Suppose them to be PQ and RS. The intersection of these two straight lines is given by the equations (L3M'2 — L'2M2)3 ß = (L'N2 — L2N'2)3 y, (L3M'2 — L'2M2)§ ß = — (L'2N2 — L3N') 3 √, which evidently give ẞ= 0, y = 0. Hence PQ, RS intersect in A. Similarly, PR, QS intersect in B, and PS, QR intersect in C. Hence, the angular points of the triangle of reference coincide with the intersections of the line joining each pair of points of intersection of the conics with the line joining the other pair. Hence also, if any number of conic sections be described about the same quadrangle*, and the diagonals of that quadrangle intersect in A, while the sides produced intersect in B and C, then A, B, C form, with respect to each of the circumscribing conics, a conjugate triad. The points A, B, C may themselves be called vertices of the quadrangle, or of the system of circumscribing conics. It will be seen, from the preceding investigation, that any two conics which intersect in four real points can be reduced, by a proper choice of the triangle of reference, to the form L3a2 + M2ß2 + N2y2 = 0. The same reduction may also be effected in every case with the reservation that if two of the points of intersection of the conics be real and two imaginary, then two of the angular points of the triangle of reference (or vertices) will be imaginary and the remaining one real. If all the points of intersection be imaginary, the vertices of the conics will be all real. This we shall prove hereafter. 16. To find the condition that a given straight line may touch the conic. Let the equation of the straight line be la + mB+ny = 0. I employ the term quadrangle in preference to quadrilateral, considering a quadrangle as a figure primarily determined by four points, a quadrilateral by four indefinite straight lines. CONDITION OF TANGENCY. Where this meets the conic, we have Ľ2 (mß + ny)2 + 1a (M2ß2 + Ñ3y) = 0, and, making the two values of ß : y equal, we get whence or (L3m2 + M3l2) (L3n2 + Ñ3l2) = L*m23n2, M2N2l2 + N2L3m2 + L3M2n2 = 0, 72 m2 n2 the required condition. bola. = 0, 49 17. To find the condition that the conic may be a para Since every parabola satisfies the analytical condition of touching the line aa+bB+cy = 0, the required condition becomes 18. To find the co-ordinates of the centre. 1 Let B,, B, be the points in which the conic is cut by CA, then, if B, B1 be bisected in Q, the line BQ will pass through the centre. Now, let f,, 0, h, be the co-ordinates of B, hg + h1 fs + fi Now f, f, are the values of a given by the equations which, eliminating ß, y, are equivalent to This gives one straight line on which the centre lies. It may be similarly proved to lie on the straight line Therefore the co-ordinates of the centre are given by the equations L'a MB Ny a Combining therewith 38 _ N b SELF-CONJUGATE CIRCLE. 51 Each of these becomes infinite when the conic is a parabola, as manifestly ought to be the case. 19. To find the equation of the circle with respect to which the triangle of reference is self-conjugate. It is a distinguishing property of the circle that the line joining the centre with any other point is perpendicular to the polar of that point. Hence the line which joins the centre with the point A, must be perpendicular to a 0. This gives (see Art. 5, p. 8) = Similarly, since the lines joining the centre with B, C are respectively perpendicular to B=0, y=0, we shall have N2 C cos C or a cos A' a cos Ab cos B Hence the equation of the required circle is a cos A. a2+b cos B.B+ccos C. y2 = 0, sin 24. a2 + sin 2B. ß2 + sin 2 C. y2 = 0. It will be remarked that this circle will be imaginary, unless one of the quantities sin 24, sin 2B, sin 2C be negative, that is, unless one of the angles 2A, 2B, 2C be greater than two right angles, or unless the triangle of reference be obtuse-angled. COR. By referring to the expressions for the co-ordinates of the centre of the conic, given in Art. 18, we see that at the centre of the circle we have a cos Aẞ cos B = y cos C. Or, the centre of the circle, with respect to which the triangle of reference is self-conjugate, coincides with the intersecof the perpendiculars drawn from the angular points to the opposite sides. This is otherwise evident from geometrical considerations. 20. To find the equation of the conic which touches two sides of the triangle of reference in the points where they meet the third. Let AB, AC be the two sides which the required conic touches in the points B, C. We then require that the constants in the equation La2 + MB2 + Ny2+2λßy + lμya + 2vaß=0 should be so related to one another, that when ẞ=0 we have the two values of a=0, and also when y = 0 the two values of a may each = Hence the two equations La2 + Ny2+2μya = 0, La+MB2+2vaẞ=0, must both be identically satisfied when a=0, and by no other value. This requires that N=0, μ=0, M=0, v=0. Hence the equation reduces to This equation, it will be observed, involves only one arbitrary constant, as ought to be the case, since when a tangent and its point of contact are given, the conic is thus subjected |