or x cos (20 + ẞ) + y sin (20 + ß) = 2p cos ẞ - x cos ẞ — y sin ß, an equation of the form whose envelope, therefore, is x2 + y2 = (x cos ẞ + y sin ß - 2p cos 6)2, the equation of a parabola having the point O for its focus. Ex. 6. Find the envelope of the line connected by the relation μ + μ' = C. A B + = 1, where the indeterminates are We may substitute for μ', C-μ, and clear of fractions; the envelope is thus found to be A2 + B2 + C2 - 2AB - 2AC – 2BC = 0, an equation to which the following form will be found to be equivalent, ±JA ± √B± JC = 0. Thus, for example,-Given vertical angle and sum of sides of a triangle to find the envelope of base. In like manner,-Given in position two conjugate diameters of an ellipse, and the sum of their squares, to find its envelope. The ellipse, therefore, must always touch four fixed right lines. 285. If the coefficients in the equation of any right line λa+μß + vy be connected by any relation of the second order in λ, μ, V, Aλ* + Bμ* + Cv3 + 2Fμv +2 Gvλ + 2Hλμ=0, the envelope of the line is a conic section. Eliminating v between the equation of the right line and the given relation, we have (Ay2 — 2 Gya + Ca3) λ2 + 2 (Hy3 — Fya – Gyß + Caß) Xμ and the envelope is +(By3 −2Fyß + CB3) μ3 = 0, (Ay3—2 (Fya+Ca2) (By3—2Fyß+Cß3)=(Hy3—Fya—Gyß+Caß)3, Expanding this equation, and dividing by y3, we get (BC − F2) x2 + (CA — G3) B2 + (AB — H3) Ÿ3 2 + (GH−AF) By + 2 (HF — BG) ya + 2 (FG − CH) aß = 0. The result of this article may be stated thus: Any tangential equation of the second order in λ, μ, v represents a conic, whose trilinear equation is found from the tangential by exactly the same process that the tangential is found from the trilinear. For it is proved (as in Art. 151) that the condition that λα + β + γ shall touch aa2 + bß* + cy2+2ƒBy + 2gya + 2haß = 0, or, in other words, the tangential equation of that conic is +2 (gh − af) μv + 2 (hƒ − bg) vλ + 2 (fg − ch) λμ = 0. Conversely, the envelope of a line whose coefficients λ, μ, v fulfil the condition last written, is the conic aa2 + &c. = 0; and this may be verified by the equation of this article. For, if we write for A, B, &c., bc-ƒ3, ca- g3, &c., the equation (BC - F2) a2 + &c. = 0 becomes (abc+2fgh—aƒ“”—bg”—ch2) (aa2+bß2+cy3+2ƒBy+2gya+ 2haß)=0. Ex. 1. We may deduce, as particular cases of the above, the results of Arts. 127, F G H 130, namely, that the envelope of a line which fulfils the condition + + =0 Τ is (Fa) + (GB) + √(Hy) = 0; and of one which fulfils the condition Ex. 2. What is the condition that λa + μß + vy should meet the conic given by the general equation in real points? Ans. The line meets in real points when the quantity (bc —ƒ2) λ2 + &c. is negative; in imaginary points when this quantity is positive; and touches when it vanishes. Ex. 3. What is the condition that the tangents drawn through a point a'ß'y' should be real? Ans. The tangents are real when the quantity (BC — F2) a22 + &c. is negative; or, in other words, when the quantities abc + 2fgh + &c. and aa"2+ bẞ'2+ &c. have opposite signs. The point will be inside the conic and the tangents imaginary when these quantities have like signs. 286. It is proved, as at Art. 76, that if the condition be fulfilled, ABC+2FGH-AF2 — BG - CH' = 0, then the equation Ax2 + Bμ2 + Cv2+2Fμv + 2 Gvλ + 2Hλμ = 0 may be resolved into two factors, and is equivalent to one of the form (a'λ + B'μ + y'v) (a′′λ + B′′μ + y′′v) = 0. And since the equation is satisfied if either factor vanish, it denotes (Art. 51) that the line λa+μẞ+vy passes through one or other of two fixed points. If, as in the last article, we write for A, bc-f2, &c., it will be found that the quantity ABC+2FGH+ &c. is the square of abc+2fgh+ &c. Ex. If a conic pass through two given points and have double contact with a fixed conic, the chord of contact passes through one or other of two fixed points. For let S be the fixed conic, and let the equation of the other be S = (λa + μß + vy)3. Then substituting the coordinates of the two given points, we have whence S' = (λa' + μß' + vy')2; S" = (λa" + μß" + vy")2; (λa' + μß' + vy') √(S′′) = ± (λa′′ + μß" + vy′′) √(S'), showing that λa + μß +vy passes through one or other of two fixed points, since S', S" are known constants. 287. To find the equation of a conic having double contact with two given conics, S and S'. Let E and F be a pair of their chords of intersection, so that S- S'EF; then represents a conic having double contact with S and S'; for it may be written μ (μE+F)2=4μS, or (μE-F)* = 4μS'. Since is of the second degree, we see that through any point can be drawn two conics of this system; and there are three such systems, since there are three pairs of chords E, F. If S' break up into right lines, there are only two pairs of chords distinct from S', and but two systems of touching conics. And when both S and S' break up into right lines there is but one such system. Ex. Find the equation of a conic touching four given lines. Ans. μ2E2 - 2μ (AC + BD) + F2 = 0, where A, B, C, D are the sides; E, F the diagonals, and AC – BD = EF. Or more symmetrically if L, M, N be the diagonals, L± M±N the sides, μ2 L2 — μ (L2 + M2 — N2) + M2 = 0. For this always touches 4L2M2 — (L2 + M2 — N2)2 = (L + M + N) (M + N − L) (L + N − M) (M + L− N), 288. The equation of a conic having double contact with two circles assumes a simpler form, viz. The chords of contact of the conic with the circles are found to be C-C'+μ=0, and C-C'-μ= 0, which are therefore parallel to each other, and equidistant from the radical axis of the circles. This equation may also be written in the form = Hence, the locus of a point, the sum or difference of whose tangents to two given circles is constant, is a conic having double contact with the two circles. If we suppose both circles infinitely small, we obtain the fundamental property of the foci of the conic. If μ be taken equal to the square of the intercept between the circles on one of their common tangents, the equation denotes a pair of common tangents to the circles. Ex. 1. Solve by this method the Examples (Arts. 113, 114) of finding common tangents to circles. Ans. Ex. 1. JC + JC' = 4 or = 2. Ans. Ex. 2. JC + √C' = 1 or = √ -79. Ex. 2. Given three circles; let L, L' be a pair of common tangents to C", C"; M, M' to C", C; N, N' to C, C'; then if L, M, N meet in a point, so will L', M', N'.* Let the equations of the pairs of common tangents be JC' + JC" =t, JC" + √C=t', √C+ √C' =t". Then the condition that L, M, N should meet in a point is ttt"; and it is obvious that when this condition is fulfilled, L', M', N' also meet in a point. Ex. 3. Three conics having double contact with a given one are met by three common chords, which do not pass all through the same point, in six points which lie on a conic. Consequently, if three of these points lie in a right line, so do the other three. Let the three conics be SL2, S- M2, S-N2; and the common chords L+ M, M + N, N + L, then the truth of the theorem appears from inspection of the equation S+ MN + NL + LM = (S − L2) + (L + M) (L + N). *This principle is employed by Steiner in his solution of Malfatti's problem, viz. "To inscribe in a triangle three circles which touch each other and each of which touches two sides of the triangle." Steiner's construction is, "Inscribe circles in the triangles formed by each side of the given triangle and the two adjacent bisectors of angles; these circles having three common tangents meeting in a point will have three other common tangents meeting in a point, and these are common tangents to the circles required. For a geometrical proof of this by Dr. Hart, see Quarterly Journal of Mathematics, vol. I., p. 219. We may extend the problem by substituting for the word "circles," "conics having double contact with a given one." In this extension, the theorem of Ex. 3, or its reciprocal, takes the place of Ex. 2. GENERAL EQUATION OF THE SECOND DEGREE. 289. There is no conic whose equation may not be written in the form aa2 + bß2 + cy2+2ƒBy + 2gya + 2haß = 0. For this equation is obviously of the second degree; and since it contains five independent constants, we can determine these constants so that the curve which it represents may pass through five given points, and therefore coincide with any given conic. The trilinear equation just written includes the ordinary Cartesian equation, if we write x and y for a and B, and if we suppose the line y at infinity, and therefore write y = 1 (see Art. 69, and note p. 72). In like manner the equation of every curve of any degree may be expressed as a homogeneous function of a, ß, y. For it can readily be proved that the number of terms in the complete equation of the nth order between two variables is the same as the number of terms in the homogeneous equation of the nth order between three variables. The two equations then, containing the same number of constants, are equally capable of representing any particular curve. 290. Since the coordinates of any point on the line joining two points a'B'y', a"B"y" are (Art. 66) of the form la+ma”, lB′ + mß", lý + my', we can find the points where this joining line meets any curve by substituting these values for a, ß, Y, and then determining the ratio : m by means of the resulting equation.* Thus (see Art. 92) the points where the line meets a conic are determined by the quadratic +ƒ(B'y'" +B”y') + g (ya” + y'a') + h (a′ß" + a′′”ß')} + m (ca + b + c +2f +2 g c +2h )=0; or, as we may write it for brevity, l'S′+2lmP+ m2 S′′ = 0. When the point a''y' is on the curve, S' vanishes, and the quadratic reduces to a simple equation. Solving it for 7: m, * This method was introduced by Joachimsthal. |