or, in other words, the tangential equation of that conic, is

+2 (gh − af) μv + 2 (hf− 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-g", &c., the equation (BC - F2) a2+&c. = 0 becomes

(abc+2fgh-af-bg”—ch”) (aa2+bB2+cy2+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 +-+ is (Fa) + √(GB) + √(Hy) = 0; and of one which fulfils the condition

μ

ע

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 aa22 + 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- AF - BG-CH2 = 0,

then the equation

Aλ2 + Bμ2 + Cv2 + 2 Fμv + 2 Gvλ + 2Hλμ = 0, may be resolved into two factors, and is equivalent to one of the (a'λ + B'μ + y'v) (a′′λ + B′′μ + y′′v) = 0.

form

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-ƒ2, &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 = (a + μμß + vy)2. Then substituting the co-ordinates of the two given points, we have

whence

S' = (λa' + μẞ' + vy')2; _S" = (\a” + μß" + vy")2;

(λa' + μß' + vy') ](S′′) = ± (λa” + μß" + vy") √(S'), showing that a + 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

(μE+F)2=4μS, or (μE-F)2 = 4μS'.

Since μ 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,

μ212 — μ (L2 + M2 − N2) + M2 = 0.

For this always touches (L2 + M2 — N2)2 — 4L2M2

=

= (L + M + N) (M + N − L) (L + N − M) (M + L − N).

Or again, the equation may be written N2 =

L2
+
cos2 sin2

M2

(see Art. 278).

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

μ3 − 2μ (C + C') + (C−C')2 = 0.

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.

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 (pp. 105, 106) of finding common tangents to circles.

Ans. Ex. 1. C + √C' = 4 or = 2. Ans. Ex. 2. JC + √C' = 1 or = √− 79. Ex. 2. Given three circles; let L, L' be the 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, √C" +√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 S-L2, 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).

GENERAL EQUATION OF THE SECOND DEGREE.

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

aa2+bB2 + 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.

*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.

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, B, 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 n' order between three variables. The two equations then, containing the same number of constants are equally capable of representing any particular curve.

th

290. Since the co-ordinates of any point on the line joining two points a'B'y', a"B"y" are (Art. 66) of the form la′+ma", IB′+mB", ly' + 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

I (ca +Bổ tay +2fgt 2 gia +2hd B)

+ 2 ca c +BB + y

+ƒ(B'y' + B′′y') + g (y'a" + y'a') + h (a'ß" + a′′ß')} thi ca +BB' + c + 2ƒß"y" +2yy"a" + 2ha"B") = 0;

112

or, as we may write it, for brevity, S'+2lmP+ m2 S" = 0. When the point a'B'y' is on the curve, S' vanishes, and the quadratic reduces to a simple equation. Solving it for 1: m, we see that the co-ordinates 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′ – 2Pß", S"y' - 2Py". These co-ordinates reduce to a'B'y' if the condition P=0 be fulfilled. Writing this at full length, we see that if a"B"y" satisfy the equation

aaa' +bBB' + cyy' +ƒ (By' + B'y) +g(y'a + ya') + h (a'ß+ aß') = 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

*This method was introduced by Joachimsthal.

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 aßy, a'B'y' of the equation just found, we infer that when a'ß'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'ß'y', a"B"y" shall be cut harmonically by the The equation of the polar may be written

curve.

aˆ (aa+hẞ+gy) + B' (ha + bB + fy) + y' (ga+fß +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

In particular, if ß', 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+BS + 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 formula of the last article, that the polar of any point with respect to the pair of lines aß is B'a+a'ß, the harmonic conjugate with respect to a, ẞ of ß'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+fß+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, ß, y