one another, as also those of AD, BC, whence it follows that the quadrilateral ABCD is projected into a parallelogram. 11. The angle EVF will be the angle between the projections of the sides AB, BC. For if the plane of projection cut the lines VA, VB, VC, VD in A, B, C, D' respectively, then the points A, B, C, D' are respectively the projections of A, B, C, D. Now the plane ABA'B' contains the points V, E, and, since the plane of projection, in which the points A', B lie, is parallel to VEF, and therefore to VE, it follows that A'B' is parallel to VE. Similarly B'C is parallel to VF, and therefore the angle A'B'C' is equal to the angle EVF, 12. Since the angle EVF may be made of any magnitude, by taking the point V anywhere on any segment of a circle of which EF is the base and which contains an angle of the required magnitude, it follows that any quadrilateral may be projected, in an infinite number of ways, into a parallelogram of which the angles are of any assigned magnitude. 13. We may now proceed to detail the application of the theory of projections to curves of the second degree. It will easily be seen that the projection of any tangent to a conic will be a tangent to the projection of the conic. Again, if any point and straight line be the pole and polar of one another with respect to a given conic, their projections will be the pole and polar of one another with respect to the projection of the conic. For, let O be any given point, XY its polar with respect to any given conic. On XY take any point T, external to Y' x Y the conic, and from T draw two tangents TP, TQ, then PQ will pass through 0. Now project the whole system, and let O', P, Q, T", X', Y' be the respective projections of O, P, Q, T, X, Y. Then TP, T Q will be tangents to the projected conic, and P'Q' will pass through O'. Hence since T" is any point on X'Y', X'Y' will be the polar of O'. 14. From the proposition just proved, it will follow that any two conics may be projected into concentric curves. For it is always possible (Arts. 5 and 7) to find one real point at least, the polar of which with respect to two given conics is the same straight line. Let then this straight line be projected to infinity, and its common pole, with respect to the two conics, will become the centre of the curves of projection. 15. It may also be proved that any two conics may be projected into similar and similarly situated curves. For it is always possible (Arts. 5 and 7) to find two straight lines which meet two given conics in the same two points, real or imaginary. Project either of these straight lines to infinity, and the conics will then be projected into curves, two of the points of intersection of which are infinitely distant, that is, into similar and similarly situated conics. These will be ellipses or hyperbolas, according as the points, in which the line projected to infinity meets the conics, are imaginary or real. If the two conics have double contact with one another, their projections will also be concentric. 16. The projections, spoken of in the last two articles, may be effected in an infinite number of ways. For any point whatever may be taken as the vertex of the cone, and if the cone be cut by a plane, parallel to that which passes through the vertex and the line which it is required to project to infinity, the required projection will be effected. 17. It hence follows that it is possible to project any two intersecting conics into hyperbolas of any assigned eccentricity. Suppose, for example, that it is required to project two conics, intersecting in points A, B, into two similar and similarly situated hyperbolas, the angle between the asymptotes of each being a. Take any point V, such that the angle ANY CONIC MAY BE PROJECTED INTO A CIRCLE. 141 AVB=Q, and describe two cones, of which V is the common vertex, passing through the two given conics. The sections of these cones made by any plane parallel to the plane VAB will be hyperbolas, of which the asymptotes are parallel to VA, VB respectively, and will therefore be similar and similarly situated to one another, and of the required form. 18. We now come to the most important and most difficult point of the theory of projections, the process by which from the properties of the circle those of conic sections in general may be deduced. We have just seen that any two conics may be projected into hyperbolas of any assigned eccentricity. Now this process, the possibility of which we have shewn by a geometrical method, of course admits of algebraical proof. And the algebraical investigation, on account of the continuity of the symbols employed, would not take any account of the restrictions introduced into the geometrical investigation, either as to the conics intersecting in real points, or as to the eccentricity of the conics into which they are projected being greater than unity. It is therefore possible, by an algebraical process, to transform the equations of any two conics whatever into those of conics of any eccentricity, and therefore into those of circles. The points and tangents common to the two given conics will be transformed into points and tangents common to their projections, and the relations of poles and polars will remain unaltered. Since all circles pass through the same two points on the line at infinity, it follows that all circles are transformed by projection into a system of conics passing through the same two points, or having a common chord. Again, since every parabola touches the line at infinity, it follows that all parabolas will project into a system of conics touching the same straight line. A system of parabolas and circles will project into a system in which all the circles will become conics passing through the same two points, and all the parabolas will become conics, having the straight line joining those two points for a common tangent, 19. We have seen (Art. 22, Chap. VI.) that the pair of imaginary tangents, drawn to a conic from any one of its four foci, satisfy the analytical conditions of being asymptotes to a circle. Hence these tangents must themselves meet the line at infinity in the two circular points. Conversely, if from the two circular points at infinity two pairs of tangents be drawn to any conic, these will form an imaginary quadrilateral, circumscribing the conic, the four angular points of which are the four foci of the curve. Hence all conics having the same focus project into conics having a pair of common tangents; and all confocal conics into conics inscribed in the same quadrilateral. The directrix is the polar of the focus, hence, if two conics have the same focus and directrix, they project into two conics having a common chord of contact for their common tangents, that is, having double contact with one another. 20. The anharmonic ratio of any pencil or range is unaltered by projection. Let the transversal PQRS cut the four straight lines OP, OQ, OR, OS. Take any point V, not lying in the plane through these straight lines, join VO, VP, VE, VR, VS, and let these lines be cut by any other plane in 0, P, Q, R, S'. Then {0'. P'Q'R'S} = [PQ'R'S] P'Q.R'S PR. QS Hence the anharmonic ratio of the given pencil and range is the same as that of their projection. a + a с 21. The following proposition is useful in the projection of theorems relating to the magnitude of angles. Any two lines which make an angle A with each other, form with the lines joining the circular points at infinity to their point of intersection, a pencil of which the anharmonic ratio is e(-2A)V-1. It will be understood that the two given lines are taken as the first and third legs of the pencil. Take the two lines as two sides of the triangle of reference, and let them be denoted by ß= 0, y=0. The lines joining their point of intersection to the circular points at infinity are given by eliminating a between the equation of the line at infinity and that of the circumscribing circle, that is, between 5 B g (B-ky) (B+k'ry) = 0 form with B= 0 and y=0 a pencil of which the anharmonic ratio is k k=- €4V-1, k' = -Av=i. AV-1 =-620V-I = T-24)V-1. e-AVCI Cor. In the case in which the lines are at right angles to one another, A=5, and the anharmonic ratio becomes unity, that is, the four lines form an harmonic pencil, This gives T |