angles SPO, NPO, and angle SPO = angle NPO by construction,

.. SO = NO,

and angle SOP = angle NOP.

Again, since SO = NO, Op common to the triangles SOp, NOP, and angle SOp angle NOp,

.. SpNp.

and pn is drawn perpendicular to the directrix,

That is, the hypothenuse of a right-angled triangle equal to one of the sides, which is impossible, .. p is not a point in the curve; and in the same manner it may be proved that no point in the straight line Tt can be in the curve, except P.

.. Tt is a tangent to the curve at P.

Cor. 1. A tangent at the vertex A, is perpendicular to the axis.

Cor. 3. Let Qq be an ordinate to the diameter PW, cutting SP in x. ૧

The subtangent to the axis is equal to twice the abscissa, That is,

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 I'

SP: SY:: SY: SA,

or,

SP: SY :: SY: SA · SP = ST by Prop. 3. cor. 2.

Cor. 1. Multiplying extremes and means,

SY2 = SP. SA.

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

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

A diameter bisects all its own ordinates.

Hence the Proposition may be thus enunciated :

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 H

B P

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.