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'ƒ+vg+u'h _ v'ƒ+u'g+wh m Putting each member of these equations uf +w'g+v'h + lk = 0, w'f+ vg+uh+mk = 0, = n determine the co-ordinates of the pole. They may also be 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'f+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'ƒ + vg+u'h) ß + (v'f+u'g + wh) y = 0, and in these points only. Hence since two conics in general intersect in four points, it follows that in this case the four points of intersection coincide two and two, that is, the conics touch one another at the two points where they meet the above-mentioned line, or have double contact with each other. The arbitrary constant k may be determined by making the conic pass through any assigned point. Suppose now that the conic is required to pass through the point (f, g, h), of which the line of contact is the polar. This gives, for the determination of k, the condition $(f, g, h) + k {(uf +w'g+v'h) ƒ + (w'ƒ + vg + u'h) g 1 +(v'f+u'g+wh) h}2 = 0, $ (f, g, h) * $ (f, g, h) $ (a, B, 7) − {(uf + w'g + v'h) a + (w'f+vg + u'h) B + (v'ƒ + u'g+ wh) y}2 = 0, represents the curve of the second degree, passing through the point (f, g, h) and touching the conic (a, B, y) = 0, at the points where the polar of this point intersects it. But this curve must evidently be coincident with the two tangents drawn from that point to the given conic & (a, B, y) = 0. This equation may be put under another form, for the coefficient of a2 will be found, by actual expansion, to be u (uf2 + vg2 + wh3 + 2u'gh + 2v'hƒ + 2w'ƒg) − (u2ƒ3 +w”g2 +v22h2 + 2v'w'gh + 2uv'hf +2uw'fg) (uv — w22) g2 + (wu — v'2) h2 + 2 (uu' — v'w') gh Wg2 + Vh2 - 2U'gh. That of 2ẞy is u' (uf2 + vg2 + wh* + 2u'gh + 2v'hf + 2w'fg) 2 = (uu' — v'w') ƒ2 + (u2 — vw) gh + (u'v' — ww') hf + (w'u' — vv′) fg = 1 Similar expressions holding for the coefficients of B2, y2, 2ya, 2aß, we obtain the equation of the two tangents under the form (Wg2+Vh2—2U'gh)a2+(Uh2+Wƒ2−2V'hƒ)B2 +(Vƒ2+Ug2 — 2W'ƒg)y2 − 2 (U'ƒ2 + Ugh — W'hf — V'ƒg) By 2 (W'h2+Wfg — V'gh — U'hf) aẞ = 0. If the point (f, g, h) be within the conic, these two tangents will be imaginary. 11. To find the co-ordinates of the centre. Since the two tangents drawn at the extremities of any chord passing through the centre, are parallel to each other, it follows that the polar of the centre is at an infinite distance, and may therefore be represented by the equation aa+bB + cy= = 0. Hence, if a, B, y, be the co-ordinates of the centre, we obtain, by an investigation similar to that of Art. 9, uvw+2u'v'w' uu'2 - vv2 — ww'2°. These equations determine the centre. -- 12. To find the equation of the asymptotes. Writing a, B, y, for f, g, h, in the investigation of Art. 10, and paying regard to equations (A) of Art. 11, the asymptotes will be found to be represented by the equation or o (a, B, y) & (α, B, y) − {(aa + bß + cy)2} k2 = 0, But, multiplying equations (A) in order by a, B', adding, we get tion or & (α, B, y) + 2A . k = 0. and Hence the asymptotes may be represented by the equa which may (a, B, y) − (a, B, y) = 0, (a, B, y) + 2A. k = 0, be put under the homogeneous form Cái thờ tây (a, B, y) +h (aa +b+cy)=0. But, by the final result of Art. 11, it may be seen that aa+bB+ cy k Ua2 + Vb2 + Wc2 + 2U'bc + 2 V'ca + 2 W'ab whence the equation of the asymptotes becomes (Ua2 + Vb2 + Wc2 +2U'bc+2V'ca + 2 W'ab) $ (a, B, v) This may also be written under the form u, ω, υ', α φ (α, β, γ) + U, CÚ, , alo (a, b, y) + w', v, u', b v', u', w, c a, b, c, 0 U, C, ói (a+b+c)=0. v', u', พ Um COR. It appears, from the preceding investigation, that if a, B, y be the co-ordinates of the centre of the conic represented by the equation $(a, B, y) = ua2 + vß2 + wy2 + Qu'By + 2v'ya + 2w'aß = 0, 13. To find the condition that the conic may be a rectangular hyperbola. If the equations of the asymptotes be la + mB + ny = 0, l'a + m'B + n'y = 0, the condition of their perpendicularity is ll' + mm' + nn' — (mn' + m'n) cos A — (nl' + n'l) cos B Ua2 + Vb2 + Wc2 + 2U'bc + 2 V'ca + 2 W'ab = D, uvw+2u'v'w' — uu'2 · vv2 — ww'2 = K, we see, by reference to Art. 12, that __ (mm' + m'n) __ } (nl' + n'l) = Du - Kbc = Dv - Kca = |