it cut the original plane in the line K'L'. Let the two straight lines AOA', BOB′ meet the lines KL, K'L' in the points A, B and A', B' respectively; and let VO meet the plane of projection in O'. Then AO' and BO' are the projections of AOA' and BOB'.

Since the planes VA'B', AO'B are parallel, and parallel planes are cut by the same plane in parallel lines, the lines VA', VB' are parallel respectively to AO, BO. The angle A'VB' is therefore equal to the angle AO'B, that is, A′ VB' is equal to the angle into which AOB is projected.

K

B

Similarly, if the straight lines CD, ED, meet K'L' in C', D' respectively, the angle C'VD' will be equal to the angle into which CDE is projected.

From the above we obtain the fundamental proposition in the theory of projections, viz.,

Any straight line can be projected to infinity, and at the same time any two angles into given angles.

For, let the straight lines bounding the two angles meet the line which is to be projected to infinity in the points

A', B' and C', D'; draw any plane through A'B'C'D', and in that plane draw segments of circles through A', B' and C', D' respectively containing angles equal to the two given angles. Either of the points of intersection of these segments of circles may be taken for the centre of projection, and the plane of projection must be taken parallel to the plane we have drawn through A'B'C'D'.

If the segments do not meet, the centre of projection is imaginary.

Ex. 1. To shew that any quadrilateral can be projected into a square. Let ABCD be the quadrilateral; and let P, Q [see figure to Art. 60] be the points of intersection of a pair of opposite sides, and let the diagonals BD, AC meet the line PQ in the points S, R. Then, if we project PQ to infinity and at the same time the angles PDQ and ROS into right angles, the projection must be a square. For, since PQ is projected to infinity, the pairs of opposite sides of the projection will be parallel, that is to say, the projection is a parallelogram; also one of the angles of the parallelogram is a right angle, and the angle between the diagonals is a right angle; hence the projection is a square.

Ex. 2. To shew that the triangle formed by the diagonals of a quadrilateral is self-polar with respect to any conic which touches the sides of the quadrilateral.

Project the quadrilateral into a square; then, the circle circumscribing the square is the director-circle of the conic, therefore the intersection of the diagonals of the square is the centre of the conic.

Now the polar of the centre is the line at infinity; hence the polar of the point of intersection of two of the diagonals is the third diagonal.

Ex. 3. If a conic be inscribed in a quadrilateral the line joining two of the points of contact will pass through one of the angular points of the triangle formed by the diagonals of the quadrilateral.

Ex. 4. If ABC be a triangle circumscribing a parabola, and the parallelograms ABA'C, BCB'A, and CAC'B be completed; then the chords of contact will pass respectively through A', B',

C'.

This is a particular case of Ex. 3, one side of the quadrilateral being the line at infinity.

Ex. 5. If the three lines joining the angular points of two triangles meet in a point, the three points of intersection of corresponding sides will lie on a straight line.

ANY CONIC PROJECTED INTO A CIRCLE. 327

Project two of the points of intersection of corresponding sides to infinity, then two pairs of corresponding sides will be parallel, and it is easy to shew that the third pair will also be parallel.

Ex. 6. Any two conics can be projected into concentric conics. [See Art. 283.]

318. Any conic can be projected into a circle having the projection of any given point for centre.

Let O be the point whose projection is to be the centre of the projected curve.

Let P be any point on the polar of O, and let OQ be the polar of P; then OP and OQ are conjugate lines.

Take OP', OQ another pair of conjugate lines.

Then project the polar of O to infinity, and the angles POQ, P'OQ' into right angles. We shall then have a conic whose centre is the projection of O, and since two pairs of conjugate diameters are at right angles, the conic is a circle.

319. A system of conics inscribed in a quadrilateral can be projected into confocal conics.

Let two of the sides of the quadrilateral intersect in the point A, and the other two in the point B. Draw any conic through the points A, B, and project this conic into a circle, the line AB being projected to infinity; then, A, B are projected into the circular points at infinity, and since the tangents from the circular points at infinity to

all the conics of the system are the same, the conics must be confocal.

Ex. 1. Conics through four given points can be projected into coaxial

circles.

For, project the line joining two of the points to infinity, and one of the conics into a circle; then all the conics will be projected into circles, for they all go through the circular points at infinity.

Ex. 2. Conics which have double contact with one another can be projected into concentric circles.

Ex. 3. The three points of intersection of opposite sides of a hexagon inscribed in a conic lie on a straight line. [Pascal's Theorem.]

Project the conic into a circle, and the line joining the points of intersection of two pairs of opposite sides to infinity; then we have to prove that if two pairs of opposite sides of a hexagon inscribed in a circle are parallel, the third pair are also parallel.

Ex. 4. Shew that all conics through four fixed points can be projected into rectangular hyperbolas.

There are three pairs of lines through the four points, and if two of the angles between these pairs of lines be projected into right angles, all the conics will be projected into rectangular hyperbolas. [Art. 187, Ex. 1.] Ex. 5. Any three chords of a conic can be projected into equal chords of a circle.

Let AA', BB', CC' be the chords; let AB', A'B meet in K, and AC', A'C in L. Project the conic into a circle, KL being projected to infinity.

Ex. 6. If two triangles are self polar with respect to a conic, their six angular points are on a conic, and their six sides touch a conic. Let the triangles be ABC, A'B'C'. Project BC to infinity, and the conic into a circle; then A is projected into the centre of the circle, and AB, AC are at right angles, since ABC is self polar; also, since A'B'C' is self polar with respect to the circle, A is the orthocentre of the triangle A'B'C'.

Now a rectangular hyperbola through A', B', C' will pass through A, and a rectangular hyperbola through B will go through C. Hence, since a rectangular hyperbola can be drawn through any four points, the six points A, B, C, A', B', C' are on a conic.

Also a parabola can be drawn to touch the four straight lines B'C', C'A', A'B', AB. And A is on the directrix of the parabola [Art. 107 (3)]; therefore AC is a tangent. Hence a conic touches the six sides of the two triangles.

CROSS RATIOS UNALTERED BY PROJECTION.

329

320. Properties of a figure which are true for any projection of that figure are called projective properties. In general such properties do not involve magnitudes. There are however some projective properties in which the magnitudes of lines and angles are involved: the most important of these is the following:

The cross ratios of pencils and ranges are unaltered by projection.

Let A, B, C, D be four points in a straight line, and A', B', C', D' be their projections. Then, if V be the centre of projection, VAA', VBB, VCC, VDD' are straight lines; and we have [Art. 55]

{ABCD} =V{ABCD} = {A'B'C'D'}.

If we have any pencil of four straight lines meeting in O, and these be cut by any transversal in A, B, C, D; then O {ABCD} = {ABCD} = V{ABCD} = {A'B'C'D'}

= 0′ {A'B'C'D'}. From the above together with Article 62 it follows that if any number of points be in involution, their projections will be in involution.

Ex. 1. Any chord of a conic through a given point O is divided harmonically by the curve and the polar of 0.

Project the polar of O to infinity, then O is the centre of the projection, the chord therefore is bisected in O, and {POQ∞ } is harmonic when PO=0Q.

Ex. 2. Conics through four fixed points are cut by any straight line in pairs of points in involution. [Desargue's Theorem].

Project two of the points into the circular points at infinity, then the conics are projected into co-axial circles, and the proposition is obvious.

321. The cross ratio of the pencil formed by four intersecting straight lines is equal to that of the range formed by their poles with respect to any conic.

Since the cross ratios of pencils and ranges are unaltered by projection, we may project the conic into a circle. Now in a circle any straight line is perpendicular to the line joining the centre of the circle to its pole with