Let P be any point of the elliptic section, draw VRPR' the generating line of the cone through P, and join SP, HP.

=

Then PS, PR are drawn from P to touch the sphere centre in S and R, therefore PS PR, and PH, PR touch the sphere centre O in H and R', therefore PH- PR. Hence SP+HP-PR+ PR=RR = KK' = AA'.

Hence the sum of the distances of any point of the ellipse from the points S, H within it is invariable and =AA'.

Hence we obtain the following construction to enable

us to describe an ellipse: Fasten the two ends of a thread to the two points S and H, and let the thread be longer than SH: then stretch it with the point of a pencil, and mark the line which is traced by moving the pencil on all sides of the points so as to keep the thread tightly stretched: the curve so traced will be an ellipse. Viewing the ellipse in its relation to the circle, S and H may be considered as a divided centre, the sum of the distances of all points in the circumference from them being the same.

4. Now let the plane through P, perpendicular to the axis of the cone, cut the cone in the circle QPQ', and the plane of the ellipse in NP at right angles to AA'.

Then we shall have

SP= PR=QK,

and the triangles QAN,

KAX are similar;

.. QK NX :: AK: AX :: AS: AX.

Hence whenever P is

situated on the ellipse, SP: NX is a constant ratio = AS: AX.

And similarly, joining HP,

HP (= PR' = Q'L') : NX :: A'H : A'X'.

The ratios AS: AX, A'H : A'X' are equal, and since A'L' < A'X', each is a ratio of lesser inequality: either of them is called the eccentricity.

So then if we draw the elliptic section in the plane of the paper, and draw through X, X' lines XZ, X'Z' at right

angles to XX, and PM, PM' perpendiculars to these lines; we have for any point P of the ellipse

SP: NX (or PM) :: AS : AX,

HP: NX (or PM') :: A'H : A'X'.

S and H are called the foci of the ellipse, XZ, X'Z the directrices.

5. The position of the foci and directrices is determined

by the following relations: CS+ CB2 = CA:

For joining the foci with the extremity of the axis minor, SB=HB= (SB+HB) = AC, and SC2+BC2 = SB2 = AC2.

Also CX is a third proportional to CS and CA,

For first, SB: CX, i.e. CA: CX:: SA: AX.

or CX is a third proportional to CS and CA.

6. Properties of the secant and tangent.

Let QSq bisect the

Let PSP, P'Sp' be two focal chords. Let the secant PP cut the directrix in F, join SF. vertical opposite angles PSP', pSp': dicular to the directrix.

PM, PM' perpen

Then SP SP' :: PM: PM' :: PF: P'F,

therefore SF bisects the angle PSp' (Euclid VI. A) and is

perpendicular to QSq; hence also the secant pp' will pass through F.

Now let the secant FPP' revolve about F, so that P, P' will approach one another; SQ still bisecting the angle PSP' will be perpendicular to SF and therefore constant in position; and P, P' will finally coincide with Q which will then be a point on the curve, and the secant will then become a tangent to the curve at that point. Similarly if the secant Fp'p turn about F, it will in its limiting position touch the curve in Sq produced. Hence the tangents at the extremity of a focal chord intersect on the directrix: and the part of any tangent between the curve and the directrix subtends a right angle at the focus.

7. The focal distances make equal angles with the tangent at any point.

Let the tangent at P meet the directrices in F, F", join

SF, HF'; PSF, PHF" will be right angles.