Also 'V' by construction = WC = QV:

hence Q'q' will equal Qq, and CV' will equal CV: QC, WV are parallelograms, and therefore

QW= V'C=VC= QW

and W is the bisection of QQ'.

And in like manner all chords between the two branches of the curves parallel to QQ' are bisected by the diameter CW.

Let QQ meet the asymptotes in S, s: then since the angle at C is a right angle and V is the middle point of Uu, CV = VU, and CV, VU are equally inclined to the asymptote; so.. are SW, WC being parallel to them:

since the angle at C is a right angle: W bisects Ss, and since it also bisects QQ',.. QS=Q's. Shewing that every straight line that cuts the curve and the asymptotes, each in two points, and the curve in one branch or both, has the intercepts between the curve and the asymptotes equal.

COR. Hence as the chord Uu moves parallel to itself, so that Q and q approach each other and finally coincide in P, the chord ultimately becomes a tangent at P and is bisected in P: shewing that the part of the tangent intercepted between the asymptotes is bisected at the point of contact.

Each half of the tangent will equal CP.

6. QV + CP = VU.

Let Ww be the chord to the asymptotes which touches the curve at P. Through P draw rPr' the double ordinate to the asymptotes. Then

QU: QR: PW: Pr: CP: Pr

QU. Qu: AC :: CP3 : AC2,

.. QÜ. Qu= CP and QV2 + CP2 = V.U2.

7. The tangents at the extremities of any chord meet in the diameter which bisects it.

Let Pp be any chord, Qq a parallel chord adjacent to it, then Vv through their middle points will pass through the centre of the curve.

Join PQ and let it meet the diameter in T.

If QV be an ordinate of any diameter CP, and QT the tangent at Q, then CP is a mean proportional between CT and CV.

Let PR be the tangent at Pintersecting QT in R, PR is parallel to QV: draw PO parallel to QT to meet QV in 0.

Join OR; it will be the diagonal of the parallelogram OPRQ and will therefore bisect the chord QP; hence (by the last proposition) RO bisecting the chord PQ and passing through the intersection of the tangents at its extremities is a diameter, and when produced will pass through C.

Hence

and

CV: CP CO: CR:: CP: CT

CV. CT-CP2.

Q. E. D.

COR. 1. If PN be the ordinate from P to the transverse axis, and the tangent at Pintersects the axis in T, we shall

COR. 2. Hence to draw tangents to an hyperbola from any point T, join CT and produce it to meet the curve in P; draw the tangent at P by the aid of COR. 1.

Produce CP to a point V such that CV is a third proportional to CT, CP: through V draw an ordinate QVq parallel to the tangent at P: join TQ, Tq; these (by the Proposition) will be the tangents required,

9. The Conjugate Hyperbola, Conjugate Diameters.

If on the conjugate axis of a rectangular hyperbola as transverse axis we draw another rectangular hyperbola, it is called the conjugate hyperbola, and it is manifest that it has the same asymptotes as the first.

Moreover if we draw semi-diameters CP, CD to meet the two curves equally inclined to the asymptotes and therefore to the axes, these will also be equal: and if we join PD it will cut the asymptotes at right angles, in O suppose. Take OR on the asymptote equal to CO and join PR, DR. Then PR = CP and therefore touches the hyperbola at P; and DR (= CD) touches the conjugate at D. Also CPRD is a parallelogram.

M

P

Moreover CP produced bisects all chords of the hyperbola parallel to PR or to CD, and similarly CD produced bisects all chords of the conjugate parallel to DR or to CP. Hence CP, CD produced each bisects the chords parallel to the other, and are called conjugate semi-diameters.