(Art. 130), we infer that the locus of the pole of λa + μß + vy with respect to a conic touching the four lines a, ß, y, Aa + Bß + Cy, is the right line λ (μβ + γ – λα) μ (υγ + λα - μβ) + B ν (λα + μβ – νγ) = 0. Or, again, since the condition that the conic should pass through a'ß'y' is J(la') + J(mß') + √(ny') = 0, the locus of the pole of λa + μß + vy with respect to a conic which touches the three lines a, ẞ, y, and passes through a point a'ß'y', is Jλa' (uß + vy —− λa)} + √{μß' (vy + λa − μß)} + J{vy' (\a + μß −— vy)} = 0, which denotes a conic touching μβ + γ – λα, νγ + λα - μβ, λα + μβ - νγ. In the case where the locus of centre is sought, these three lines are the lines joining the middle points of the sides of the triangle formed by a, ß, y. Ex. 3. To find the coordinates of the pole of λa+uß +vy with respect to Iẞy + mya+naß. The tangential equation in this case being, Art. 127, [2x2 + m2μ2 + n2v2 — 2mnμv — 2nlvd the coordinates of the pole are =3 2ίπλμ = 0, a = 1 (D - mu - ni), B = m (mu – n − DX), y = n (n whence my' + nẞ' = — 2lmnλ, na' + ly' = — 21mnμ, lẞ' + ma' = and, as in the last example, we find l, m, n respectively proportional to 2lmny; α' (μβ ́ + νγ' - λα'), β' (υγ' + λα - μβ'), γ' (λα + μβ' - νγ'). Thus, then, since the condition that a conic circumscribing aßy should pass through a fourth point a'ß'y' is + + 0, the locus of the pole of λa + μß + vy, with a' m n which, when the locus of centre is sought, denotes a conic passing through the middle points of the sides of the triangle. The condition that the conic should touch Aa + BB + Cy being J(Al) + √(Bm) + √(Cn) = 0, the locus of the pole of λa + μß + vy, with regard to a conic passing through three points and touching a fixed line, is J{Aa (uß + vy — λa)} + J{Bß (vy + λa − μß)} + √Cy (λa + μß − vy) = 0, which, in general, represents a curve of the fourth degree. 294. If a"B"y" be any point on any of the tangents drawn to a curve from a fixed point a'B'y', the line joining a’'B'y', a′′B”y′′ meets the curve in two coincident points, and the equation in 7: m (Art. 290), which determines the points where the joining line meets the curve, will have equal roots. To find, then, the equation of all the tangents which can be drawn through a'B'y', we must substitute la + ma', 18+mB', ly+my' in the equation of the curve, and form the condition that the resulting equation in 1: m shall have equal roots. Thus (see Art. 92) the equation of the pair of tangents to a conic is SSP, where S= aa2 + &c., S'aa'2+ &c., P= aaa' + &c. = This equation may also be written in another form; for since any point on either tangent through a'B'y evidently possesses the property that the line joining it to a'B'y' touches the curve, we have only to express the condition that the line joining two points (Art. 65) - a (B'y” — B'y') + B (ya” — y'a') + y (a ́ß" — a′′ B') = 0 should touch the curve, and then consider a"B"y" variable, when we shall have the equation of the pair of tangents. In other words, we are to substitute By- B'y, ya' — y'a, aßaß for λ, μ, v in the condition of Art. 285, Αλ + Β' + εν + 2 μν + 2 Gνλ + 2 Ηλμ = 0. Attending to the values given (Art. 285) for A, B, &c., it may easily be verified that (aa2 + &c.) (aa2 + &c.) − (aaa' + &c.)2 = A (By' — B′y)2 + &c. Ex. To find the locus of intersection of tangents which cut at right angles to a conic given by the general equation (see Ex. 4, p. 169). We see now that the equation of the pair of tangents through any point (Art. 147) may also be written — 2F (x − x′) (xy' — yx′) + 2G (y — y) (xy' — x'y) — 2H (x − x') (y − y) = 0. This will represent two right lines at right angles when the sum of the coefficients of x and y2 vanishes, which gives for the equation of the locus C(x2+ y2)-2Gx - 2Fy + A + B = 0. This circle has been called the director circle of the conic. When the curve is a parabola, C = 0, and we see that the equation of the directrix is Gx + Fy = § (A + B). 295. It follows, as a particular case of the last, that the pairs of tangents from By, ya, aß are By2+ CB2 – 2Fßy, · Ca3 + Ay3 – 2 Gya, Aß2 + Ba3 − 2Hαß, as indeed might be seen directly by throwing the equation of the curve into the form (aa+hB+gy)2 + (CB2 + By2 − 2Fßy) = 0. Now if the pair of tangents through By be ẞ – ky, B-k'y, it appears from these expressions that kk' = B and that the corre sponding quantities for the other pairs of tangents are = CA A' B' and these three multiplied together are 1. Hence, recollecting the meaning of k (Art. 54), we learn that if A, F, B, D, C, E be the angles of a circumscribing hexagon, = .1. sin EAB.sin FAB.sin FBC.sin DBC.sin DCA. sin ECA sin EAC.sin FAC. sin FBA.sin DBA.sin DCB.sin ECB Hence also three pairs of lines will touch the same conic if their equations can be thrown into the form M2+ N2+ 2ƒ'MN= 0, N2+ L2+ 2g′NL = 0, L2 + M2 + 2h′EM = 0, for the equations of the three pairs of tangents, already found can be thrown into this form by writing L√(A) for &c. 296. If we wish to form the equations of the lines joining to a'B'y all the points of intersection of two curves, we have only to substitute la + ma', 13 + mß', ly+my' in both equations, and eliminate : m from the resulting equations. For any point on any of the lines in question evidently possesses the property that the line joining it to a'B'y meets both curves in the same point; therefore the equations in 7: m, which determine the points where one of these lines meets both curves, must have a common root; and therefore the result of elimination between them is satisfied. Thus, the equation of the pair of lines joining to a'B'y' the points where any right line L meets S, is L*S-2LL'P+ L'S=0. If the point a''y be on the curve the equation reduces to L'S- 2LP=0. Ex. A chord which subtends a right angle at a given point on the curve passes through a fixed point (Ex. 2, Art. 181). We use the general equation, and by the formula last given, form the equation of the lines joining the given point to the intersection of the conic with Ax+uy+v. The coordinates being supposed rectangular, these lines will be at right angles if the sum of the coefficients of z2 and y2 vanish, which gives the condition And since λ, μ, v enter in the first degree, the chord passes through a fixed point, α- b b-a viz. b+ a a+ x', by'. If the point on the curve vary, this other point will describe a conic. If the angle subtended at the given point be not a right angle, or if the angle be a right angle, but the given point not on the curve, the condition found in like manner will contain λ, μ, in the second degree, and the chord will envelope a conic. 297. Since the equation of the polar of a point involves the coefficients of the equation in the first degree, if an indeterminate enter in the first degree into the equation of a conic it will enter in the first degree into the equation of the polar. Thus, if P and P be the polars of a point with regard to two conics S, S', then the polar of the same point with regard to S+kS will be P+kP'. For (a + ka') aa' + &c. = aaa' + &c. + k {a'aa' + &c.}. Hence, given four points on a conic, the polar of any given point passes through a fixed point (Ex. 2, Art. 151). If Q and be the polars of another point with regard to S and S, then the polar of this second point with regard to S+kS' is Q+kQ. Thus, then (see Art. 59), the polars of two points with regard to a system of conics through four points form two homographic pencils of lines. Given two homographic pencils of lines, the locus of the intersection of the corresponding lines of the pencils is a conic through the vertices of the pencils. For, if we eliminate k between P+kP', Q+kQ', we get PQ'P'Q. In the particular case under consideration, the intersection of P+kP, Q+kQ is the pole with respect to S+kS' of the line joining the two given points. And we see that, given four points on a conic, the locus of the pole of a given line is a conic (Ex. 1, Art. 278). If an indeterminate enter in the second degree into the equation of a conic, it must also enter in the second degree into the equation of the polar of a given point, which will then envelope a conic. Thus, if a conic have double contact with two fixed conics, the polar of a fixed point will envelope one of three fixed conics; for the equation of each system of conics in Art. 287 contains μ in the second degree. We shall in another chapter enter into fuller details respecting the general equation, and here add a few examples illustrative of the principles already explained. Ex. 1. A point moves along a fixed line; find the locus of the intersection of its polars with regard to two fixed conics. If the polars of any two points a'ß'y', a"ß"y" on the given line with respect to the two conics be P', P"; Q', Q"; then any other point on the line is λa' + pa", λß′ + μß", λy' + μy"; and its polars AP' +μP” AQ'′+uQ", which intersect on the conic P'Q" = P"Qʻ. Ex. 2. The anharmonic ratio of four points on a right line is the same as that of their four polars. For the anharmonic ratio of the four points la' + ma", l'a' + m'a", l''a' + m"a", l'"'a' + m""a", is evidently the same as that of the four lines IP' + mP", l'P' + m'P", l''P' +m"P", l'"'P' + m'"P". Ex. 3. To find the equation of the pair of tangents at the points where a conic S is met by the line y. The equation of the polar of any point on y is (Art. 291) a'S, + ß'S2 = 0. But the points where y meets the curve are found by making y = 0 in the general equation, whence aa” + 2ha + bß” = 0. Eliminating a', ' between these equations, we get for the equation of the pair of tangents Thus the equation of the asymptotes of a conic (given by the Cartesian equation) is for the asymptotes are the tangents at the points where the curve is met by the line at infinity z. Ex. 4. Given three points on a conic: if one asymptote pass through a fixed point, the other will envelope a conic touching the sides of the given triangle. If t, to be the asymptotes, and aa + bẞ+cy the line at infinity, the equation of the conic is tt (aa + bß + cy)2. But since it passes through ẞy, ya, aß, the equation must not contain the terms a2, 62, y2. If therefore t, be λa + μß + vy, to must a2 b2 be a+ B+ Y; and if t, pass through a'ß'y', then (Ex. 1, Art. 285) t, touches μ a J(aa') + b ↓(BB′) + c √(vy') = 0. The same argument proves that if a conic pass through three fixed points, and if one of its chords of intersection with a conic given b by the general equation aa2 + &c. = 0 be λa+μß+vy, the other will be a+ a B+-y. V Ex. 5. Given a self conjugate triangle with regard to a conic: if one chord of intersection with a fixed conic (given by the general equation) pass through a fixed point, the other will envelope a conic [Mr. Burnside]. The terms aß, ßy, ya are now to disappear from the equation, whence if one chord be λa + μß + vy, the other is found to be λa (ug + vh − λf) + μß (vh + \ƒ − μg) + vy (\ƒ + μg — vh). Ex. 6. A and A' (α1ẞ1Y1, α2ß2Y1⁄2) are the points of contact of a common tangent to two conics U, V; P and P' are variable points, one on each conic; find the locus of C, the intersection of AP, A'P', if PP' pass through a fixed point O on the common tangent [Mr. Williamson]. Let P and Q denote the polars of aẞ1Y1, a2ß272, with respect to U and V respectively; then (Art. 290) if aßy be the coordinates of C, those of the point P where AC meets the conic again, are Ua, – 2Pa, Uß, – 2Pß, Uy, — 2Py; and those of the point P' are, in like manner, Va2 - 2Qa, &c. If the line joining these points pass through 0, which we choose as the intersection of a, ß, we must have and when A, A', O are unrestricted in position, the locus is a curve of the fourth order. If, however, these points be in a right line, we may choose this for the line a, and making a, and a2 = 0, the preceding equation becomes divisible by a, and reduces to the curve of the third order PVB, QUB1. Further, if the given points = |