*

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 By, ya, aß. We shall, as before, write S1, S2, S, for the three quantities, aa + hẞ+gy, ha + bẞ +fy, ga+fß +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 coordinates are S'a - 2P'a', S'B-2P'B', S'y - 2P'y. Now if the point a'ß'y' be the intersection of lines ẞ, Y, we may take a' = 1, B′ = 0, y' = 0, which gives S′ = a, P' S1, and the coordinates of the point where the line joining aßy to By meets the curve, are aa 281, άβ, αγ. In like manner, the line joining aẞy to ya, meets the curve again in ba, bß – 281⁄2 by. The line joining these two points will pass through aß, if

=

which is the condition to be fulfilled by the coordinates of the vertex. Writing in this equation aa = S1 — hẞ – gy, bẞ = S2 - ha -fy, it becomes

h (aS1 + BS2) + y (ƒS, + gS2) = 0.

But since aẞy is on the curve, aS1 + ßS1⁄2 + yS2 = 0, and the equation last written reduces to

y (ƒS1+gS2 − hS2) = 0.

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 ƒS1+gS2 − h$3. It can be verified immediately that ƒS1 = gS2 = hS ̧ denote the lines joining the corresponding vertices of the triangles aßy, S1SS3. Consequently (see Ex. 2, Art. 60), the line ƒ§1 + g§1⁄2 – hS3 is constructed as follows: "Form the triangle DEF whose sides

E

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, Art. 146). Conversely, then, AL passes through the pole of BC, and L is on the polar of A (Ex. 1, Art. 146).

This construction becomes indeterminate if the triangle is selfconjugate in which case the problem admits of an infinity of solutions.

a

NN.

Ex. 8. If two conics have double contact, any tangent to the one is cut harmonically 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′′ẞ"y" satisfy the equation S = 0), we get

(IR' + mR'')2 + 2lm P = 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'ß + C'y.

Ans. (la)* + (mß)* + (ny)*, where 1, m, n are determined by the conditions

Ex. 10. Find the equation of the conic touching the five lines, a, ẞ, y, a + B + y, 2a + B-Y.

We have l+m + n = 0, fl + m − n = 0: hence the required equation is

2 (− a)*+ (3ß)* + (y)* = 0.

Ex. 11. Find the equation of the conic touching a, ß, y, at their middle points. Ans. (aa)* + (b,ß)2 + (cy)* = 0.

Ex. 12. Find the condition that (la) + (mp)* + (ny)* = 0 should represent a para

bola.

m n
с

Ans. The curve touches the line at infinity when + + = 0.

a

Ex. 13. To find the locus of the focus of a parabola touching a, ß, y. Generally, if the coordinates 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 lines 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

Ex. 12) these values satisfy both the equations,

a sin A + ẞ siu B+ y sin C = 0, Jla + JmB + Jny = 0.
sin2A sin B

The coordinates, then, of the finite focus are

sin2C

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 coordinates have just been given, we find

la (sin2B + sin C-sin3A)+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 cos C = 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 By cos C') = 0; hence the directrix always passes through the intersection of the perpendiculars of the triangle (see Ex. 3, Art. 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 coordinates of one focus a'ß'y'd', but also by those 1 1 1 1 The locus is therefore the curve of the third degree.

of the other' '' ö

CHAPTER XV.

THE PRINCIPLE OF DUALITY; AND THE METHOD OF
RECIPROCAI. POLARS.

298. THE methods of abridged notation, explained in the last chapter, apply equally to tangential equations. Thus, if the constants, μ, 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

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 coγδ, δα. ordinates 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=kB2 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 y. The points a, y are therefore on this conic, and B 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 manner as at Art. 270, and

=

k3ß may be treated in the same any point on the curve may be

represented by μ3a + 2μkß+y, while the tangent at that point joins the points μa + kB, μkß + y.*

Ex. 1. To find the locus of the centre of conics touching four given lines. Let Σ= 0, '0 be the tangential equations of any two conics touching the four lines; then, by Art. 298, the tangential equation of any other is Σ + kE' = 0. And (see G+kG' F+kF Art. 151) the coordinates of the centre are the form of which C+kC'' C+kC" ' shows (Art. 7) that the centre of the variable conic is on the line joining the centres G FG' F' of the two assumed conics, whose coordinates are; and that it divides the distance between them in the ratio C: kC'.

Ex. 2. To find the locus of the foci of conics touching four given lines. We have only in the equations (Ex. Art. 258a) which determine the foci to substitute A + KA' for A, &c., and then eliminate & between them, when we get the result in the form {C (x2 - y2) + 2Fy - 2Gx + A - B} {C'xy - F'x - G'y + H'}

We have then C and C' both

= {C" (x2 − y2) + 2F′y − 2G'x + A' — B'} {Cxy — Fx – Gy + H}. This represents a curve of the third degree (see Ex. 15, p. 275), the terms of higher order mutually destroying. If, however, Σ and E' be parabolas, Σ + kΣ' denotes a system of parabolas having three tangents common. = 0, and the locus of foci reduces to a circle. Again, if the conics be concentric, taking the centre as origin, we have F, F, G, G' all 0. In this case Σ + kƐ' represents a system of conics touching the four sides of a parallelogram and the locus of foci is an equilateral hyperbola.t

Ex. 3. The director circles of conics touching four fixed lines have a common radical axis. This is apparent from what was proved, p. 270, that the equation of the director circle is a linear function of the coefficients A, B, &c., and that therefore when we substitute A+ kA' for A, &c. it will be of the form S+kS' = 0. This theorem includes as a particular case, "The circles having for diameters the three diagonals of a complete quadrilateral have a common radical axis."

299. Thus we see (as in Art. 70) that each of the equations used in the last chapter is capable of a double interpretation, according as it is considered as an equation in trilinear or in tangential coordinates. And the equations used in the last chapter, to establish any theorem, will, if interpreted as equations

* In other words, if in any system x'y'z', x'y'"z", be the coordinates of any two points on a conic, and ""y""z"" those of the pole of the line joining them, the coordinates of any point on the curve may be written

μax' + 2μkx”” + x', μ2y' + 2μky""' + y'', μ2z' + 2μkz""' + z′′,

while the tangent at that point divides the two fixed tangents in the ratios μ: k, uk: 1. When k = 1, the curve is a parabola. Want of space prevents us from giving illustrations of the great use of this principle in solving examples. The reader may try the question :-To find the locus of the point where a tangent meeting two fixed tangents is cut in a given ratio.

† It is proved in like manner that the locus of foci of conics passing through four fixed points, which is in general of the sixth degree, reduces to the fourth when the points form a parallelogram.