Ex. To find the locus of the intersection of normals to a conic, at the extremities x2 of a chord which passes through a given point aß. Let the curve be S = +32-1; y2 a2 b2 then the points whose normals pass through a given point x'y' are determined (Art. 181, Ex. 1) as the intersections of S with the hyperbola S′ = 2 (c2xy + b2y'x — a2x'y). We can then, by this article, form the equation of the six chords which join the feet of normals through x'y', and expressing that this equation is satisfied for the point aß, we have the locus required. 1 a262, We have A =— → = 0, 0′ = − (a2x22 + b2y'2 — c1), ▲′ = — 2a2b2c2x'y'. The equation of the locus is then 8 a262 B2 (a2ßx — b3ay — c3aß)3 + 2 (a2x2 + b2y2 — c1) (a2ßx − b3⁄4ay — c2aß) (a2 b2 a2 + + 2a2b2c3xy (~32 + 2 − 1)* = 0, b2 which represents a curve of the third degree. If the given point be on either axis, the locus reduces to a conic, as may be seen by making a = 0 in the preceding equa tion. It is also geometrically evident, that in this case the axis is part of the locus. The locus also reduces to a conic if the point be infinitely distant; that is to say, when the problem is to find the locus of the intersection of normals at the extremities of a chord parallel to a given line. 371. If on transforming to any new set of coordinates, Cartesian or trilinear, S and S become S and S, it is manifest that kS+S becomes kS+ S', and that the coefficient k is not affected. It follows that the values of k, for which kS+ S' represents right lines, must be the same, no matter in what system of coordinates S and S are expressed. Hence, then, the ratio between any two coefficients in the cubic for k, found in the last Article, remains unaltered when we transform from any one set of coordinates to another. The quantities ▲, ☺, O', A' are on this account called invariants of the system of conics. If then, in the case of any two given conics, having by transformation brought S and S to their simplest form, and having calculated ▲, O, O', ▲', we find any homogeneous relation existing between them, we can predict that the same relation will exist between these quantities, no matter to what axes the equations are referred. It will be found possible to express in It may be proved by actual transformation that if in S and S' we substitute for x, y, z; lx + my + nz, l'x + m'y + n'z, l'x + m'y + n"z, the quantities ▲, O, O' A' for the transformed system, are equal to those for the old, respectively multiplied by the square of the determinant terms of the same four quantities the condition that the conics should be connected by any relation, independent of the position of the axes, as is illustrated in the next Article. The following exercises in calculating the invariants A, O, O', A', include some of the cases of most frequent occurrence. Ex. 1. Calculate the invariants when the conics are referred to their common self-conjugate triangle. We may take S = ax2 + by2 + cz2, S′ = a'x2 + b'y2 + c'z2 ; and we may further simplify the equations by writing x, y, z, instead of x √(a'), y (b'), z √(c'), so as to bring S' to the form x2 + y2+z2. We have then And S+kS' will represent right lines, if k3 + k2 (a + b + c) + k (bc + ca + ab) + abc = 0. And it is otherwise evident that the three values for which S+kS' represents right lines are - a, b, — c. * = Ex. 2. Let S', as before, be x2 + y2 + 22, and let S represent the general equation. Ans. (bc - ƒ2) + (ca − g2) + (ab − h2) = A + B + C ; O' = a + b + c. Ex. 3. Let S and S' represent two circles x2 + y2 — p2, (x − a)2 + (y — B)2 — p′2 ̧ Ans. ▲ = r2, − = a2 + ß2 — 2r2 — p′2, 0′ = a2 + ß2 — r2 — 2r22, ▲′ = - r'2. So that if D be the distance between the centres of the circles, S + kS' will represent right lines if p2 + (2p2 + poo2 − D2) k + (μ2 + 2p12 — D2) k2 + r′2k3 — 0. Now since we know that S-S' represents two right lines (one finite, the other infinitely distant), it is evident that - 1 must be a root of this equation. And it is in fact divisible by k + 1, the quotient being Ex. 5. Let S represent the parabola y2 — 4mx, and S' the circle as before. Ans. A-4m2, 0 =—· 372. To find the condition that two conics S and S'should touch each other. When two points, A, B, of the four intersections of two conics coincide, it is plain that the pair of lines AC, BD is identical with the pair AD, BC. In this case, then, the cubic Ak3 + Ok2 + O'k + ▲′ = 0, must have two equal roots. But it can readily be proved that the condition that this should be the case is (☺☺′ — 9AA′)2 = 4 (℗2 — 340′) (℗” — 3A′O), or Θ*Θ" + 184 Δ' ΘΘ' - 27 Δ Δ' - 4ΔΘ" - 440 = 0, which is the required condition that the conics should touch. It is proved, in works on the theory of equations, that the left-hand member of the equation last written is proportional to the product of the squares of the differences of the roots of the equation in k; and that when it is positive the roots of the equation in k are all real, but that when it is negative two of these roots are imaginary. In the latter case (see Art. 282), S and S'intersect in two real and two imaginary points: in the former case, they intersect either in four real or four imaginary points. These last two cases have not been distinguished by any simple criterion. If three points A, B, C coincide the conics osculate and in this case the three pairs of right lines are all identical so that the cubic must be a perfect cube; the condition for this are 3A = Θ' 3A' The conditions for double contact are of a different kind and will be got further on. Ex. 1. To find by this method the condition that two circles shall touch. Forming the condition that the reduced equation (Ex. 3, Art. 371), r2 + (p2+p22— D2)k+r'2k2=0, should have equal roots, we get r2 + p12 − D2 = ± 2rr' ; D=r±r' as is geometrically evident. Ex. 2. The conditions for contact between two conics can be shortly found in the cases of trinomial equations by identifying the equations of tangents at any point given Arts. 127, 130, and are for fyz + gzx + hxy = 0, √(lx) + √(my) + √(nz) = 0, (fl)3 + (gm)3 + (hn)} = 0, for √(lx) + √(my) + √(ne) = 0, ax2 + by2+ cz2 = 0, (~~)3 + (m2)* + (123) * = for = 0, ax2 + by2 + cz2 = 0, fyz + gzx + hxy = 0, (aƒ2)} + (bg2)} + (ch2)} = 0. Ex. 3. Find the locus of the centre of a circle of constant radius touching a given conic. We have only to write for A, A', O, O' in the equation of this article, the values Ex. 4 and 5, Art. 371; and to consider a, ẞ as the running coordinates. The locus is in general a curve of the eighth degree, but reduces to the sixth in the case of the parabola. This curve is the same which we should find by measuring from the curve on each normal, a constant length, equal to r. It is sometimes called the curve parallel to the given conic. Its evolute is the same as that of the conic. The following are the equations of the parallel curves given at full length, which may also be regarded as equations giving the length of the normal distances from any point to the curve. The parallel to the parabola is p6 − (3y2 + x2 + 8mx − 8m2) p2 + {3y* + y2 (2x2 - 2mx + 20m2) - +8mx3 + 8m2x2 - 32m3x + 16m1} r2 — (y2 — 4mx)2 {32 + (x − m)2} = 0, + p1 {ca (aa + 4a2b2 + b1) — 2c2 (a* — a2b2 + 3b1) x2 + 2c2 (3aa — a2b2 + b1) ya + (a* - 6a2b2 + 6b3) x2 + (6aa — 6a2b2 + b1) ya + (6a* — 10a2b2 + 6b1) x2y2} + p2 {− 2a2b2c1 (a2 + b2) + 2c2x2b2 (3aa — a2b2 + b1) — 2c2y2a2 (aa — a2b2 + 3b4) — b2x2 (6a1 — 10a2b2 + 6b1) — a2y1 (6aa — 10a2b2 + 6b1) + x2y2 (4a6 — 6a4b2 6a2b⭑+466) + 2b2 (a2 − 2b2) x® − 2 (aa — a2b2 + 3b1) xay2 — 2 (3aa — a2b2+ b1) x2y*+ 2a2 (b2 — 2a2) y®} + (b2x2 + a2y2 — a2b2)2 {(x — c)2 + y2} {(x + c)2 + y2} = 0. Thus the locus of a point is a conic, if the sum of squares of its normal distances to the curve be given. If we form the condition that the equation in r2 should have equal roots, we get the squares of the axes multiplied by the cube of the evolute. If we make r = 0, we find the foci appearing as points whose normal distance to the curve vanishes. This is to be accounted for by remembering that the distance from the origin vanishes of any point on either of the lines x2 + y2 = 0. Ex. 4. To find the equation of the evolute of an ellipse. Since two of the normals coincide which can be drawn through every point on the evolute, we have only to express the condition that in Ex. Art. 370 the curves S and S' touch. Now when the term k2 is absent from an equation, the condition that Ak3 + O'k + ▲′ should have equal roots reduces to 27AA'2 + 40′3 = 0. The equation of the evolute is therefore (a2x2 + b2y2 — c1)3 + 27a2b2c1x2y2 = 0. (See Art. 248). Ex. 5. To find the equation of the evolute of a parabola. We have here S = y2 - 4mx, S′ = 2xy + 2 (2m — x′) y — 4my', A= - 4m2, 0=0, A'=- 4m (2m - x), A' = 4my, and the equation of the evolute is 27my2 = 4 (x — 2m)3. It is to be observed, that the intersections of S and S' include not only the feet of the three normals which can be drawn through any point, but also the point at infinity on y. And the six chords of intersection of S and S' consist of three chords joining the feet of the normals, and three parallels to the axis through these feet. Consequently the method used (Ex., Art. 370) is not the simplest for solving the corresponding problem in the case of the parabola. We get thus the equation found (Ex. 12, Art. 227), but multiplied by the factor 4m (2my + y'x — 2my') — y'3. 373. If S" break up into two right lines we have ▲'=0, and we proceed to examine the meaning in this case of and '. Let us suppose the two right lines to be x and y; and, by the principles already laid down, any property of the invariants, true when the lines of reference are so chosen, will be true in general. The discriminant of S+ 2kxy is got by writing h+k for h in A, and is ▲ + 2k (fg – ch) - ck2. Now the coefficient of k vanishes when c=0; that is, when the point xy lies on the curve S. The coefficient of k vanishes when fg = ch; that is (see Ex. 3, Art. 228), when the lines x and y are conjugate with respect to S. Thus, then, when S' represents two right lines, A' vanishes; '=0 represents the condition that the intersection of the two lines should lie on S; and 0 is the condition that the two lines should be conjugate with respect to S. = The condition that ▲ + k + 'k should be a perfect square is = 440', which, according to the last Article, is the condition that either of the two lines represented by S" should touch S. This is easily verified in the example chosen, where *- 44' is found to be equal to (bc-ƒ3) (ca – g3). Ex. 1. Given five conics S1, S2, &c., it is of course possible in an infinity of ways to determine the constants 1, 2, &c., so that 11S1 + 12S2 + 13S3 + 1484 + 15S5 may be either a perfect square L2, or the product of two lines MN: prove that the lines L all touch a fixed conic V, and that the lines M, N are conjugate with regard to V. We can determine V so that the invariant → shall vanish for V and each of the five conics, since we have five equations of the form Aa1+ Bb1+Cc1 + 2Ff1 + 2Gg1 + 2Hh1 = 0, which are sufficient to determine the mutual ratios of A, B, &c., the coefficients in the tangential equation of V. Now if we have separately Aa, +&c. =0, Aa2+&c. = 0, Aa, + &c. = 0, &c., we have plainly also A (la1 + 12α2 + 13α3 + 1⁄4α4 + 15α5) + &c. = 0 ; that is to say, vanishes for V and every conic of the system 11S1 + 12S2 + 13S3 + 14S4 + 15S59 whence by this article the theorem stated immediately follows. If the line M be given, N passes through a fixed point; namely, the pole of M with respect to V. Ex. 2. If six lines x, y, z, u, v, w all touch the same conic, the squares are connected by a linear relation 4x2 + 12y2 + 1 ̧22 + 1⁄4u2 + 15v2 + %w2 = 0. This is a particular case of the last example, but may be also proved as follows: Write down the conditions, Art. 151, that the six lines should touch a conic, and eliminate the unknown quantities A, B, &c., and the condition that the lines should touch the same conic is found to be the vanishing of the determinant But this is also the condition that the squares should be connected by a linear relation. Ex. 3. If we are only given four conics S1, S2, S3, S4, and seek to determine V, as in Ex. 1, so that ✪ shall vanish, then, since we have only four conditions, one of the tangential coefficients A, &c. remains indeterminate, but we can determine all the rest in terms of that; so that the tangential equation of V is of the form Σ + kΣ' = 0, or V touches four fixed lines. We shall afterwards show directly that in four ways we can determine the constants so that S, + 12S2 + 13S3 + 48 may be a perfect square. It is easy to see (by taking for M the line at infinity) that if M be a given line it is a definite problem admitting of but one solution to determine the constants, so that S, &c. shall be of the form MN. And Ex. 1 shows that N is the locus of the pole of M with regard to V. Compare Ex. 8, Art. 228. |