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288. The equation of a conic having double contact with two circles assumes a simpler form, viz.

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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 √ C±√C' = √μ.

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.

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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. C + JC' = 4 or = 2. Ans. Ex. 2. JC + JC' = 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=ť′, JC + √C' = e.

Then the condition that L, M, N should meet in a point is t+tt"; 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 S-L2, S M2, S-N2; and the common chords L+ M, M + N, N+ L, then the truth of the theorem appears from inspec tion of the equation

8+ MN+NL+ LM = (S − L2) + (L + M) (L + N).

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* 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,

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GENERAL EQUATION OF THE SECOND DEGREE.

289. There is no conic whose equation may not be written in the form

aa2 + bß* + 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).

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In like manner the equation of every curve of any degree may be expressed as a homogeneous function of a, B, y. For it can readily be proved that the number of terms in the complete equation of the n' 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 d'B'y', a′′ß"y" are (Art. 66) of the form la +ma”, IB′ + mẞ", lý + my”, we can find the points where this joining line meets any curve by substituting these values for a, B, 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

ľ2 (aa” + bß" + cy"2 + 2ƒB'y' + 2gy'a + 2ha'B')

+2lm {aa ́a" + bß'ß" + cý'y"

+ƒ(B'y′′ +B′′y') + g (y'a′′ + y′′a') + h (a′ B′′ + a′′ß′)} + m (aa +b + c +213 +2 g c +2h 3) = 0;

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or, as we may write it for brevity, l'S'+2lm P+ m2Ś" = 0. When the point d'B'y' is on the curve, S' vanishes, and the quadratic reduces to a simple equation. Solving it for 1: m,

*This method was introduced by Joachimsthal.

we see that the coordinates of the point where the conic is met again by the line joining a"B"y" to a point on the conic a'B'y', are S'a-2Pa", S'B' -2PB", S"y-2Py". These coordinates reduce to a'B'y if the condition P=0 be fulfilled. Writing this at full length, we see that if a′′ẞ"y" satisfy the equation aaa'+bBB' + cyy'+ƒ (By'+ B’y) + g (ya' + y ́a) + h (aß′ + a′ß) = 0, then the line joining a"B"y" to a'B'y meets the curve in two points coincident with a'B'y'; in other words, a"B"y" lies on the tangent at a'B'y. The equation just written is therefore the equation of the tangent.

291. Arguing, as at Art. 89, from the symmetry between ay, a'B'y' of the equation just found, we infer that when a'B'y is not supposed to be on the curve, the equation represents the polar of that point. The same conclusion may be drawn from observing, as at Art. 91, that P=0 expresses the condition that the line joining a’B'y', a′′B”"y" shall be cut harmonically by the curve. The equation of the polar may be written

a' (aa+hẞ+gy) + B′ (ha+bB +ƒy) + y′ (ga +ƒB + cy) = 0. But the quantities which multiply a', B', y' respectively, are half the differential coefficients of the equation of the conic with respect to a, B, y. We shall for shortness write S, S., S, instead ds ds dS

of

da' d' dy; and we see that the equation of the polar is

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In particular, if B', y both vanish, the polar of the point By is S, or the equation of the polar of the intersection of two of the lines of reference is the differential coefficient of the equation of the conic considered as a function of the third. The equation of the polar being unaltered by interchanging aßy, a'B'y', may also be written aS,' + ßS,' + y§ ̧' = 0.

292. When a conic breaks up into two right lines, the polar of any point whatever passes through the intersection of the right lines. Geometrically, it is evident that the locus of harmonic means of radii drawn through the point is the fourth harmonic to the pair of lines and the line joining their intersection to the given point. And we might also infer, from the

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formula of the last article, that the polar of any point with respect to the pair of lines aß is B'a+aß, the harmonic conjugate with respect to a, B of B'a-a'ß, the line joining aß to the given point. If then the general equation represent a pair of lines, the polars of the three points By, ya, aß,

aa+hẞ+gy = 0, ha+bB+fy=0, ga+ƒB+cy=0,

are three lines meeting in a point. Expressing, as in Art. 38, the condition that this should be the case, by eliminating a, B, Y between these equations, we get the condition, already found by other methods, that the equation should represent right lines, which we now see may be written in the form of a determinant, ah, g h, b, f

or, expanded,

g, f, c=0;

abc+2fgh-af-bg" — ch2 = 0.

The left-hand side of this equation is called the discriminant of the equation of the conic. We shall denote it in what follows by the letter A.

293. To find the coordinates of the pole of any line λα + μβ + νγ. Let as'y be the sought coordinates, then we

must have

aa' + hB′ + gy′ = λ, ha' +bB′ +ƒŸ' = μ, ga' +ƒ‚3′ + cy' = v. Solving these equations for a', B', y', we get

▲a' = λ (bc −ƒ3) + μ (fg − ch) + v (hƒ — bg),

AB' =λ(fg− ch) + μ (ca − g3) + v (gh− af),

▲y' =λ (hf — bg) + μ (gh− af) + v (ab − h2) ;

μ

or, if we use A, B, C,† &c. in the same sense as in Art. 151, we find the coordinates of the pole respectively proportional to Ax+ Hμ+ Gv, Hλ+ Bμ + Fv, Gλ+ Fp + Cv.

Since the pole of any tangent to a conic is a point on that tangent, we can get the condition that λa+uß + vy may touch the conic, by expressing the condition that the coordinates just found satisfy a + μẞ+vy = 0. We find thus, as in Art. 285, Aλ2 + Bμ2 + Cv2 + 2Fμv + 2 Gvλ + 2Hλμ = 0.

* See Lessons on Modern Higher Algebra, Lesson XI.

† A, B, C, &c. are the minors of the determinant of the last article.

If we write this equation coordinates of the pole are 2,, 22, 2, that is to say, the differential coefficients of Σ with respect to λ, μ, v. Just, then, as the equation of the polar of any point is aS,' + BS,' +yS2 = 0, so the condition that λa+μß+vy may pass through the pole of X'a + μ'B+vy (or, in other words, the tangential equation of this pole) is 2, + μΣ,' + vΣ,' = 0. And again, the condition that two lines λα + μβ + νγ, λ'α+ μ'β + ν'y may be conjugate with respect to the conic, that is to say, may be such that the pole of either lies on the other, may obviously be written in either of the equivalent forms

=0, it will be observed that the

Χ'Σ + μ'Σ + ν'Σ = 0, λΣ, + μΣ, + vΣ = 0.

From the manner in which Σ was here formed, it appears that Σ is the result of eliminating a', B', ', p between the equations cá thểt gitpr=0, há +ơ tổn + PH=0,

ga' +ƒB' + cy' + pv = 0, λa' + μß'+ vý′ = 0;

in other words, that Σ may be written as the determinant

Αλ* Β

λ, μ, v, 0 |= Aλ2 + Bμ3 + Cr2 + 2Fμv +2 Gvλ + 2Hλμ.

2Ηλμ.

a, h, g, λ

h, b, f, μ

g, f, c, v

Ex. 1. To find the coordinates of the pole of λa + μß +vy with respect to J(la) + √(mB) + (ny). The tangential equation in this case (Art. 130) being

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a' = mv + nμ, ß' = nλ + lv, y' = lμ + mλ.

Ex. 2. To find the locus of the pole of λa + μß + vy with respect to a conic being given three tangents, and one other condition.*

Solving the preceding equations for l, m, n, we find l, m, n proportional to

λ (uß' + vy' — λa'), μ (vy' + λa' — μß′), v (λa' + μß' — vy').

Now (la) +mß) + √(ny) denotes a conic touching the three lines a, ß, y; and any fourth condition establishes a relation between l, m, n, in which, if we substitute the values just found, we shall have the locus of the pole of λa + μß + vy. If we write for A, μ, v the sides of the triangle of reference a, b, c, we shall have the locus of the pole of the line at infinity aa + bß + cy, that is, the locus of centre. Thus the condition that the conic should touch Aa + Bß + Cy being m n + + =0

A B

The method here used is taken from Hearn's Researches on Conic Sections.

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