Ex. 1. To prove the harmonic properties of a complete quadrilateral.

[§ 9, p. 183. Project the quadrilateral into a square. Then the diagonals form a harmonic pencil with the straight lines drawn through their intersection parallel to the sides.

Hence the pencil O is harmonic, &c.

Ex. 2. To prove Pascal's Theorem.

[§ 8, p. 182.

[fig., p. 183.

[p. 186.

Let the conic be projected into a circle, the straight line which joins the intersections of two pairs of opposite sides of the inscribed hexagon being unprojected. Then the proposition is reduced (§ 16) to the following, which is easily proved.

A hexagon inscribed in a circle has two pairs of opposite sides parallel; prove that the remaining sides are parallel.

Ex. 3. Two sides of a triangle inscribed in a conic pass through fixed points; prove that the envelope of the third is a conic.

Let the conic be projected into a circle, the line which joins the fixed points being unprojected. Then two sides of the projected triangle are parallel to fixed lines.

Hence the third side, subtending a constant angle at the circumference and therefore at the centre, envelops a concentric circle.

ORTHOGONAL PROJECTION.

18. If from all points of any plane figure perpendiculars be drawn to a given plane, the feet of the perpendiculars trace out what is called the Orthogonal Projection of the original figure.

Orthogonal projection is a particular case of conical projection, since, when the vertex of the cone is removed to infinity, the generating lines become parallel.

P

19. Parallel straight lines project orthogonally into parallel straight lines.

For planes drawn through the lines perpendicular to the plane of projection are parallel, and therefore cut the plane of projection in parallel lines.

20. Parallel straight lines are to their projections in a

constant ratio.

For if PQ be any straight line (fig., p. 144) and NM its orthogonal projection, then, the inclination of PQ, MN being constant, their ratio is also constant.

21. In general, all theorems which are true of Corresponding Points hold also in Orthogonal Projection. The connection between the methods may be exhibited as follows:

Let a circle be supposed to be orthogonally projected into an ellipse. Let the planes of the curves first move parallel to themselves until the axis of the ellipse coincides with a diameter of the circle, and then revolve about that diameter until they coincide. The distance of any point on the ellipse from the axis will then bear a constant ratio to the distance from the axis of the point of which it is the projection. Hence points and their projections are thus made to Correspond.

EXAMPLES.

1. The projection of a conic will be a hyperbola or an ellipse according as the unprojected line cuts or does not cut the conic.

2. Project two conics into concentric conics.

3. Project a quadrilateral inscribed in a conic into a rectangle inscribed in a circle.

4. Hence prove that the intersection of the diagonals of a quadrilateral inscribed in a conic is the pole of the line joining the intersections of its opposite sides; and that, if a second quadrilateral be formed by drawing tangents at the vertices of the first, the diagonals of the two will meet in a point.

5. Project a conic inscribed in a quadrilateral into a parabola inscribed in an equilateral triangle.

6. Project a conic circumscribing a triangle into a parabola circumscribing an equilateral triangle.

7. Project any conic into a parabola having a given focus. 8. If two triangles be such that the intersections of their corresponding sides lie on a straight line, the straight lines which join their corresponding vertices meet in a point.

9. If a quadrilateral be divided by any straight line, the diagonals of the original and the component quadrilaterals intersect in three points which lie on a straight line.

10. If a triangle be inscribed in a conic, the tangents at the vertices will intersect the opposite sides in three points which lie on a straight line.

11. Show how to project two conics into similar conics.

12. Under what circumstances is it possible to project any number of conics into similar and concentric conics?

13. Project any two conics into conics whose axes are parallel.

14. In a given conic inscribe a triangle whose sides shall pass through fixed points.

15. If through a fixed point O a line be drawn meeting a conic in P, Q, and if {OPQR} be constant, the locus of R will be a conic having double contact with the former.

Project the fixed point to infinity and the conic into a circle. What is the result when {OPQR}=1?

16. If a tangent to a conic meet two fixed tangents in P, Q, and a fixed straight line in R, the locus of a point S, so taken that {PQRS} is constant, will be a conic passing through the intersections of the fixed tangents with the fixed straight line. What does this theorem become when the fixed line is projected to infinity?

17. POP', QOQ, ROR', SOS' being chords of a conic, the conics which pass through O, P, Q, R, S; O, P, Q, R', S' respectively have a common tangent at 0.

18. If Pp, Qq Rr be intersecting chords of a conic, another conic can be described touching the six lines PQ, QR, Rp, pq, qr, rP.

19. Show that there are two solutions of the problem of projecting a conic into a circle having a given centre.

20. Given a cone described upon a conic as base; determine the planes of circular section, and show that they are parallel to one of two fixed planes.

Many of the examples of previous chapters will serve also as examples on projection.

APPENDIX.

In the following articles an Ellipse is considered to be defined as the locus of a point, the sum of whose distances from two fixed points is constant; and a Hyperbola as the locus of a point the difference of whose distances from two fixed points is constant. In either case it is easily seen that the constant length is equal to AA' or 2CA, with the usual notation.

The fixed points are called Foci, and will be denoted by S, H.

I. The straight line which bisects the exterior angle between the focal distances of any point P on the ellipse is the tangent at that point.

Draw YPZ (fig., p. 58) bisecting the angle between HP produced and SP, and let SY, drawn perpendicular to PY, meet HP in R. Then PR = SP.

Also, if Z be any point on the line PY, then SZ RZ.
Therefore

HZ+SZ=HZ+ RZ.

Therefore HZ+ SZ is greater than HR or AA', except when Z coincides with P. Hence the line PY meets the ellipse in one point only, and therefore touches it.

II. It follows that the bisector of the angle SPH, being at right angles to PY, is the normal at P.