is QkQ'. 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, p. 243). 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' + μa", \ß' + μß", λy' + μy"; and its polars AP' + μP", AQ' +μQ", 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 comic S is met by the line y. The equation of the polar of any point on y is (Art. 291) a'S1 + B'S2 = 0. But the points where y meets the curve, are found by making y = 0 in the general equation, whence aa2+2ha'B' + bß^2 = 0. Eliminating a', ß', between these equations, we get for the equation of the pair of tangents a S2 - 2h SS2 + b‹S12 = 0. 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 t1, to be the asymptotes, and aa + bẞ + cy the line at infinity, the equation of the conic is tit2 = (aa + bẞ+cy)2. But since it passes through By, ya, aß, the equation must not contain the terms a2, 62, y2. If therefore t1 be λa + μß + vy, t2 must a2 b2 c2 be a+ ī μ B+ y; and if to pass through a'ß'y' then (Ex. 1, p. 250) t, touches ע a Jaa') + b √(BB′) + c √(yy') = 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 be λα + μβ + νγ, the other will be a+ β μ α с B+ r. 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 - Xƒ) + μß (vh + \ƒ − μg) + vy (\ƒ + μg − vh). Ex. 6. A and A′ (α1ß1Y1, α1⁄2ß2Y2) 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 0 on the common tangent [Mr. Williamson]. Let P and Q denote the polars of aẞ171, a2ß2Y2, with respect to U and V respectively; then (Art. 290) if aßy be the co-ordinates of C, those of the point P where AC meets the conic again, are Uα1 – 2Pα, Uẞ1 — 2Pß, Uî, — 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 a1 and a2 = 0, the preceding equation becomes divisible by a, and reduces to the curve of the third order PVß2 = QUB1, Further, if the given points are points of contact of a common tangent, P and Q represent the same line; and another factor divides out of the equation which reduces to one of the form U = kV, representing a conic through the intersection of the given conics. Ex. 7. To inscribe in a conic, given by the general equation, a triangle whose sides pass through the three points ßy, ya, aß. We shall, as before, write S1, S2, S3 for the three quantities, aa + hẞ+gy, ha + bß +ƒ¥, ga+ƒB+cy. Now we have seen, in general, that the line joining any point on the curve aẞy to another point a'ẞ'y' meets the curve again in a point, whose co-ordinates are S'a - 2P'a', S'ß – 2P'ß', S'y – 2P'y. Now if the point a’ß'y' be the intersection of lines ß, y, we may take a' = 1, ẞ' = 0, y' = 0, which gives S' = a, PS1, and the co-ordinates of the point where the line joining aßy to ẞy meets the curve, are aa – 281, aß, aɣ. In like manner, the line joining aßy to ya, meets the curve again in ba, bß – 2S1⁄2, by. The line joining these two points will pass through aß, if which is the condition to be fulfilled by the co-ordinates of the vertex. Writing in this equation aa = S1 − hß − gy, bß = S2 — ha − ƒy, it becomes - h (αS1 + BS2) + y (ƒSx+gS2) = 0. But since aẞy is on the curve, aS1 + ßS1⁄2 + y§,= 0, and the equation last written reduces to y (fS1+gS2-hS3), = 0. E Now the factory may be set aside as irrelevant to the geometric solution of the problem; for although either of the points where y meets the curve fulfils the condition which we have expressed analytically, namely, that if it be joined to ẞy and to ya, the joining lines meet the curve again in points which lie on a line with aß; yet, since these joining lines coincide, they cannot be sides of a triangle. The vertex of the sought triangle is therefore either of the points where the curve is met by fS1+ gs2-hS. It can be verified immediately that ƒS1 = gS2 = hS, denote the lines joining the corresponding vertices of the triangles aßy, S1S2S3. Consequently (see Ex. 2, p. 58), the line ƒS + gS2-hS2 is constructed as follows: "Form the triangle DEF whose sides are the polars of the given points A, B, C'; let the lines joining the corresponding vertices of the two triangles meet the opposite sides of the polar triangle in L, M, N; then the lines LM, MN, NL pass through the vertices of the required triangles." F L M N The truth of this construction is easily shown geometrically: for if we suppose that we have drawn the two triangles 123, 456 which can be drawn through the points A, B, C; then applying Pascal's theorem to the hexagon 123456, we see that the line BC passes through the intersection of 16, 34. But this latter point is the pole of AL (Ex. 1, p. 143). Conversely, then AL passes through the pole of BC, and L is on the polar of A (Ex. 1, p. 143). This construction becomes indeterminate if the triangle is self conjugate in which case the problem admits of an infinity of solutions. Ex. 8. If two conics have double contact, any tangent to the one is cut har monically at its point of contact, the points where it meets the other, and where it meets the chord of contact. If in the equation S + R2 = 0, we substitute la' + ma”, lß' + mß", ly' +my", for a, ß, y, (where the points a'ß'y', a"B"y" satisfy the equation S= 0), we get (IR′+mR")2 + 2lmP = 0. Now, if the line joining a'ß'y', a"ß"y", touch S+R2, this equation must be a perfect square and it is evident that the only way this can happen is if P−−2R′R", when the equation becomes (IR' - mR")2 = 0; when the truth of the theorem is manifest. Ex. 9. Find the equation of the conic touching five lines, viz. a, ß, y, Aa+BB+CY, A'a + B'B+C'y. Ans. (la) + (mp)2 + (ny), where l, m, n are determined by the conditions Ex. 10. Find the equation of the conic touching the five lines, a, ß, y, a + B + 7, 2a + B-Y. We have l+m+ n = 0, ¿l + m - n = 0: hence the required equation is 2 (− a)3 + (3,3)3 + (y)3 = 0. Ex. 11. Find the equation of the conic touching a, ß, y, at their middle points. Ans. (aa) + (BB) + (y) = 0. Ex. 12. Find the condition that (lu) + (mp)2 + (ny) = 0 should represent a para bola. m n Ans. The curve touches the line at infinity when + + = 0. a b с Ex. 13. To find the locus of the focus of a parabola touching a, ß, y. Generally, if the co-ordinates of one focus of a conic inscribed in the triangle aßy be a'ß'y', the lines joining it to the vertices of the triangle will be and since the lines to the other focus make equal angles with the sides of the triangle (Art. 189), these line will be (Art. 55) Hence, if we are given the equation of any locus described by one focus, we can at once write down the equation of the locus described by the other; and if the second focus be at infinity, that is, if a" sin A + ẞ" sin B + y" sin C = 0, the first sin A sin B must lie on the circle a parabola at infinity are α B' + 7 m sin C + = 0. The co-ordinates of the focus of n sin2Ã' sin2 B' sin2C, since (remembering the relation in Ex. 12) these values satisfy both the equations, a sin A+ẞ sin B+ y sin C = 0, √la + √mß + √ny = 0. sin2A sin2 B sin2C The co-ordinates, then, of the finite focus are m n Ex. 14. To find the equation of the directrix of this parabola. Forming, by Art. 291, the equation of the polar of the point whose co-ordinates have just been given, we find la (sin2B+ sin2C — sin2A) +mẞ (sin2C′+ sin2A — sin2B) + ny (sin2A + sin2B — sin2C') = 0, la sin B sin C cos A + mẞ sin C sin A cos B + ny sin A sin B cosC = 0. Substituting for n from Ex. 12, the equation becomes or 7 sin B sin C (a cos A· y cos C) +m sin C sin A (ẞ cos B − y cos C') = 0; hence the directrix always passes through the intersection of the perpendiculars of the triangle (see Ex. 3, p. 54). Ex. 15. Given four tangents to a conic find the locus of the foci. Let the four tangents be a, ß, y, d, then, since any line can be expressed in terms of three others, these must be connected by an identical relation aa + bß + cy + do = 0. This relation must be satisfied, not only by the co-ordinates of one focus a'ß'y'd', but also by those 1 1 1 of the other CHAPTER XV. THE PRINCIPLE OF DUALITY; AND THE METHOD OF 298. THE methods of abridged notation, explained in the last chapter, apply equally to tangential equations. Thus, if the constants A, u, v in the equation of a line be connected by the relation (aλ+bμ+ cv) (a'λ +b'μ+c'v) = (a′′λ+b"μ+c"v) (a"λ+b′′μ+c"v), the line (Art. 285) touches a conic. Now it is evident that one line which satisfies the given relation is that whose λ, μ, v are determined by the equations μ aλ + bμ+ cv = 0, a′′λ+b′′μ+c′′v = 0. That is to say, the line joining the points which these last equations represent (Art. 70), touches the conic in question. If then a, B, y, & represent equations of points, (that is to say, functions of the first degree in λ, μ, v) ay=kßd is the tangential equation of a conic touched by the four lines aß, By, yd, da. More generally, if S and S' in tangential coordinates represent any two curves, S-kS" represents a curve touched by every tangent common to S and S'. For, whatever values of λ, μ, v make both S=0 and S'=0, must also make S-kS'0. Thus, then, if S represent a conic, S-kaß represents a conic having common with S the pairs of tangents drawn from the points a, B. Again, the equation ay = kß3 represents a conic such that the two tangents which can be drawn from the point a coincide with the line aß; and those which can be drawn from y coincide with the line ys. The points a, y are therefore on this conic, and ẞ is the pole of the line joining them. In like manner, S-a represents a conic having double contact with S, and the tangents at the points of contact meet in a; or, in other words, a is the pole of the chord of contact. So again, the equation ay = k2ß may be treated in the same manner as at Art. 270, and any point on the curve may be |