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any two straight lines DQ, DR, to terminate in the assymptotes CQ, CS, and from any other point N let NP be drawn parallel to DQ, and NS parallel to DR, to terminate in the same lines. Then RD DQPN NS.

For, let LDK and MNO be drawn parallel to the conjugate axis, to meet the assymptotes. Then, from the similar triangles RDK, SNO, RD: DK = NS: NO, and from the similar triangles, LDQ, MNP, DQ: LD PN: MN.

Therefore RD DQ : LD DK =

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PN NS: MN NO. But LD DK MN NO (III. 9, Cor. 4). Consequently RD DQ = PN.NS.

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COR. 1.-If from two points in the same or in opposite hyperbolas, straight lines be drawn, each parallel to one assymptote, and meeting the other, they are to one another inversely as the parts they intercept from the centre. For QD, DR, being parallel respectively to PN and NS, therefore QDDR = NS NP, and consequently QD: NS NP: DR, or QD: NSCS: CQ.

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COR. 2.-If from any point in a given hyperbola, two straight lines be drawn parallel to the assymptotes, the parallelogram formed thereby is of a given magnitude. For (Pl. Ge. VI. 23, Cor. 1) QR: PS = (DQ: NS, DR: NP) DQ DR: NS NP; but DQ DRNS NP, therefore QRPS, and any other rectangle similarly formed is shown in the same manner to be equal to PS.

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COR. 3.-Every sector of an hyperbola is equal to the quadrilateral figure contained by the curve, by one assymptote, and by parallels to the other, through the extremities of the base of the sector.

For, since the parallelograms RQ, PS, are equal, the two

triangles CQD, CSN, are together equal to PS, and these equals being taken from the figure CQDNS, there remains the sector CDN equal to the quadrilateral figure PQDN.

PROPOSITION XIV.

If in one of the assymptotes of an hyperbola, any number of points be assumed, such, that their distances from the centre be in continued proportion, and straight lines be drawn from these points to the curve, parallel to the other assymptote, the mixtilineal quadrilateral figures formed thereby will be equal.

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H, F, meet CQ in N, L, C A B and CS in O, M. Then, because OG

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GN (III. 9, Cor. 3), OB BC, and because MH = FL (III. 9, Cor. 2), MD = AC; therefore MD: OB = CB: CD DH: BG (III. 13, Cor. 1). Hence, the triangles MDH, OBG, which have the angles at D and B equal, are equiangular, and LM is parallel to NO. The diameter CG, therefore, bisects the chord HF, and every chord parallel to HF (III. 10). Consequently CG bisects the segment FGH of the hyperbola. But it also bisects the triangle FCH; therefore the sector CFG=CGH, and the quadrilateral AFGB = BGHD (III. 13, Cor. 3).

Schol. The ninth and twelfth propositions of the ellipse may be applied to the hyperbola, and demonstrated in the

same manner.

EXERCISES.

1. The square of any semi-diameter of an hyperbola is equal to the rectangle under the distances of its vertex from

the foci, added to the difference of the squares of the semitransverse and semi-conjugate axis.

2. Every tangent of an hyperbola is harmonically divided by the transverse axis and perpendiculars falling upon

it from the foci.

3. The difference of the squares of any two conjugate diameters of an hyperbola, is equal to the difference of the squares of the two axes.

4. If from any point in an hyperbola, straight lines be drawn through the vertices of a diameter, to limit the tangents at these points, the rectangle under the tangents will be equal to the square of the semi-conjugate diameter.

5. A semi-ordinate to any diameter is a mean proportional between its segments, intercepted from the diameter by two straight lines intersecting each other in any point of the curve, and passing through the vertices of the diameter.

6. If a quadrilateral figure be formed by tangents to the four hyperbolas, a straight line through the centre, parallel to that which joins two opposite points of contact, will divide the two opposite sides of the figure, so that the segments of the one shall be inversely proportional to the segments of the other.

7. Also, the straight line which joins the middle points of its diagonals will pass through the centre of the hyperbolas. 8. If through a fixed point, any straight line be drawn, to meet the hyperbola, or opposite hyperbolas, in two points, the rectangle under its segments, from the fixed point, will be to the rectangle under its segments, intercepted by the assymptotes, from either point in the curve, in a constant ratio.

9. If from a point in one of the assymptotes of an hyperbola, any straight line be drawn to intersect the curve (or opposite curves), in two points, and from the points of section, lines parallel to the same assymptote be drawn to meet the other, the sum (or difference) of the parallels will always be of the same magnitude.

The propositions in the exercises to the preceding book may be applied to the hyperbola.

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FOURTH BOOK.

DEFINITIONS.

1. Let there be a circle, and a fixed point, without its plane, and let a straight line, always passing through that point, and indefinitely extended both ways, revolve about the circle in its circumference; the two surfaces thus described are each of them named a conic surface, the fixed point its vertex, the circle its base, and the straight line passing through the vertex and the centre its axis (first figure to proposition 1).

2. A cone is a solid bounded by a circle and a conic surface. The fixed point is called the vertex, and the circle the base of the cone. Also, the straight line passing through the fixed point, and the centre of the circle, is named the axis of the cone.

3. Let there be a circle, and any straight line intersecting its plane at the centre, and let another straight line, always parallel to the former, revolve about the circle in its circumference; the surface thus described is named a cylindric surface, of which the circle is the base, and the straight line through its centre the axis (second figure to proposition 1).

4. A cylinder is a solid bounded by two equal and parallel circles, and a cylindric surface. Either of the circles is named the base, and the straight line joining their centres the axis of the cylinder.

5. A cone or cylinder is termed right or oblique, according as the axis is perpendicular or oblique to the base.

6. The section of a cone or cylinder is termed parallel or oblique, according as its plane is parallel or inclined to the

base.

7. When a cone or cylinder is cut by a plane touching the axis, and perpendicular to the base, and by another plane perpendicular to the former, in such a manner, that the common section of the two planes make angles with the common sections of the first plane and the surface, alter

nately equal to those which the common section of the first plane and the base makes with the same lines on the same side, the section of the second plane is called a subcontrary section (figures to Prop. 2).

COROLLARIES FROM THE DEFINITIONS.

COR. 1.-A straight line joining the vertex, and any point in a conic surface, lies wholly in that surface; its continuation one way is in the same, and its continuation the other way in the opposite surface; also, every such line meets the circumference of the base.

COR. 2.-Any plane touching the axis of a conic surface, cuts that surface in two straight lines, as AD, AF (fig. to Prop. 1).

COR. 3. If a cone be cut by a plane touching the axis, or by a plane through the vertex, and any two points in the circumference of the base, the section is a triangle, as AFD (first figure to Prop. 1).

COR. 4.-If a plane, touching a tangent to the base, pass through the vertex, it touches the conic surface in the straight line joining the vertex and the point of contact, every point in the plane, except in that straight line, being without the surface.

COR. 5.-A straight line, drawn from any point in a cylindric surface, parallel to the axis, lies wholly in that surface.

COR. 6.-Any plane touching the axis of a cylindric surface, cuts that surface in two parallel straight lines.

PROPOSITION I.

If a conic or cylindric surface be cut by a plane parallel to the base, their line of common section is the circumference of a circle, having its centre in the axis.

1. Let ABDCA be a conic surface, of which BCD is the base, and AE the axis, and let it be cut by the plane GOL, parallel to BCD, the line of common section GKHL is the circumference of a circle having its centre in AE.

For, let any two planes, ABC, AFD, touching the axis AE, cut the surfaces in the straight lines AB, AC, AF,

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