Definition of the Tangential Co-ordinates of a Straight Line Interpretation of the Negative Sign. Equation of certain points 3. General Equation of a Point . Identical relation between the co-ordinates of any straight 5. An equation of the nth degree represents a curve of the nth 6. Equation of a Conic, touching the three sides of the triangle of 7. Equation of circumscribed Conic 8. Equation of the Pole of a given straight line, and of the centre. 9. Circular points at infinity II-13. Tangential rectangular Co-ordinates ON THE INTERSECTION OF CONICS, ON PROJECTIONS, AND ON THE 1-3. Any two conics intersect in four points, real or imaginary. Vertices of the quadrangle formed by these points 4—7. If the four points of intersection be all real, or all imaginary, all the vertices are real. If two of the points of intersection be real, and two imaginary, one vertex only is real. If the four points of intersection be all real, all the common chords are real; if not, one pair only is real 10-12. Any quadrilateral may be projected, in an infinite number of ways, into a parallelogram of which the angles are of any 15. Also into similar and similarly situated curves 16. These projections may be effected in an infinite number of ways ib. 17. Any two intersecting conics may be projected into hyperbolas 18. Any two conics may be projected into conics of any eccentricity, 19. Projection of the foci and directrices of a Conic The anharmonic ratio of any pencil or range is unaltered by 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 The anharmonic ratio of any four points on, or any four tan- 144 23, 24. Any system of Points in involution projects into a system in involution, and the foci of one system project into the foci of 25 A system of Conics, passing through four given points, cut any . If five points be given, one conic only can be drawn If four points and one tangent be given, two conics can be If three points and two tangents be given, four conics can be If two points and three tangents be given, four conics can be If one point and four tangents be given, two conics can be If five tangents be given, one conic only can be drawn ib. CHAPTER I. TRILINEAR CO-ORDINATES. EQUATION OF A STRAIGHT LINE. 1. In the system of co-ordinates ordinarily used, the position of a point in a plane is determined by means of its distances from two given straight lines. In the system of which we are about to treat, the position of a point in a plane will be determined by the ratios of its distances from three given straight lines in that plane, these straight lines not passing through the same point. The triangle formed by these three straight lines is called the triangle of reference, its sides, lines of reference, and the distances of a point from its three sides will be called the trilinear co-ordinates of that point. We shall usually denote the angular points of the triangle of reference by the letters A, B, C, the lengths of the sides respectively opposite to them by a, b, c, and the distances of any point from BC,.CA, AB respectively by the letters a, b, y. When two points lie on opposite sides of a line of reference, the distance of one of these points from that line may be considered as positive, and that of the other as negative. We shall consider a, the distance of a point from the line BC, as positive if the point lie on the same side of that line as the point A does, negative if on the other side; and similarly for 7. It thus appears that the trilinear co-ordinates of any point within the triangle of reference are all positive; while no point has all its co-ordinates negative. 2. Between the trilinear co-ordinates of any point an important relation exists, which we proceed to investigate. If A denote the area of the triangle of reference, a, , y, the trilinear co-ordinates of any point, then ac + b3 + cy=24. Let P be the given point, and first suppose it to lie within B and the triangle of reference (fig. 1). Join PA, PB, PC, and draw PD perpendicular to BC. Then PD=a, and aa=twice the area of the triangle PBC. Fig. 1. : Similarly bB = twice the area of PCA, cy=twice the area of PAB. Adding these equations, we get aa + b3 + cy=2A. Next, suppose P to lie between AB, AC produced, and on the side of BC remote from A (fig. 2). Then a will be Fig. 2. D negative, while B, y are positive. Hence, twice the area PBC will be represented by — ad, and we shall therefore have as before aa + b3 + cy=2A. Thirdly, let P lie between AB, AC, produced backwards (fig. 3), so that B, y are negative while a is positive. Twice Fig. 3. B the areas of PBC, PCA, PAB, are now represented by aa, - bß, - cy respectively, so that we still have aa + b3 + cy=2A. In all cases, therefore, aa + b3 + cy=2A. The importance of the above proposition arises from its enabling us to express any equation in a form homogeneous with respect to the trilinear co-ordinates of any point to which it relates. Any locus may be represented, as in the ordinary system, by means of a relation between two coordinates, ß and y for example, and this may be made homo aa+bB+ oy geneous in a, b, y by multiplying each term by 24 |