PROPOSITION IV. THEOREM.

The subtangent to the axis is equal to twice the abscissa.

The subnormal is equal to one half of the latus rectum. That is,

If a straight line be drawn from the focus perpendicular to the tangent at any point, it will be a mean proportional between the distance from the focus to that point, and the distance from the focus to the vertex.

That is, if SY be a perpendicular let fall from S upon Tt the tangent at any point P

Join A, Y.

SP: SY::SY : SA.

.. AY is perpendicular to AM. Hence the triangles SYA, SYT, are similar,

..

ST: SY:: SY: SA

or, SP: SY:: SY: SA.. SP=ST by Prop. III, cor. 2. Cor. 1. Multiplying extremes and means,

SY' SP. SA.

Cor. 2. SP: SA:: SP2: SY

And since SA is constant for the same parabola,

SP proportional to SY.

The square of any semi-ordinate to the axis is equal to the rectangle under the latus rectum and the abscissa.

That is, if P be any point in the curve

For,

=

PM2

= L. AM.

PM' SP-SM' (Prop. XXIV. B. IV. El.

Geom.)

=

(AM=AS)-(AM — AS)' ..SP=AM+AS(Prop. 1,) & SM=AM-AS = 4 AS. AM. (Prop. X. and XI, B. IV. El. Geum.)

= L. AM.

Prop. I.

Cor. 1. Since L is constant for the same parabola PM' proportional to AM,

That is, The abscissæ are propotional to the squares of the ordinates.

PROPOSITION VIII.

THEOREM.

If Qq be an ordinate to the diameter PW and Pv, the corresponding abscissa, then,

Qu2 = 4SP x Pv.

Draw PM an ordinate to the axis. T
Join S, Q; and through Q draw

DQN perpendicular to the axis.

A

S

From S let fall SY perpendicular

on the tangent at P.

The triangles SPY, QDv, are similar.
Qu2: QD2:: SP2

:: SP.

: SY'

:SA, Prop, VI. Cor. 2.

The triangles PTM, QDv, are also similar;

But,

QD

Dv: PM

:: PM2

:: 4AS

: MT

: PM.MT

: 2PM

:: 4AS.AM: 2PM. AM

.. 2PM. QD = 4AS. Dv

PM' — QN'= 4ÀS. AM-4AS. AN=4AS(AM-AN) = 4AS. MN

And, PM'-QN'= PM+QN) (PM—QN)

= (PM+QN). QD

.. (PM+QN). QD 4AS. MN 4AS. DP

=

But, 2 PM. QD = 4AS. Dv

.. (PM—QN) . QD

Or, QD'

=

4AS. Pv

= 4AS. Pv

=

Qu2 4AS. Pv :: SP: SA.

Qu2 4SP. Pv.

=

Cor. 1. In like manner it may be proved, that

Hence, Qu = qu; and since the same may be proved for any ordinate, it follows, that

A diameter bisects all its own ordinates.

The square of the semi-ordinate to any diameter is equal to the rectangle under the parameter and abscissa

It will be seen, that Prop. VII. is a particular case of the present proposition.

ELLIPSE.

DEFINITIONS.

1. AN ELLIPSE is a plane curve, such that, if from any point in the curve two straight lines be drawn to two given fixed points, the sum of these straight lines will always be the same. 2. The two given fixed points are called the foci.

Thus, let ABa be an ellipse, S and

3. If a straight line be drawn joining the foci and bisected, the point

of bisection is called the centre.

B P1

P3

Ha

4. The distance from the centre to either focus is called the eccentricity.

5. Any straight line drawn through the centre, and terminated both ways by the curve, is called a diameter.

6. The points in which any diameter meets the curve are called the vertices of that diameter.

7. The diameter which passes through the foci is called the axis major, and the points in which it meets the curve are called the principal vertices.

8. The diameter at right angles to the axis major is called the axis minor.

Thus, let ABa be an ellipse, S'and H the foci.

Join S,H; bisect the straight line SH in C, and produce it to meet at the curve in A and a.

Through C draw any straight line Pp, terminated by the curve in the points P, p.

Through C draw Bb at right angles to Aa.

S

B

Then, C is the centre, CS or CH the eccentricity. Pp is a diameter, P and p its vertices, Aa is the major axis, Bb is the minor axis.

9. A straight line which meets the curve in any point, but which, being produced both ways, does not cut it, is called a tangent to the curve at that point.

10. A diameter drawn parallel to the tangent at the vertex of any diameter, is called the conjugate diameter to the latter, and the two diameters are called a pair of conjugate diameters. 11. Any straight line drawn parallel to the tangent at the vertex of any diameter and terminated both ways by the curve, is called an ordinate to that diameter.

12. The segments into which any diameter is divided by one of its own ordinates are called the abscissa of the diameter.

13. The ordinate to any diameter, which passes through the focus, is called the parameter of that diameter. Thus, let Pp be any diameter, and

Tt a tangent at P.

Draw the diameter Dd parallel to Tt.

Take any point Q in the curve, draw Qq parallel to Tt, cutting Pp

R

D

ย

Then, Dd is the conjugate diameter to Pp.

T

Qq is the ordinate to the diameter Pp, corresponding to the point Q.

Pv, up are the abscissæ of the diameter Pp, corresponding to the point Q.

Rr is the parameter of the diameter Pp.

14. Any straight line drawn at right angles to the major axis, and terminated both ways by the curve, is called an

ordinate to the axis.

15. The segments into which the major axis is divided by one of its own ordinates are called the abscissa to the axis. 16. The ordinate to the axis which passes through either focus is called the latus rectum.

(It will be proved in Prop. IV., that the tangents at the principal vertices are perpendicular to the major axis; hence, definitions 14, 15, 16, are in reality included in the three which immediately precede them.)

17. If a tangent be drawn at the extremity of the latus rectum and produced to meet the major axis, and if a straight line be drawn through the point of intersection at right angles to the major axis, the tangent is called the focal tangent, and the straight line the directrix.