Hence

PG.PF-PN. PR Cn. Ct=BC2.

Similarly a circle on rg as diameter will pass through n and F, and PF. Pg = Pn. Pr = CN .CT= AC2.

11. NG: NC:: BC2: AC2.

If we draw a circle on the axis major of the ellipse as diameter it is called the auxiliary circle, and will be equal to that which projects into the ellipse; and if the projected circle were made to revolve about its diameter AA', it would come to coincide with the auxiliary circle: and the point which, when the circle was projected, projected into P will now be found in the ordinate NP produced, and if this point be called Q we shall have NP: NQ :: BC: AC. Also the tangent to the circle which projects into the tangent PT passes through T, and when made to revolve about AA' will come into the position QT: hence the tangent to the auxiliary circle at Q will meet AA' produced in the same point as the tangent to the ellipse at P.

Hence we shall have

А

NG.NT-NP2 and NC. NT=NQ2 ;

A'

.. NG: NC:: NP: NQ:: BC: AC.

12. By the reasoning

of the last article we see that if CP, CD be conjugate semi-diameters, Q, R the points in which the ordinates NP, MD produced meet the auxiliary circle, QCR will be a right angle.

M

D

N

Hence the triangles CQN, RCM will be equal in all respects. Hence

=

CM QN, and CN2 + CM2 = CN2 + QN2 = CQ2 = A C2.

Also QN CM, RM= CN; .. QN2 + RM2 = AC2, and PN2+DM2: QN2+RM2 :: BC2 : AC2; ..PN2 + DM2= BC2, hence CP2+ CD2 = CN2 + CM2 + PN2 + DM2 = AC2+BC2.

13. In the circle if the chords Qq, Rr intersect in O, we have QO. Oq = RO. Or; and if we draw radii CP, CD parallel to the chords we have

Q0. Oq: CP2:: RO. Or; CD2.

D

PR

2

These ratios are not affected by projection, and the same proportion holds in the ellipse: it is independent of the position of O, but depends only on the direction in which the chords are drawn. It is equally true if O be a point exterior to the ellipse. And if tangents be drawn to an ellipse from any point, their lengths will be proportional to the parallel semi-diameters.

Hence if a circle intersect an ellipse in four points and the common chords be drawn, it will easily appear that the

J. C. S.

2

semi-diameters to which these chords are parallel must be equal, and therefore they and the chords parallel to them equally inclined to the axes.

14. The area of the parallelogram circumscribing an ellipse and touching it at the extremities of conjugate diameters = 4AC. BC.

We have seen that the tangents at the extremities of any diameter are parallel, and if these be drawn at the extremities of two conjugate diameters they will form a parallelogram which is the projection of a square circumscribing the circle whose diameter = 2A C, and its area therefore = 44 C. But by projection this area is diminished in the ratio of AC: BC. Hence the area of the parallelogram circumscribing the ellipse having its sides parallel to any pair of conjugate diameters = 4AC. BC.

COR. If PF be the perpendicular from P on the diameter DCd conjugate to CP, then PF. CD=AC.BC.

EXAMPLES.

1. If an ellipse be projected on a plane intersecting its plane in BC, and having with its plane the maximum ratio of diminution AC: BC, it will project into a circle on BC as diameter.

2. If from any point P of an ellipse PQ be drawn parallel to the minor axis to meet the auxiliary circle in Q, and PR parallel to the major axis to meet the circle on the minor axis in R, QR produced will pass through the common centre.

3. A rectilinear figure circumscribing an ellipse and having its sides bisected at the points of contact is the projection of one regular polygon, and will project into another.

4. All such rectilinear figures of a given number of sides are equal in area however placed about the curve.

5. Two triangles circumscribing an ellipse have their sides bisected at the points of contact; the triangular corners cut off from the two triangles, from each by the sides of the other, are equal.

6. A rectilinear figure inscribed in an ellipse has one of its sides bisected at V, and CV produced meets the curve in P; if CV: CP is a constant ratio the same for all the sides, the figure is the projection of one regular polygon and will project into another.

7. If PCP', DCD' are conjugate diameters, then PD, PD' are proportional to the diameters parallel to them.

8. All triangles circumscribing an ellipse which have the line joining each angular point with the point of contact of the opposite side passing through the centre, are equal and are less than any other that has not this property.

9. Any two diameters bisecting supplemental chords are conjugate.

10. Diameters which coincide with the diagonals of the parallelogram on the axes are equal and conjugate.

11. A straight line parallel to the axis major joins the extremities of equal conjugate diameters and intersects the circle on the axis minor in two points. The lines from the centre to these points are at right angles to one another.

12. The length of the line joining the extremities of equal conjugate diameters is independent of the magnitude of that axis to which it is at right angles.

13. The diagonals of any parallelogram formed by tangents at the extremities of conjugate diameters coincide with conjugate diameters.

14. The angular points of these parallelograms lie on one ellipse concentric, similar and similarly situated with the original.

15. In a circle, if the tangents be drawn at the extremities of a chord which passes through a fixed point within the circle, the point of intersection of the pairs of tangents lies always on a straight line. This is called the polar of the point which is called the pole. If be the pole, the polar is at right angles to CQ produced, and if CQ produced meets the polar in T, then CQ . CT: square on the radius. Prove this, and transfer the property to the ellipse, shewing that if CQT meets the ellipse in P, CQ. CT = CP2.

16. If in each of two ellipses the two axes are in the same ratio to one another, shew that radii which are equally inclined to either (say the major) axis in both bear to one another a constant ratio, i.e. the curves are similar.

17. Parallel elliptic sections of the same cone are similar.

18. If in Ex. 15, Q lies always upon an ellipse similar and similarly situated to the original ellipse, intersecting it and having its tangents at the points of intersection meeting in C, then T will be found upon the part of the second ellipse which lies without the curve.

19. The rectangle of the segments of any tangent intercepted between two parallel tangents made by the point of contact = square on the parallel semi-diameter.

.

20. The rectangle of the intercepts on parallel tangents made by any other tangent = square on the semi-diameter parallel to them.

21. If any tangent meet any two conjugate diameters, the rectangle under its segments = square on the parallel semi-diameter.

22. If from the extremity of each of any two semi-diameters ordinates be drawn to the other, the two triangles so formed will be equal in area.