EQUATION OF THE TANGENT. 75 we get (uf+w'g+v'h) a + (w'f+vg+u'h) B + (v'f+u'g+wh) y uf2 + vg3 + wh2 + 2u'gh + 2v'hƒ + 2w'fg = = 0, since (f, g, h) is a point on the conic. The tangent, therefore, at (f, g, h) is represented by the equation (uf+w'g+v'h) a + (w'f+vg+u'h) B + (v'f+u'g+wh) y = 0. OBS. Those who are acquainted with the Differential Calculus will remark that this equation may be written thus, 4. To find the condition that a given straight line may touch the conic. Let the equation of the given straight line be la + mB+ny = 0. Let (f, g, h) be the co-ordinates of its point of contact; then, comparing this with the equation of the tangent just investigated, we see that we must have uf+w'g+v'h_w'f+vg+u'h _ v'f+u'g+wh Also, since (f, g, h) is a point on the given line, Eliminating f, g, h, k, between (1), (2), (3), (4), we as the necessary condition that the line (l, m, n) should touch the conic (f, g, h) = 0. Expanding the determinant, this may be written (vw — u'2) 12 + (wu — v′2) m2 + (uv — w’2) n2 + 2 (v'w' — uu') mn + 2 (w'u' — vv') nl + 2 (u'v' — ww') lm = 0. 5. The coefficients of l2, m3, n2, 2mn, 2nl, 2lm, in the above equation, will be observed to be the several minors of the determinant u, w', w', v, u' v', u', w They will frequently present themselves in subsequent investigations, and it will be convenient, therefore, to denote each by a single letter. We shall adopt the following notation: vw - u2=U, wu — v'2 = V, 12 uv-w" = W, v'w' - uu'=U', w'u' - vv' = V', u'v' — ww' = W'. The condition of tangency investigated in Art. 4 may then be written, Ul2 + Vm2 + Wn2 + 2 U'mn + 2 V'nl + 2 W'lm = 0, CONDITION OF TANGENCY. 77 the same condition, it will be observed, as that which must hold, in order that the point (l, m, n) may lie on the conic Uu2 + Vß2 + Wy2 + 2U'ßy + 2 V'yx + 2 W'aß = 0. 6. 1o find the condition that the conic may be a parabola. Since every parabola touches the line at infinity, the condition required will be obtained by writing a, b, c respectively in place of l, m, n, in the condition of tangency. This gives, as the necessary and sufficient relation among the coefficients, u, w', v', a = 0 w', v, u', b v', u', w, C a, b, C, 0 or Ua2 + Vb2+ We2 + 2 U'bc + 2 V'ca + 2 W'ab = 0. 7. To find the condition that the conic may break up into two straight lines, real or imaginary. For this purpose it is necessary and sufficient that the expression (a, B, y) should break up into two factors. The condition for this has been shewn in Art. 9, Chap. III. to be 8. To find the equation of the polar of a given point. If through a given point any straight line be drawn cutting a conic in two points, and at each point of section a tangent be drawn to the curve, the locus of the intersection of these tangents is the polar of the given point. We proceed to find the equation of the polar of (f, g, h). 2 Let f, g, h1; fa, 92, h2 be the co-ordinates of the points in which any straight line drawn through (f, g, h) meets the conic. Then, since (f, g, h), (f1, I1, h1), (ƒ2, I21 h) lie in the same straight line, we have ƒ(g1h2—gh1)+g (h‚ƒ‚— h ̧ƒ1) + h (ƒ12 —ƒ291) = 0...... (1), (see Art. 12, Chap. 1.). Again, the equations of the tangents at (f 91, h), (fa, 92, h) respectively, are 1 f1 (ua+w'ß + v'y) +9, (w'a+vß +u' y) + h ̧ (v'a+u'ß+wy) = 0, ƒ2 (ua+w'ß+v'y) +9, (w'a + vß +u'y) +h ̧ (v'a+u'ß+wy) = 0. Where these intersect, we have ua + w'ß + v'y__ w'a + vß + u'y__ v'a + u'ß + wy = Combining this with equation (1), we get (2). f(ua + w'ß + v′y) + g (w'a + vß + u'y) + h (v'a + u'ß + wy) = 0, or (uf+w'g+v'h) a + (w'f+vg+u'h) B+ (v'f+u'g+wh) y = 0, as a relation which holds at the intersection of the tangents; and which, since it is independent of the values of f, g, h,; f, g, h,; must be the equation of the locus of the point of intersection of the tangents drawn at the extremities of any chord passing through (f, g, h), that is, it is the equation of the polar of (f, g, h). It will be remarked that this equation is identical in form with that already investigated for the tangent at a point (f, g, h) of the curve. In fact, when the point (f, g, h) is on the curve, the polar and the tangent become identical. 9. To find the co-ordinates of the pole of a given straight line. Let the equation of the given straight line be If (f, g, h) be the co-ordinates of its pole, we must have, applying the equation just investigated for the polar of (f, g, h), uf+w'g + v'h_w'f+vg+u'h _ v'f+u'g+wh _ determine the co-ordinates of the pole. They may also be written f = h Ul+W'm+V'n " W'l+Vm+U'n ̄ V'l + U'm +Wn' 10. To find the equation of the pair of tangents drawn to the conic from a given external point. Consider the equation $ (a, B, y) + k {(uf + w'g + v'h) a + (w'f+vg + u'h) B where k is an arbitrary constant. +(v'ƒ+u'g+wh) y}2 = 0, This, being of the second degree in a, B, y, represents a conic; and meets the conic (a, B, y) = 0 in the two points in which that conic meets the line (uf+w'g+v'h) a + (w'f + vg+u'h) B + (v'f +u'g+wh) y = 0, |