledge of them is now indispensable to any one who seeks to acquaint himself with the remarkable physical discoveries of the present age.
Apollonius and all the earlier writers on conic sections derived the elementary properties of the curves from the nature of the cone; but in 1665 Dr Wallis, in his De Sectionibus Conicis, laid aside the consideration of the cone, deriving the properties of the curves from a description in piano. In the following treatise, as has been already stated, the properties of the conic sections are deduced from their description in a plane.
An assemblage of points, all of which satisfy some condition, whether or not they form a continuous curve, is called a locus ; as, for example, we could define a circle as the locus of a point whose distance from a fixed point is constant, or a conic section as the locus of a point whose distance from a fixed point always bears a constant ratio to its distance from a fixed straight line.
The following is a proposition which is very useful in the discussion of the properties of conic sections.
Lemma. —The locus of a point in a plane whose distances from two fixed points in the plane always bear a constant ratio to one another is a circle.
Let A, B (fig. 1) be the two fixed points, m the common ratio, and P any point on the locus.
Divide BA internally and exter∙ in the given ratio, so that CA DA_ _PA CB - DB ~ m, - PB *
Join PC, PD.
Then, because PA: PB = CA: CB,
PC is the internal bisector of the angle APB (Eucl. vi. 3); and because PA: PB = DA: DB ,
PD is the external bisector of the angle APB (Euch vi. A).
Therefore the angle CPD is a right angle, and the locus of P a circle described on the line CD as diameter.
PART I.—THE PARABOLA.
Definitions.
A straight line perpendicular to the directrix, terminated at one extremity by the parabola, and produced indefinitely within it, is called a diameter.
The point in which the diameter meets the parabola is called its vertex.
The diameter which passes through the focus is called the axis of the parabola; and the vertex of the axis is called the principal vertex. Corollary. A perpendicular drawn from the focus to the directrix is bisected at the vertex of the axis.
A straight line terminated both ways by the parabola, and bisected by a diameter, is called an ordinate to that diameter.
The segment of a diameter between its vertex and an ordinate, is called an abscissa.
A straight line meeting the parabola in two points P, Q, is called a chord.
The focal chord which is bisected by a diameter is called the parameter of that diameter.
The limiting position of the chord PQ, which it assumes when the point Q moves up to and coincides with P, is called the tangent at P.
A line through P at right angles to the tangent is called the normal at P.
Proposition T.
To find where a parabola of given focus and directrix is cut by a straight line parallel to the directrix.
Let S (fig. 2) be the focus, and XK the directrix. Draw SX perpendicular to KX, and bisect SX in A; draw AZ at right angles to SX, and equal to AS. Join XZ.
Let QN be any straight line parallel to the directrix, cutting XZ in Q and the axis in N.
With centre S and radius equal to QN, describe a circle cutting QÑ in P and P'; these will be points on the parabola, because
SP: XN = QN: XN = ZA: AX = 1: 1.
.∙. SP = XN = distance of P from the directrix.
It is clear that if the point P exists, the point P' on the opposite side of the axis also exists, and therefore the parabola is symmetrical with respect to the axis.
Again, the point P will exist, or, in other words, the circle will cut QN, as long as SP or QN is greater than SN, which is always the case as long as QN lies on the same side of AZ as the focus.
The whole of the curve, therefore, lies to the right of AZ, and branches off to an infinite distance from the directrix.
Prop. II.
To find where a parabola of given focus and directrix is cut by a straight line parallel to the axis.
Let S (fig. 3) υe the focus, and XK the directrix; draw AY bisecting SX at right angles.
Let KQ be any line parallel to the axis cutting the directrix , in K. Join SK cutting AY in Y, and draw YP at right angles to SK cutting KQ in P. P will be a point on the curve.
It is easily shown that the triangles SPY, KP Y are equal in all respects, and that SP = PK.
Now, the point P will exist, or, in other words, YP will intersect KQ, for all positions of KQ.
The parabola, therefore, branches off on either side of the axis to an infinite distance, and is cut by a straight line parallel to the axis in one point only.
It appears from what has gone before that the general shape of the curve is of the form given in fig. 4, which shows the focus, directrix, and axis.
It can be easily seen that all points within the curve are nearer to the focus than to the directrix, and all points without the curve are nearer to the directrix than to the focus.
A parabola can be described mechanically in the following planner (see fig. 5) :—
Suppose a bar KQ to move always parallel to itself, with its end K on a line at right angles to it; then, if a string of length equal to KQ, attached to the bar at Q, and also to a fixed point S, be always kept tight by means of a ring P sliding on KQ, a pencil at P would, trace a parabola whose focus is S and directrix XK.
Prop. III.
If a chord PQ (fig. 6) intersect the directrix in Z, then SZ will be the external bisector of the angle PSQ.
Join SP, SQ, and draw PM, QN perpendicular to the directrix.
Then, because the triangles PMZ, QNZ are similar,
PZ: QZ = PM: QN = SP: SQ.
.∙. (Eucl. vi. A) SZ bisects the external angle of the triangle PSQ.
Corollary.— If the point Q move up to and become coincident with P, or if, in other words, the chord PQ become the tangent to the parabola at P, then the angle PSZ will become a right angle.
Prop. IV.
The tangent at any point of a parabola bisects the angle between the focal distance of the point and the perpendicular from the point on the directrix.
Let PZ (fig. 7) be the tangent at P, meeting the directrix in Z; then, if PM be drawn perpendicular to the directrix, it is easily seen that the two triangles SPZ, MPZ are equal in all respects, and the angle SPZ equal to the angle MPZ.
If SM be joined, it can be shown that it is bisected at right angles by PZ, and that its middle point is the point Y in Prop. ii.
The line AY, it will be observed, is the tangent to the parabola at the vertex A.
It appears, therefore, that the locus of the foot of the perpendicular from the focus on the tangent at any point is the tangent at the vertex.
It can also be seen that, if the tangent at P meet the axis in T, then SP= ST. For the angles STP, SPT are each equal to the angle MPT, and therefore (Euch i. 6) SP, ST are equal.
It may further be remarked that, if 0 be any point in the tangent at P, then the triangles SPO, MPO are equal in all respects.
If PN be drawn perpendicular to the axis to meet it in N, then it will be seen that PN = 2AY and TN = 2AN = 2AT.
Now, in the right-angled triangle TYS,—
TA: AY = AY: AS (Euch vi. 8), and therefore YA 2 = TA . AS
Therefore PN 2 = 4YA 2 = 4TA. AS
= 4AS.AN.
If the normal PG be drawn meeting the axis in G, then the triangles PNG, YAS are similar, and therefore—
NG-AS = PN:YA = 2:1
.∙. NG = 2AS
Prop. V,
To draw a tangent to a parabola at a point on the curve.
First Method.— Take a point T in the axis (fig. 7), such that ST is equal to SP, and join TP. Then STP will be the tangent at P.
Second Method.— Draw SZ at right angles to SP, meeting the directrix in Z. ZP is the tangent at P.
Third Method— On SP as diameter describe a circle; this will touch the tangent at the vertex AY in a point Y. YP is the tangent r,t P.
Prop. VI.
To draw a pair of tangents to a parabola from an external point.
First Method.— Let 0 (fig. 8) be the point. Join OS, and with centre 0 and radius OS describe a circle cutting the directrix (which it will always do) in Μ and M'.
Draw MP, M'P', parallel to the axis, cutting the parabola in P and P'. Join OP, OP ’. They are tangents to the parabola at P and P'.
Join SP, SP'.
In the triangles OPS, OPM,
OP, PM = OP, PS each to each, and OM = OS by the construction; therefore the angles OPS, OPM are equal, and therefore OP is a tangent to the curve at P (Prop, iv.)
Second Method.— Let 0 (fig. 9) he the point. Upon OS as diameter describe a circle cutting the tangent at the vertex (which it will always do) in Y and Y'.
Join YO, YO, and, if necessarj produce them to meet the curve in P, P'. They will be tangents to the curve at P, Pi
Because OYS is a semicircle, therefore the - angle OYS is a right angle, and therefore YO is a tangent to the parabola (Prop, iv.)
Prop. VII.
If OP, OP' (fig. 8) be tangents to the parabola at P, P', then the triangles OSP, P'SO are similar, and SO 2 = SP. SP'.
Because the angle OSP=angle 0MP=angle 0M'P'=angle OSP', and the angle OPS = angle 0PM= angle SMM ’ =⅛ angle SOM' (Eucl. ιii. 20) = angle SOP' (Prop, vi.); therefore the remaining angles OPS, P'OS are equal, and the triangles OPS, P'OS similar; and therefore (Euch vi. 4)
SP:SO = SO:SP', or
SO 2 - SP. SP'.
Pkop. VIII.
If 0 be the intersection of OP, OP', the tangents to the parabola at P and P', then OV drawn parallel to the axis will bisect PP'. From fig. 8 we see that, if a line through 0 meet MM' in Z, MM' is bisected in Z; and, because MP. Z0V, and M'P' are parallel, therefore PP' is bisected in V.
Prop. IX.
The angle between two tangents is equal to half the angle subtended at the focus by the chord of contact.
From fig. 9 we see that angle YOY' = angle YSY' = angle YSA - angle Y'SA = ⅜ anglo PSA - ⅛ angle P'SA = ⅛ anglo PSP'.
It may be shown, by means of this proposition, that the circle which is described about the triangle formed by any three tangents to a parabola passes through the focus.
Prop. X.
If OV (fig. 10) meet the parabola in Q, the tangent at Q is parallel to PP', and OV will be bisected in Q.
Draw the tangent RQR' at Q, meeting OP,OP r in R, R ’. Join PQ, and let RW be drawn parallel to OQV meeting PQ in W.
Then PW = WQ (Prop, viii.) Therefore OR = RP (Euch vi. 2). Similarly we can shew that OR' ≡ R'P'. Therefore
OR:RP≈OR':R'F, and therefore RR' is parallel to PP' (Euch vi. 2); and also .∙. OQ = QV. OQ:QV-OB:BP.
Prop. XL
If V be the middle point of a chord PP', and Q be the point at which the tangent is parallel to PP', then PV 2 -=4SO. QV.
Suppose in fig. 10 SQ, SR joined, and PO produced to meet the axis in T.
Then angle ORQ = J angle QSP (Prop, ix.) =angle QSR (Prop, yn.), and angle QRS =angle RPS (Prop, vii.) =angle STP (Prop, iv.) = angle QOR (Eucl. i. 29); therefore the two triangles 00E, SQR are similar, and
OQ:QR≈QR;SQ
therefore QR 2 = SQ. OQ.
But PV=2QR, and OQ=QV, therefore PV 2 =4SQ. QV.
Pkop XII.
The^parameter of the diameter QV is
If the tangent at Q (fig. 11) meets the axis in T,
SQ = ST = QV.
Therefore the equality PV 2 = 4SQ. QV becomes PV 2 = 4SQ 2 therefore PV-2SQ
or PP' = 4SQ.
Prop. XIII.
If POP' (fig. 12) ⅛e any chord, and OR be drawn parallel to the axis through any point O to meet the curve in R, then PO. OP'≡ 4SQ . Rθ, where 4SQ is the parameter of diameter PP'.
Draw RW parallel to PP' to meet QV in W.
Then PV 2 =4SQ.QV
and RW 2 = 4SQ.QW, and RO = W V and RW=OV.
Therefore PO. OP'= PV 2 - OV 2 (Eucl.ii. 5).
= PV 2 -RW 2
= 4SQ.QV-4SQ.QW
≈4SQ. WV=4SQ. RO Prop. XIV.
If POP', pθp' be any two chords intersecting in 0, and Q, q are the points of contact of the tangents parallel to them, then PO . OP': pθ. Op'=SQ: Sq.
By Prop, xiii., PO . OP'=4SQ. RO and similarly pθ. Op'=4Sq. RO. Therefore
PO . OP': pθ . Op'=4SQ. RO: 4Sq. RO = SQ: Sq Pkop. XV.
The area included between any chord of a parabola and the curve is two-thirds of the area of the triangle formed by the chord and the tangents to the curve at its extremities.
It is easily seen in fig. 10 that the area of the triangle ORR' is one quarter the area of the triangle OPP', and therefore one half the area of the triangle QPP'.
Now if we draw tangents where RW, R'w meet the curve, we shall have two pairs of triangles whose areas are in the ratio 1: 2, and so we may go on indefinitely. The sum of all the external triangles will be half the sum of all the internal triangles.
The sum of the external triangles is the curvilinear area OPQPi and the sum of the internal triangles is the curvilinear area PQpζ Therefore
2 × area OPQP'=area PQP'
.∙. 2 × triangle OPP'=3 area PQP ’ or area PQP =f triangle OPP'.
PART II.-THE ELLIPSE.
Definitions.
A straight line passing through the centre, and terminated both ways by the ellipse, is called a diameter.
The extremities of a diameter are called its vertices.
The diameter which passes through the foci is called the transverse axis, also the major axis.
The diameter which is perpendicular to the transverse axis is called the conjugate axis, also the minor axis.
Any straight line not passing through the centre, but terminated both ways by the ellipse, and bisected by a diameter, is called an ordinate to that diameter.
Each of the segments of a diameter intercepted between its vertices and an ordinate, is called an abscissa.
A chord, tangent, and normal are defined exactly in the same words as in the case of the parabola.
Prop. I.
To find where an ellipse of given focus, directrix, and eccentricity is cut by a straight line parallel to the directrix.
Let S (fig. 13) be the focus, XK the directrix, and ¢ the eccentricity.
Draw SX perpendicular to XK, and divide it internally and externally in the ratio e : 1 in the points A, A', so that SA: AX = SA>: A'X = e: 1.
It is clear that A and A' will both be on the same side of X as S is, because the eccentricity e is less than 1.
Draw AZ at right angles to SX and equal to AS; and join XZ.
Let QN be any straight line parallel to the directrix, cutting XZ in Q and the axis in N. With centre S and radius equal to QN describe a circle cutting QN in P and P'; these will be points on the ellipse; for we have
SP: XN = QN :XN≈ZA: AX = SA: AX≈e: 1.
It is clear that if this point P exists, the point P' on the opposite side of the axis also exists, and therefore the ellipse is symmetrical with respect to the axis. Again, the point P will exist, or, in other words, the circle will cut QN as long as SP or QN is greater than SN, which is always the case as long as the angle QSN is greater than half a right angle. Now if SL, A'Z' be drawn at right angles to SX, cutting XZ in L, Z', then (Eucl. vi. 4)
Z'A': A'X = ZA: AX.
But SA': A'X = SA: AX
and SA = Z A
SA'= Z'A'
Prom which it is easily seen that the angle Z'SA' is half a right ancte. The whole curve therefore lies between the two lines AZ, A'Z'.
Prop. II.
To find where an ellipse of given focus, directrix, and eccentricity is cut by a straight line perpendicular to the directrix.
Let S (fig. 14) be the focus, XK the directrix, and e the eccentricity.
Draw SX perpendicular to XK, and divide SX in A, A', so that
SA: AX = SA': A'X=β: 1; and draw AR, A'R' at right angles to SX.
Let KQ be any line parallel to the axis cutting the directrix in K. Join SK cutting AR, A'R' in R, R', and upon RR' as diameter describe a circle cutting KQ in P, P'; these will be points on the curve.
For SR: RK = SA: AX = e: 1 = SA': A'X
≈SR':R'K.
Therefore by the Lemma in the introduction
SP: PK = SP': P'K = SR: RK = e: 1.
Therefore P and P' are points on the ellipse.
Now, if the point P exists, the point P' also exists, and it is easily seen that the middle point L of PP' lies on a straight line bisecting AA' at right angles. The curve therefore is symmetrical, not only with respect to the axis AA', but also with respect to this line bisecting AA'at right angles. The middle point C of AA' is called the centre of the curve, from the fact that every straight line through C is bisected at the point.
It is evident from what has preceded that, if we measure CS'= CS, and CX' = CX, in the opposite direction to CS and CX, and draw X'K' parallel to XK, the ellipse might be described with S' for focus, X'K' for directrix, and eccentricity e.
The ellipse, therefore, has two foci and two directrices.
Now, since SP: PK = SP': P'K = e: 1.
Therefore SP + SP': PK +P'K=β: 1, but PK + P'K = 2KL=2CX.
.∙. SP + SP'=2e .CX.
Now, it is easily seen that SP'≈S'P; therefore SP + S'P = 2e. CX; or the sum of the focal distances of any point on the curve is constant.
Again SA: AX = SA': A'X = e: 1
therefore SA + SA': AX + A'X=β: 1.
But AX + A'X = 2CX,
.∙. SA+SA' = SP + S'P.
And SA'=S'A, therefore SP+S'P = SA + S'A
-AA'; or the sum of the focal distances of any point on the ellipse is equal to the major axis.
The point P will exist, or, in other words, the circle on RR' as diameter will intersect KQ, if OL is less than OR when 0 is the middle point of RR'.
Now SX: SK = CA: OR,
.∙. OR.SX≈SK.CA; and OL=OC+CL≈KX. ∣ξ + KX.
Therefore the point exists if
KX. SC + KX. SX<SK. CA; i.e., if KX.CX<SK.CA; if KX 2 .CX 2 <SK 3 .CA2 j <(SX 2 + KX 2 ).CA s j if KX 2 .(CX 2 -CA 2 )<SX 2 .CA 2 5 if KX≈<≡g^½orifKX<CB, when CX 2 - CA 2 : CA 2 =SX 2 : CB 2 .
When KX has this value the points P, P ’ coincide in CL, say at B or B'.
It appears therefore that the ellipse lies wholly within a certain rectangle, and that its general shape is of the form given in fig. 15 which shews the centre C, the foci S, S', and directrices XK and X'K'.
It can easily be shown that the sum of the focal distances of any point λvithin the ellipse is less than, and the sum of any point without greater than, AA', and also that the ratio of the focal distance of any point within the ellipse to its distance from the corresponding directrix is less than, and the ratio for any point without greater than, the eccentricity.
An ellipse can be described mechanically in the following manner.
If an endless string be placed over two small fixed pegs S, S' and be kept tight so as to form a triangle PSS', then a pencil at P would trace out an ellipse whose foci are S, S', and whose major axis is equal to the length of the string minus the distance between the pegs.
Prop. III.
If a chord PQ (fig. 16) intersect in Z the directrix corresponding to the focus S, then SZ will be the external bisector of the angle PSQ.
Join SP, SQ; and draw PM, QN perpendicular to the directrix. Then because the triangles PMZ, QNZ are similar,
PZ:QZ = PM:QN =SP:SQ
.∙. (Eucl. vi. A) SZ bisects the external angle of the triangle PSQ.
Corollary. —If the point Q moves up to and coincides with P, or, in other words, the chord PQ becomes the tangent to the ellipse at P, then the angle PSZ will become a right angle.
Prop. IV.
The foot of the perpendicular from the focus on the tangent always lies on the circle described on the major axis as diameter.
Let PZ (fig. 17) be the tangent at P, meeting the directrix in Z. Let S be the corresponding focus. Join SP, SZ, and draw PM, SY, perpendicular to the directrix and. the tangent respectively. Join YX, SM.
Because the angles PMZ, PSZ are right angles, a circle will circumscribe PSZM; and because the angles SYZ, SXZ are right angles, a circle will circumscribe SYZX.
Therefore angle SYX≈ angle SZX
= supplement of angle SZM=angle SPM; and angle SX Y=angle SZ Y=angle SMP. Therefore the triangles SYX, SPM are similar, and SY: YX — SP: PM —SA: AX.
Therefore the locus of Y is a circle on AA' as diameter.
(Lemma, Introduction.) Prop. V.
The product of the perpendiculars from the foci on the tangent is constant.
If SY, S'Y' (fig. 18) be the perpendiculars from the foci on any tangent, then it is easily seen that, if YS be produced to meet the circle on AA' as diameter again in Z, ZCY' is a straight line, and SΥ'≈SZ.
Therefore SY. S'Y' = SY. SZ - AS. SA' (Euch iii. 35).
Prop. VI.
The tangent at any point of an ellipse makes equal angles with the focal distances of the point. Let the tangent ZPZ' (fig. 19) at the point P meet the two directrices in Z,Z'. Join ZS, SP, PS', S'Z', and draw Μ PM' parallel to the axis, to meet the directrices in Μ, M'. Then because
SP: PZ = ePM: PZ = ePM': PZ' = S'P: PZ', and the angles PSZ, PS'Z' are right angles (Prop, vii.), therefore the triangles PSZ, PS'Z' are similar (Euch vi. 7). Therefore the angle SPZ = angle S'PZ', Prop. VII.
To draw a tangent to an ellipse at a point on the curve.
First Method.— Join SP, S'P, and draw a line bisecting the external angle of SPS'. This line is the tangent at P. (Prop, vi.)
Second Method.— Draw SZ at right angles to SP, meeting the corresponding directrix in Z. ZP is the tangent at P. (Prop, iii.)
Third Method.— On SP as diameter describe a circle which will touch the circle on AA' as diameter in a point Y. YP is the tangent at P. (Prop, iv.)
Prop. VIII.
To draw a pair of tangents to an ellipse from an external point. First Method. —Let 0 (fig. 20) be the point, and S, S' the foci. Join OS. With centre 0 and radius
OS describe a circle; and with centre S' and radius equal to A A' describe another circle. It can be shown that these two circles will always intersect in two points Μ, M'.
Join S'M, S'M', cutting the curve in P 1 P , . Then OP, OP' will be tangents to the curve.
Join SP, SP'.
Now SP + PS' = A A'=MS' = MP + PS' therefore SP = MP; and OS = OM; therefore the two triangles OPS, OPM are equal in all respects, and the angle OPS = angle OPM.
Therefore OP is a tangent to the ellipse ar P (Prop, vi.) Second Method. — Let 0 (fig. 21) be the point, and S, S' the foci.
Join OS, and upon it as diameter describe a circle, cutting the circle described on AA' as diameter (which it will always do) in Y, Y'.
Join OY, OY', and produce them if necessary to meet the curve in P, P'. They will be tangents to the curve at P, P ’.
Because OYS is a semicircle, the angle OYS is a right angle, and therefore OY is a tangent to the ellipse (Prop, iv.)
Prop. IX.
If OP, OP' be tangents to the ellipse at P, P', and S be a focus, the angles OSP, OSP ’ are equal.
In fig. 20 we have angle OSP = angle OMP
>= angle M'MS'- angle M'MO
—angle MM'S' - angle MM'O =angle 0 M'S'= angle OSP'.
Prop. X.
If OP, OP ’ be two tangents to the ellipse, and S, S' be the foci, the angles SOP, S'OP ’ are equal.
In fig. 21 suppose S'Z, S'Z' be drawn perpendicular to OP, OP respectively, then
SY.S'Z = SY'.S'Z' (Prop, v.) therefore SY: SY'-S'Z': S'Z.
Also the angle YSY'= supplement of angle POP'= Z'SZ.
Therefore the triangles YSY', Z'SZ are similar, and the angle Y'YS = S'Z'Z, and angle Y'YS=angle Y'OS, and angle S'Z'Z = angle S'OZ. ö
Therefore angle SOP = angle SOP'.
Prop. XL
If C (fig. 22) be the middle point of A A', then CA 2 = CS, CX. SA': A'X=c: 1=SA: AX.
.∙. SA'+SA: SA = A'X +AX: AX, or AA': SA = XX': AX
.∙. AA'; XX' = SA: AX=≈e: 1. . . (η
Again,
SA'-SA: SA = A'X-AX: AX.
or SS': SA = AA': AX
.∙. SS':AA'≈SA :AX=e:l. . . . (2)
From (1) and (2),
AA': XX' = SS': AA', or CA:CX≈CS:CA
.∙. CA 2 = CS.CX (3)
Also CS : CX-CS 2 : CA 2 fc = CA 2 :CX 2 . . . . (4)
Prop. XII.
L PN be an ordinate of the ellipse, then PN 2 always bears a constant ratio to AN. NA'.
By similar triangles PNA', Z'XA' (fig. 22),
PN: NA'=Z'X: XA', and by similar triangles PNA, ZXA, PN :NA = ZX: XA.
Therefore PN 2 : AN. NA'=ZX. XZ': XA. XA'.
Now it appears from Prop. iii. that the angle Z'SZ is a right angle; therefore ZX . XZ'=SX 2 (Euch vi. 8.); and XA. XA'=CX 2 - CA 2 .
Therefore N 2 : AN. NA'=SX 2 .’: CX 2 - CA 2
-CB 2 : CA 2 (Prop, ii.)
Prop. XIII.
The ordinates of the ellipse and of the circle described on AA' as diameter are in a constant ratio.
If, in fig. 22, NP be produced to meet the circle on AA' as diameter in Q, then
QN 2 = AN ’. NA' and PN 2 : AN . NA'-=CB 2 : CA 2 (Prop, xii.)
.∙. PN 2 : QN 2 = CB 2 : CA 2 and PN: QN = CB: CA.
Corollary. —The ordinates of two ellipses which have a common major axis are in a constant ratio.
It can easily be shown from the last result, if QPN, Q'P'N' be two common ordinates of the circle and ellipse,—(1) that the chords PP', QQ' will meet the axis in the same point; (2) that the tangents at P, Q will meet the axis at the same point; (3) that the intersection of the tangents at P, P' and the intersection of the tangents at Q, Q' will lie on a straight line perpendicular to the axis; (4) that the middle points of PP' and QQ' lie on a straight line perpendicular to the axis.
It can also be shown by means of this proposition that the area of the circle is to the area of the ellipse as AC to BC, and that the area of the parallelogram formed by the four tangents at the extremities of two conjugate diameters (see definition below) is constant, and is equal to AC. BC.
Prop. XIV.
The middle points of all parallel chords in an ellipse lie on a straight line through the centre. Let QPN, Q'P'N (fig.
23) be two common ordinates of the eiιcle on A A' as diameter and the ellipse.
Let W, V be the middle points of QQ', PP'∙ W, V lie on a straight line which bisects N N ' at right angles.
Now, as long as PP' remains parallel to itself, QQ' must remain parallel to itself, and therefore its middle point W lies on a fixed straight line, the diameter at right angles to QQ'. Therefore V lies on a fixed straight line through C, since WM: VM = BC: AC.
The tangents at the points where CV cuts the ellipse will be parallel to the chords PP’ which CV bisects.
Definition.— If CD be drawn parallel to the tangent at P, then CD is said to be conjugate to CP.
Now it is evident that CD and CP will correspond to two radii at right angles in the circle on AA' as diameter, and therefore if CD is conjugate to CP, CP will also be conjugate to CD.
Prop. XV.
If CP, CD (fig, 24) be semi-conjugate diameters, then CP 2 +CD 2 = AC 2 + BC 2 .
Draw QPN, Q'DN' common ordinates of the circle and the ellipse.
Then CQ, CQ' will be at right angles, and therefore the two triangles QCN, CQ'N' will be equal in all respects.
.∙. NC = Q'N' and QN=CN'.
.∙. QN 2 + Q'N' 2 =QN 2 +NC 2 = AC 2 .
.∙. PN 2 +DN' 2 =BC 2 (Prop. xii.).
.∙. CP 2 + CD 2 = CN 2 + NP 2 + CN' 2 + N'D 2 = Q'N' 2 +NP 2 + Q N 2 + N'D 2 =∙AC 2 + BC 2 .
It follows that if the tangent at P meets the axes in T, T' (fig. 25), then PT. PT'= CD 2 .
Draw ordinates PM, DM' to the major axis, and PN to the minor axis.
Then PT: PM ≈ CD: DM'
and PT': PN = CD: CM'
.∙. PT. PT ’: PM. PN = CD 2 : CM'. DM'. . ∙. PT. PT': CD 2 = PM. PN: CM'. DM'.
_PM .DM' = CM' ’ CM ≈52 ι 52=l ι l AC AC
.∙. PT. PT' = CD 2 .
Prop. XVI.
Jf CP, CD be semi-conjugates, and QV be an ordinate parallel to CD, then QV 2 : PV . VP'=CD 2 : CP 2 .
Draw QR an ordinate parallel to CP, and draw UQW the tangent to the ellipse at Q, meeting CP, CD in U, W.
Then CR.CW=CD 2
.∙. CR 2 :CD 2 = CR:CW ≈UV:CIT.
Again CU.CV=CP 2
.∙. CU: CV = CP 2 : CV 2 .
.∙. CU-CV: CU=CP 2 - CV 2 cCP 2 or UV:CU≈PV. VP':CP 2 .
Hence CR 2 : CD 2 =PV. VP': CP 2 or QV 2 : PV. VP'=CD 2 : CP 2 .
Prop. XVIΓ.
If POP' (fig. 26) be any chord, and ROCR' the diameter through 0, then
PO. 0P ’: RO. 0R'=CD 2 : CR 2 , where CD is the semi-diameter parallel to PP'.
Draw CVWQ conjugate to PP', meeting the curve in Q, and the ordinate through R in W. ë
Then PV 2 : CQ 2 - CV 2 = CD 2 : CQ 2 and RW 2 : CQ 2 - CW 2 =CD 2 ; CQ 3 (Prop. χvi. )
.∙. PV 2 - RW 2 .: CQ 2 - CV a - (CQ 2 - CW 2 )
= CD 2 :CQ 2 .
Now PV» - RW 2 . ∣∑ζ = PV 2 - OV*=PO. OP'
and CQ 2 - CV 2 - (CQ 2 - CW 2 ) 2Xζ
G W z
-c Q ..(¾gv·)
-CQ 1 ∙ c -¾^*∙
Therefore PO. OP': CQ 2 .5θiθ5' = CD 2 ∙ CQ a or PO. 0P ’: RO. OR' = CD 2 : CR 2 .
Prop. XVIΠ.
If POP', pθp' be any two chords, and CD, Cd the semi-diameters parallel to them, then
PO. OP': pθ. Op'=CD 2 : Cd 2 .
From the last proposition we have
PO . OP': RO. OR'=CD 2 : CR β and pθ . Op'=RO . OR' = Cd 2 : CR 2 .
Therefore PO . OP': CD 2 = RO. OR': CR 2
≈pO.Op':Cd 2 or PO. 0P ’: pθ. Op' - CD 2 : Cd a .
Prop. XIX.
If the two extremities of a rod slide along two fixed straight lines at right angles to one another, any fixed point in the rod will describe an ellipse.
Let OM, ON (fig. 27) be the two fixed straight lines, and MPN any position of the rod, and P the tracing point.
Complete the rectangle QMON and join OQ, and draw, parallel to ON, RPH to meet OQ in R and OM in H.
Then it can easily be shown that OR = NP and that RH: PH = OR: PM ≈PN:PM.
Tne locus of R is a circle whose centre is 0
and radius PN. And the locus therefore of P is an ellipse whose axes are in OM and ON and equal to PN, PM respectively.
Prop. XX.
If a circle roll on the inside of a fixed circle of double the radius, any fixed point in the circumference of the moving circle will trace out a diameter of the fixed circle, and any other point in the plane of the moving circle will trace out an ellipse.
If the point Μ (fig. 28) coincided with A at the beginning of the motion and the circle now touch at Q, the arcs MQ, QA must be equal. Therefore if C and 0 be the centres, the angle QCM is double the angle QOM, and therefore OCQ is always a straight line; as also MCN. .
It is clear therefore that the motion of a point P in MN is exactly the same as in that of a point in the moving rod. (Prop. xix. )
PART III.—THE HYPERBOLA.
Definitions.
A straight line passing through the centre, and terminated by the hyperbola, is called a diameter.
The extremities of a diameter arc called its vertices.
The diameter which passes through the foci is called the transverse axis.
A straight line BCB' passing through the centre, perpendicular to the transverse axis, such that
BC 2 = B'C 2 = SC 2 -AC 2 is called the conjugate axis.
Any straight line terminated both ways by the hyperbola, and bisected by a diameter produced, is called an ordinate to that diameter.
Each of the segments of a transverse diameter produced, intercepted by its vertices and an ordinate, is called an abscissa.
A chord, tangent, and normal are defined exactly in the same words as in the case of the parabola.
Prop. I.
To find where an hyperbola of given focus, directrix, and eccentricity is cut by a straight line parallel to the directrix.
Let S (fig. 29) be the focus, XK the directrix, and e the eccentricity.
Draw SX perpendicular to XR, and divide it internally and externally in the ratio e to 1 in the points A, A', so that
SA :AX≈SA': A'X=β :l.
It is clear that A will lie between S and X, and A' without SX on the side remote from S.
Draw AZ at right angles to SX and equal to AS, and join XZ.
Let QN be any straight line parallel to the directrix, cutting XZ in Q and the axis in N.
With centre S and radius equal to QN, describe a circle cutting QN in P and P'; these will be points on the hyperbola.
It is clear that if the point P exists, the point P ’ on the opposite side of the axis also exists, and therefore the hyperbola is symmetrical with respect to the axis.
Again, the point P will exist, or, in other words, the circle will cut QN as long as SP or QN is greater than SN, which is always the case as long as the angle QSN is greater than half a right angle.
Now, if SL, A'Z' be drawn at right angles to SX, cutting XZ in L, Z', then (Eucl. vi. 4)
Z'A': A'X = ZA: AX, but SA': A'X = SA: AX, and SA=ZA,
.∙. SA'= Z'A', from which it is easily seen that the angle Z'SA' is half a right angle.
The whole curve therefore lies without the two lines AZ, A'Z'.
Prop. II.
To find where an hyperbola of given focus, directrix, and eccentricity is cut by a straight line perpendicular to the directrix.
Let S (fig. 30) be the focus, XK the directrix, and e the eccentricity.
Draw SX perpendicular to XK, and divide SX in A, A', so that
SA: AX=SA': A'X = e :l; and draw AR, A'R' at right angles to SX.
Let KQ be any line parallel to the axis, cutting the direc∙ trix in K. Join SK, cutting AR, A'R' in R, R', and upon RR' as diameter describe a circle cutting KQ in P, P'; these will be points on the curve. Now
SR :RK = SA:AX=e: 1 = SA': A'X =SR': R'K.
Therefore by the Lemma in the introduction
SP: PK = SP': P'K=SR: RK = e: 1.
Therefore P and P' are points on the hyperbola.
Now, if the point P exists, the point P ’ also exists, and it is easily seen that the middle point L lies on a straight line bisecting AA' at right angles. The curve, therefore, is symmetrical, not only with respect to the axis AA', but also with respect to the line bisecting AA' at right angles. The middle point C of AA' is called the centre of the curve, from the fact that every straight line through C is bisected at the point.
It is evident from what has preceded that if we measure CS'≈ CS and CX'=CX, in the opposite direction to CS and CX, and draw X'K' parallel to XK, the hyperbola might be described with S' for focus, X'K' for directrix, and eccentricity e.
The hyperbola therefore has two foci and two directrices.
Now, since
SP :PK = SP': P'K=e: 1, therefore SP ’- SP: P'K - PK=e: 1, but P'K-PK = 2LK = 2CX
.∙. SP'-SP = 2e . CX.
Now it is easily seen that SP' = S'P, therefore S'P-SP = 2e . CX, or the difference of the focal distances of any point on the curve is constant.
Again SA: AX=SA': A'X=e: 1
.∙. SA'-SA: A'X-AX=e: 1, but SA'-SA=AA'
and A'X-AX = 2CX∙
.∙. SP'-SP = SA'- SA=AA', or the difference of the focal distances of any point on the hyperbola is equal to th¢ transverse axis.
The point P will always exist, or in other words, the circle on
R, R' as diameter will always intersect KQ, because R, R' are on opposite sides of KQ.
Any straight line therefore parallel to the axis cuts the curve, and the curve must take the form given in fig. 31 which shows the centre C, the foci S, S', and the directrices XK, X'K'.
It can easily be shown that the difference of the focal distances of any point on the concave side of either of the two branches (which is called within the curve) is greater than AA', and the difference of the focal distances of any point outside the curve is less than AA'; and also that the ratio of the focal distance of any point within the hyperbola to its distance from the corresponding directrix is less, and the ratio for any point without is greater, than the eccentricity.
If a line BCB' be drawn perpendicular to ACA', and points B, B' taken in it such that CB 2 = CB' 2 = CS 2 - CA 2 , then AA' is called the transverse axis, and BB' the conjugate axis of the hyperbola.
If an hyperbola be described with BB' for transverse axis, and AA' for conjugate axis, then this hyperbola is said to be conjugate to the first one.
It is clear that the foci of the conjugate hyperbola will be in BCB' at the same distance from C as S and S'.
An hyperbola can be described mechanically in the following manner :—
Suppose a bar SQ (fig. 32) to revolve round its extremity S which is fixed; then if a string of given length, attached to the bar at Q, and also to a fixed point S', be always kept tight by means of a ring P sliding on SQ, a pencil at P would trace a hyperbola whose foci are S, S', and whose transverse axis is equal to the length of the rod minus the length of the string.
Prop. III.
If a chord PQ intersect in Z the directrix corresponding to the focus S, then SZ will be the external bisector of PSQ if P, Q both lie on the same branch of the hyperbola, and SZ will be the internal bisector of the angle PSQ, if P, Q lie on different branches.
It can be shewn exactly as in Prop. iii. on the ellipse that PZ:QZ≈SP:PQ, which proves the proposition. .
Corollary. —If the point Q moves up to and coincides with P, or in other words, the chord PQ becomes the tangent to the hyperbola at P, then the angle PSZ will become a right angle.
Prop. IV.
The foot of the perpendicular from the focus on the tangent always lies on the circle described on the transverse axis as diameter.
This proposition is proved in exactly the same way as Prop. iv. on the ellipse.
Prop. V.
The product of the j∣erpendiculars from the foci on the tangent is constant.
This proposition is proved exactly in the same way as Prop. v. on the ellipse, the only difference in the figure being that S, S' lie without the circle instead of within as in the case of the ellipse.
Prop. VI
The tangent at any point of an hyperbola makes equal angles with the focal distances of the point.
Let the tangent PZZ' at the point P (fig. 33) meet the two directrices in Z, Z'.
Join ZS, SP, PS', S'Z', and draw PMM' parallel to the axis to meet the directrices in Μ, M'.
Then because
SP:PZ = ePM:PZ ≈ePM':PZ' ≈PS':PZ', and the angles PSZ, PS'Z' are right angles (Prop, iii.), therefore the triangles PSZ, PS'Z' are similar (Euch vi. 7). Therefore the angle SPZ = angle S'PZ'.
Prop. VII.
To draw a tangent to an hyperbola at a point on the curve.
First Method. —Join SP, S'P, and draw a line bisecting the angle SPS': this line is the tangent at P. (Prop, vi.)
Second Method. — Draw SZ at right angles to SP meeting the corresponding directrix in Z: ZP is the tangent at P. (Prop, iii.)
Third Method.— On SP as diameter describe a circle which will touch the circle on A A' as diameter in some point Y: YP is the tangent at P. (Prop, iv.)
Prop. VIII.
To draw a pair of tangents to an hyperbola from an external point. First Method.— Let 0 (fig. 34) be the point, and S, S' the foci.
Join OS; with centre 0, and radius OS, describe a circle; and with centre S' and radius equal to AA' describe another circle. It can be shown that these circles will always intersect in two points Μ, M'.
Join S'M, S'M' cutting the curve in P, P. Then OP, OP' will be tangents to the curve.
Join SP, SP'.
Now S'P-SP≈=AA'=
S'M = S'P-MP, therefore SP≈MP; and OS≈OM; therefore the two triangles OPS, OPM are equal in all respects, and the angle OPS =
Therefore OP is a tangent to the hyperbola at P (Prop, vi.) Second Method— Let 0 (fig. 35) be the point, and S, S' the foci. Join OS, and upon it as diameter describe a circle cutting the circle described on AA' as diameter (which it will always do) in Y, Y'.
Join OY, 0Y', and produce them if necessary to meet the curve in P, P'. They will be tangents to the curve at P, P ’.
Because OYS is a semicircle, the angle OYS is a right angle, and therefore OY is a tangent to the hyperbola (Prop, iv.)
Prop. IX.
If OP, 0P ’ be tangents to the same branch of an hyperbola at P, Γ v and S be a focus, the angles OSP, OSP' are equal: if OP, OP' be tangents to different branches, the angles OSP, OSP ’ are supplementary.
This proposition is proved in the same manner as Prop. ix. on the ellipse.
Prop. X.
If OP, OP' be two tangents to the hyperbola, and S, S' be the foci, the angles, SOP, SOP' are equal.
This proposition is proved exactly in the same way as Prop. x. on the ellipse.
Prop. XI.
If C be the middle point of A A', then CA 2 =CS. CX.
SA':A'X≈e:l≈SA: AX.
.∙. SA'-SA: SA = A'X-AX: AX.
.∙. AA':XX'≈SA: AX-e:l. . (1.)
Again SA' + SA: SA = A'X + AX: AX.
or SS':SA—AA': AX.
.∙. SS': AA'≈SA:AX≈e:l. . . (2.)
From (1) and (2)
AA':XX'≈SS':AA' or CA:CX — CS:CA
.∙. CA’-CS.CX. . . . κ 3∙)
Also OS: CX-CS 2 : CA 8
=CA*:CX*. , . (4.)
Prop. XII.
If PN be an ordinate of the hyperbola, then PN 2 always bears a constant ratio to AN .NA'.
It is proved exactly as in the case of the ellipse Prop, xii∙, that PN 2 : AN . NA'=CB 2 : CA 2 .
Prop. XIII.
The ordinates of two hyperbolas which have the same transverse axis are in a constant ratio.
Let PP ’ N be the common ordinate of two hyperbolas, whose transverse axis is AA', and whose conjugate axes are CB, CB'.
Then PN 2 : AN . NA'=CB 2 : CA 2 and P'N 2 : AN . NA'=CB' 2 : CA 2 .
Therefore PN 2 : CB 2 = AN . NA': CA a
= P'N 2 : CB' 2 .
.∙. PN:CB≈PN: CB'
or PN: P'N = CB: CB'.
Definition.
The diagonals of the rectangle formed by the tangents to a hyperbola and its conjugate at their vertices are called asymptotes. Prop. XIV.
If a straight line be drawn through any point Q on one of the asymptotes perpendicular to the transverse axis, meeting the hyperbola in P, P z and the other asymptote in Q' (fig. 36), then QP. QP' = BC 2 .
From similar triangles QNC, EAC, we have QN 2 : CN 2 = BC 2 : AC 2 .
and from prop. xii. ∙
PN 2 : AN.NA'=BC 2 : AC 2 .
. ∙. ON 2 -PN 2 : CN 2 -AN. NA' = BC 2 : AC 2 ∙
but QN 2 - PN 2 = QP , . QP, and AN . NA'=CN 2 -CA 2 ;
.∙. QP.QP ’ fCA a = BC 2 : CA 2 ,
QP . QP'≈BC 2 .
It is easily seen ιl∙at
QP = Q'P', therefore PQ.PQ' = BC 2 .
Similarly it can be shewn that if RQR' be drawn parallel to AA', to meet the hyperbola in R, R', then
QR.QR'=≈AC 2 .
It is clear that the further the point Q moves away the greater the line PQ' becomes, and it can be made greater than any assignable quantity, however large; and since PQ.PQ' = BC 2 , therefore the line PQ becomes smaller and smaller, and can be made less than any assignable quantity, however small. Hence the asymptote never actually reaches the curve, though the distance between them constantly decreases, and can be made smaller than any assignable quantity.
It can easily be shewn that if the asymptote cuts a directrix in the point F, then CF = CA.
As the asymptote may be considered as the tangent to the hyperbola at a point at an infinite distance, the foot of the perpendicular from the focus on the asymptote must lie on the circle whose diameter is AA' (Prop, iv.)
SF therefore must be perpendicular to the asymptote, as appeal s from other reasons (from Prop, iii., for example. )
Prop. XV.
If QPP'Q' (fig. 37) be any chord cutting the asymptotes in Q, Q and the curve in P, P', then QP = P'Q'. andQP. PQ'=CD 2 , where CD is the semi-diameter in the conjugate hyperbola parallel to PP'.
Draw RPR', DWW' perpendicular to the transverse axis, meeting the asymptotes in R, R', and W. w.
Then from similar triangles PRQ, DWG
PQ: PR = CD: DW and from similar triangles PR'Q', DW'C PQ': PR' = DC: DW'.
Therefore PQ. PQ': PR, PR' = CD 2 : DW . DW'; but DW. DW' = BC e = PR. PR' (Prop. xiv.) ∙
therefore PQ.PQ' = CD 2
= P'Q. P'Q'.
Now, if V be the middle point of QQ', then
PQ.PQ' = QV 2 -PV 2 and P'Q. P'Q'=QV 2 -P'V 2 .
Therefore PV = P'V.
Thus V is the middle point of PP', as well as of QQ', or in other words PQ = Q'P'.
It is clear that when the points PP' coincide, or we have the tangent parallel to PP', say qp q, then g ’ 2>=j>∕=CD, and also that the line Cp will bisect all chords parallel to the tangent at p.
PV is called an ordinate to the diameter Cp. Definition.
A chord which is parallel to the tangent at P is said to be conjugate to CP.
If a diameter CD be drawn parallel to the tangent at P to meet the conjugate hyperbola in D, CP, CD are said to be conjugate semi-diameters.
It is clear from Prop. xv. that a diameter is conjugate to the chords which it bisects.
Prop. XVΓ.
If CD be conjugate to CP, then CP is conjugate to CD.
Let the tangent at P meet the asymptote in L, then PL is parallel to CD; it is also equal to CD (Prop, xv.); therefore DL is equal and parallel to CP (Euclid i. 33).
Therefore LD is the tangent to the conjugate hyperbola at D, and therefore CP is conjugate to CD.
It is easily seen that PD is bisected by one asymptote, and parallel to the other asymptote.
Prop. XVII.
If P'CPV (fig. 38) be a diameter, and QV be an ordinate to CP, then QV s : PV.P'V=CD 2 : CP 2 .
Draw PL the tangent at P to meet an asymptote in L, and let QV produced meet the asymptotes in R, R'.
Then RV 2 - QV 2 = RQ . QR' = PL 2 .
Therefore QV 2 = RV 2 -PL 2
-(⅛*->)rt∙
-CV ’-Cf CD·
CP 2 vυ
- PV∙P ’ v cd ≡
CP 2 cιj Therefore QV 2 : PV. P z V - CD 2 : CP≡.
Prop. XVIII.
If POP' (fig. 39) be any chord, and ORCR' the diameter through 0, then
PO. OP': RO . 0R'=CD 2 : CR 2 , where CD is the semi-diameter parallel to ΓP ’. Draw CQWV conjugate to PP' meeting the curve in Q, and the ordinate through R in W.
Then PO.OP'=PV 2 -OV 2
= PV 2 -RW 2 .2^ a .
Now PV 2 : CV 2 - CQ 2 =CD 2 : CQ 2 and RW 2 : CW 2 - CQ 2 = CD 2 : CQ 2
. ∙. PV’ - RW 2 , £ïì: CV 2 - CQ 2 - (C W 2 - CQ 2 ) ^=CD 2 : CQ’.
or PO. OP': CQ√2∑* - lλ = CD 2 : CQ 2 . ∖CW j ∕
.∙. PO .OP': CD 2 = CV 2 -CW 2 : CW 2
= CO 2 -CR 2 : CR 2
= R0.0R': CR 2 . Therefore
PO . OP': RO. OR' - CD 2 : CR 2 .
Prop. XIX.
If POP', pQp ’ be any two chords, and CD, Cd the semi-diameters parallel to them, then
PO . OP': pθ . O∕∕=CD 2 : Cd 2 .
From the last proposition we have
PO . OP': RO. OR'=CD 2 : CR 2 , and also pθ . Op': RO. OR' = Ctí 2 : CR 2 . Therefore PO. OP ’: CD 2 =RO . OR': CR 2 ≈pO.Op':Cd 2 , or PO.OP ’:j?O.Op'=CD 2 :Cd 2 .
Prop. XX.
If from a point Q on one asymptote (fig. 40) ordinates QPM, QDN be drawn to two conjugate hyperbolas in P, D, PD will be parallel to the other asymptote.
QM 2 : QN 2 = QM 2 : CM 2 = BC 2 : AC 2 and QM 2 - PM 2 : QN 2 - DN 2 = BC 2 : AC 2 .
.∙. PM 2 : DN 2 = BC 2 : AC 2 = QM 2 : QN 2 or PM:DN-QM:QN.
Therefore DP is parallel to NM (Eucl. vi. 2), and NM is parallel to BA, and therefore to the other asymptote.
Corollary. —It follows therefore that CP, CD are conjugate (Prop, xvi.)
Prop. XXI.
If CP, CD be conjugate semi-diameters, CP 2 - CD 2 = CA 2 - CB 2 .
Let ordinates PM, DN (fig. 41) in the two hyperbolas be produced, they will meet in a point Q on the asymptote (Prop, xx., Cor.) Then CP 2 -CD 2 = CM 2 + MP 2 -CN 2 -ND 2
= QN 2 + MP 2 - QM 2 - DN 2 = QN 2 - DN 2 - (QM 2 - PM 2 ) =CA 2 -CB 2 (Prop, xiv.) It follows that if the tangent at P meets the axes in T, T', then
PT. PT'=CD 2 .
For PT:CD≈PM:CN
and PT':CD = CM:DN
.∙. PT.PT':CD 2 = PM.CM :CN.DN
=PM.QN: QM.DN
= 1: 1
.∙. PT.PT'=CD 2 .
Prop. XXII.
If CP, CD be conjugate semi-diameters, the area of the triangle CPQ is constant.
Produce QP (fig. 42) to meet the other asymptote in Q': and ioin MON, PLD. They are parallel to CQ'.
.∙. QO: LO = QM: PM (Eucl. vi. 2.) QO 2 : LO 2 = QM 2 : PM 2
.∙. QO 2 -LO 2 : QO 2 = QM 2 -PM 2 : QM 2 or QL .LC: QO 2 = BC 2 : QM 2
.∙. 4CL.LQ ιCQ 2 = BC 2 : QM 2 or 4CL. LQ: BC 2 = CQ 2 : QM 2
= CA 2 + CB 2 : CB 2
= CS 2 : CB 2
.∙. 4CL.LQ = CS 2 .
Now in the right angled triangle PQD, L is the middle point of the hypotenuse; therefore PL = LD = LQ.
.∙. 4CL.LP=CS 2 .
If PL'be drawn parallel to CL to meet the other asymptote,
4PL.PL'-CS 2 , and the area of the quadrilateral CLPL' is constant.
It follows that, if the tangent at P meets the asymptotes in K, K', the area of the triangle CKK' is constant; also that the area of the quadrilateral formed by the tangents at the extremities of two conjugate diameters is constant.
PART IV.—THE CONE AND ITS SECTIONS.
Definitions.
If through the point V, without the plane of the circle ADB (fig. 43), a straight line AV be drawn, and produced indefinitely both ways, and if the point V remain fixed while the straight line AV is moved round the whole circumference of the circle, a superficies of two sheets, which is called a cone, will be generated by its motion.
The fixed point V is called the vertex of the cone.
The circle ADB is called the base of the cone.
Any straight line drawn from the vertex to the circumference of thg base is called a side of the cone.
A straight line VC drawn through the vertex of the cone, and the centre of the base, is called the axis of the cone.
If the axis of the cone be perpendicular to the base, it is called a right cone.
If the axis of the cone be not perpendicular to the bas⅛ it is called a scalene cone. Prop. I.
If a cone be cut by a plane passing through the vertex, the section will be a triangle.
Let ADVB be a cone of which VC is the axis; let AD be the common section of the base of the cone and the cutting plane; join VA, VD. When the generating line comes to the points A and D, it is evident that it will coincide with the straight lines VA, VD; they are therefore in the surface of the cone, and they are in the plane which passes through the points V, A, D, therefore the triangle VAD is the common section of the cone and the plane which passes through its vertex.
Prop. II.
If a cone be cut by a plane parallel to its base, the section will be a circle, the centre of which is in the axis.
Let EFG be the section made by a plane parallel to the base of the cone, and VAB, VCD two sections of the cone made by any two planes passing through the axis VC; let EG, HF be the common sections of the plane EFG and the planes ÝAB, VCD. Because the planes EFG, ADB are parallel, HE, HF will be parallel to CA, CD, and
AC: EH = (VC: VH = )CD: HF; but AC=CD, therefore EH=∙HF. For the same reason GH = HF; therefore EFG is a circle of which H is the centre and EG the diameter.
Prop. III.
If a scalene cone ADBV (fig. 44) be cut through the axis by a plane perpendicular to the base, making the triangle VAB, and from any point H in the straight line AV a straight line HK be drawn in the plane of the triangle VAB, so that the angle VHK may be equal to the angle VBA, and. the cone be cut by another plane passing through HK perpendicular to the plane of the triangle ABV, the common section HFKN of this plane and the cone will be a circle.
Take any point L in the straight line HK, and through L draw EG parallel to AB, and let EFGN be a section parallel to the base, passing through EG; then the two planes HFKN, EFGN being perpendicular to the plane VAB, their common section FLN is perpendicular to ELG, and since EFGN is a circle (by last Prop.), and EG its diameter, the square of FL is equal to the rectangle contained by EL and LG (Eucl. iii. 35); but since the angle VHK is equal to VBA or VGE, the angles EHK,EGKare equal, therefore the points E, H, G, K, are ìl the circumference of a circle (Eucl. iii. 21), and HL. LK = EL . LG (Eucl. iii. 35) =FL 2 , therefore the section HFKN is a circle of which HLK is a diameter (Eucl. iii. 35).
This section is called a Subcontrary Section.
Prop. IV.
If a cone be cut by a plane which does not pass through the vertex, and which is neither parallel to the base nor 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 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 perpendicular to OP, meeting the circumference of the base in A and B; let a plane pass through V, A, and B, meeting the plane OVP in the line NV, 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; or it will meet VA, produced beyond the vertex, in H, as in fig. 47.
Let EFGM be a section of the cone parallel to the base, meeting the plane VAB in EG, and the plane FKM in FM, and let L be the intersection of EG and FM; then EG will be parallel to NB, and FM will he parallel to PO, and therefore will make the same angle with LK, wherever the lines FM, LK cut each other; and since BN is perpendicular to PO, EG is perpendicular to F Μ. Now the section EFGM is a circle of which EG is the diameter (Prop, ii.), therefore FM is bisected at L, and FL 2 = EL. LG.
Case 1. Let the line PNO be without the base of the cone. Through K and H (fig, 45) draw KR and HQ parallel to AB. The triangles KLG, KHQ are similar, as also HLE, HKR; therefore KL: LG=KH: HQ, and HL:LE≈KH:KR; 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 constant ratio to LF 2 , consequently (Prop. xii. on the ellipse) the section HFKM is an ellipse, of which HK is a diameter and MF an ordinate.
Case 2. Next, suppose the line ONP to touch the circumference 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 = Al. IB: EL. LG.
But since EG is parallel to AB, and IK parallel to AV, Al is equal to EL, and
IB:LG=KI:KL, therefore DI 2 : 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 fc that HL.LK has to LF 2 a constant ratio, therefore the section DFKMS is an hyperbola, of which KH is a transverse 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 parabola; 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 ot an hyperbola be supposed to retain a constant ratio to each other, and, at the same time, to diminish 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 parabola, 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 (Euch 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 parallel 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 1.—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 coinciding 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 up to and coinciding with P.
Prop. II.
To draw the circle of curvature at any point of a conic.
Draw the tangent at P, of which methods have been given above. Draw PQ equally inclined to the axis, cutting the conic again in Q Draw PO at right angles to the tangent, and make the angle PQO equal to the angle QPO.
This gives 0 the centre of the circle of curvature.
Pkop. III.
The focal chord of curvature in the parabola is equal to 4SP.
Let the common chord of the circle of curvature and the parabola be PQ, cutting the axis at F (fig. 48).
Draw the double ordinate PNP', cutting the axis at N; the tangents at P, P' will meet the axis in the same point T. Then angle PFT=angle PTF = angle PTF.
. ∙. TP', PQ are parallel.
.∙. PQ = 2PV = 4PF≈4PT.
Now let PS produced cut the circle again in U; join UQ.
Thenangle UQP = angleTPU(EucI. iii. 32) = angle PTS=angle PFT; therefore UQ is parallel to SF
.∙. PU: PS = PQ: PF = 4: 1, or PU = 4SP.
Prop. IV.
To find an expression for the chord of curvature at any point of a parabola drawn in any direction.
Using the same construction as in Prop, iii., let PW (fig. 49) be the chord required.
Draw SY parallel to the given direction to meet the tangent at Pin Y.
Then angle PWU = angle SPY (Eucl. iii. 32). and angle UPW = angle YSP. Therefore the triangles UWP, YSP are similar, and
PW: PU = SP: SY or PW.SY = PU.SP = 4SP 2 .
.∙. PW=1≡5 2 . SY Corollary.— The diameter of 4SP 2 curvature = —- where SY is the perpendicular on the tangent.
Prop. V.
If the chord of intersection PQ (fig. 50) of an ellipse or hyperbola with the circle of curvature at P meet CD the semi-diameter conjugate to CP in K, then ∙
PQ . PK = 2CD 2
Draw the double ordinate P,NI v ; the tangents at P, P' meet in the axis at T, and the tangent at P' is parallel to PQ, and therefore CP ’ bisects PQ in V.
Let PQ meet the axes in F, F', then
PV :PF-PV: P'C = TF: TC =≈PF: PK since CD is parallel to PT.
Therefore PV . PK = PF . PF'
= PT. PT'=CD 2
(Ellipse, Prop. xv. and Hyperbola, Prop, xxi.) and .∙. PQ. PK = 2CD 2 .
Prop. VI.
If the chord of curvature PQ' (fig. 51) of an ellipse or hyperbola in any direction meet CD in K', then
PQ'.PK' = 2CD 2 .
The angle Q'QP = angle TPK' = angle PKK'; therefore the triangles FKK', PQ'Q are similar, and PQ': PQ = PK: PK'
.∙. PQ'.PK'≈PQ . PK = 2CD 2 .
If PQ" be the chord of curvature through the focus, then PK" = CA
and PQ". CA = 2CD 2 .
If PQ'" be the chord of curvature through the centre PK'"=CP
and PQ'". CP=2CD 2 .
If PQ"" be the diameter of curvature
PK"". CD = CA.CB
and PQ"". PK"" = 2CD 2 .
.∙. PQ"". CA.CB = 2CD 3 ∙.
For other powerful methods of investigating the properties of th· conic sections which have been much developed of late reference is made to Geometry and other headings. (H. Μ. T.) of a promise made to his dying friend. In 1852 he commenced, in conjunction with Mr Goldwin Smith, a complete edition of Virgil with a commentary, of which the first volume appeared in 1858, the second in 1864, and the third soon after his death. Mr Goldwin Smith was compelled to withdraw from the work at an early stage, and in the last volume his place was taken by Mr Nettleship. In 1863 appeared Conington’s translation of the Odes and Carmen Seczιlare of Horace. This was followed in 1866 by the work by which its author is best known to the general public, the translation of the Aeneid of Virgil into the octosyllabic metre of Scott, which deservedly takes almost the highest rank in its own department. The version of Dryden is the work of a stronger artist; but for fidelity of rendering, for happy use of the principle of compensation so as to preserve the general effect of the original, and for beauty as an independent poem, Conington’s version is unrivalled. That the measure chosen does not reproduce the majestic sweep of the Virgilian verse is a fault in the conception and not in the execution of the task, and Conington maintained that his choice had advantages which more than counterbalanced this defect. His last effort in his favourite task of translation was his rendering of the Satires, Epistles, and Art of Poetry of Horace, which was published in 1869. He died at Boston on the 23d October 1869. His edition of Persius, with a commentary and a spirited prose translation, was published posthumously in 1872. In the same year appeared his Miscellaneous Writings, edited by Symonds, with a memoir by Professor H. J. S. Smith.
