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It may be shown, as in Art. 128, that this represents a circle, by throwing it into the form

(≈ sinA+y sin B+z sin C) (

+

A+B+2H cos C

=

sin C

=

B+C+2F cosA C+A+2G cos B

sin A

Θ

x +

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sinA sinB sin C (yzsin A+zxsin B+xysin C'),

where 0 is the condition (Art. 382) that the curve should be a parabola. When = 0, this equation gives the equation of the directrix.

384. In general, Σ+k' denotes a conic touching the four tangents common to Σ and '; and when k is determined so that + k represents a pair of points, those points are two opposite vertices of the quadrilateral formed by the common tangents. In the case where ' denotes the circular points at infinity, when Σ+k' represents a pair of points, these points are the foci (Art. 258a). If, then, it be required to find the foci of a conic, given by a numerical equation in Cartesian coordinates, we first determine k from the quadratic

(ab — h3) k2 + ▲ (a + b) k + ▲2 = 0.

Then, substituting either value of k in 2+ (x2+μ"), it breaks up into factors (λx' + μy' + vz') (λx" +μy" + vz"); and the foci

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x" y"

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2. One value of ✯ gives the two real foci,

k

and the other two imaginary foci. The same process is applicable to trilinear coordinates.

In general, + (x2+ μ3) represents tangentially a conic confocal with the given one. Forming, by Art. 285, the corresponding Cartesian equation, we find that the general equation of a conic confocal with the given one is

▲ S + k { C (x2 + y3) − 2 Gx − 2Fy + A + B} + l2 = 0. From this we can deduce that the equation of common tangents is

{ C (x2 + y3) − 2 Gx − 2Fy + A + B}2 = 4^ S.

By resolving this into a pair of factors

{(x − a)2 + (y − B)3} {(x − a')2 + (y − B')"},

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we can also get a, B; a', B' the coordinates of the foci.

Z Z.

Ex. 1. Find the foci of 2x2 - 2xy + 2y2 - 2x8y+11. The quadratic here is 3k2 + 4k▲ + A2 = 0, whose roots are k=A, k=- JA, But A=-9. Using the value k = 3,

6λ2+21μ2 + 3v2 + 18μv + 12vλ + 30λμ + 3 (λ2 + μ2) = 3 (λ + 2μ + v) (3λ +4μ + v), showing that the foci are 1, 2; 3, 4. The value 9 gives the imaginary foci 2 ± √(− 1), 3 F √(− 1).

Ex. 2. Find the coordinates of the focus of a parabola given by a Cartesian equation. The quadratic here reduces to a simple equation, and we find that

(a + b) {Aλ2 + Bμ2 + 2Fμv + 2Gvλ + 2Hλμ} − ▲ (12 + μ2)

is resolvable into factors. But these evidently must be

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(a + b) B-A

2 (a + b) G 2 (a + b) Fμ + v.

The first factor gives the infinitely distant focus, and shows that the axis of the curve is parallel to Fx - Gy. The second factor shows that the coordinates of the focus

are the coefficients of A and μ in that factor.

Ex. 3. Find the coordinates of the focus of a parabola given by the trilinear equation. The equation which represents the pair of faci is

O'Σ = ▲ (12 + μ2 + v2 – 2μv cos A - 2vλ cos B - 2λu cos C).

But the coordinates of the infinitely distant focus are known, from Art. 293, since it is the pole of the line at infinity. Hence those of the finite focus are

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A sin A+ H sin B + G sin C' H sin A + B sin B+ F sin C'

O'C-A

G sin A+F sin B + C sin C'

385. The condition (Art. 61) that two lines should be mutually perpendicular,

λλ' + μμ' + ν' - (μν' + μ'ν) cos 4 - (νλ' + ν'λ) cos B

— (λμ' +λ'μ) cos C=0, is easily seen to be the same as the condition (Art. 293) that the lines should be conjugate with respect to

x2+μ* + v* − 2 μv cos A-2vλ cos B-2λu cos C=0.

The relation, then, between two mutually perpendicular lines is a particular case of the relation between two lines conjugate with regard to a fixed conic. Thus, the theorem that the three perpendiculars of a triangle meet in a point is a particular case of the theorem that the lines meet in a point which join the corresponding vertices of two triangles conjugate with respect to a fixed conic, &c. It is proved (Geometry of Three Dimensions, Chap. IX.) that, in spherical geometry, the two imaginary circular points at infinity are replaced by a fixed

imaginary conic; that all circles on a sphere are to be considered as conics having double contact with a fixed conic, the centre of the circle being the pole of the chord of contact; that two lines are perpendicular if each pass through the pole of the other with respect to that conic, &c. The theorems then, which, in the Chapter on Projection, were extended by substituting, for the two imaginary points at infinity, two points situated anywhere, may be still further extended by substituting for these two points a conic section. Only these extensions are theorems suggested, not proved. Thus the theorem that the intersection of perpendiculars of a triangle inscribed in an equilateral hyperbola is on the curve, suggested the property of conics connected by the relation =0, proved at the end

of Art. 375.

It has been proved (Art.306) that to several theorems concerning systems of circles, correspond theorems concerning systems of conics having double contact with a fixed conic. We give now some analytical investigations concerning the latter class of systems.

386. To form the condition that the line λx+μy + vz may touch S+(x+ p'y + víz)". We are to substitute in Σ, a +X"2, b+μ", &c. for a, b, &c. The result may be written

12

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where the quantity within the brackets is intended to denote the result of substituting in S μν' - μ'ν, νλ' - ν'λ, λμ' - λ'μ for x, y, z. This result may be otherwise written. For it was proved (Art. 294) that

(ax2 + &c.) (ax2 + &c.) − (axx' + &c.)2 = A (yz′ — y ́z)2 + &c.

And it follows, by parity of reasoning, and can be proved in like manner,

that

(Ax2+&c.) (AX"3 + &c.) − (Aλλ'+ &c.)3 = ▲ {a (μv'— μ'v)3+ &c.},

where AXX' + &c. is the condition that the lines λx + μy + vz, Xx + μ'y + v'z may be conjugate; or

Aλλ'+ Вμμ'+ Cvv + F (μv'+ μ'v) + G (vX' + v′λ) + H (λμ'+λ'μ).; If then we denote AX"+ &c. by E', and Aλλ'+&c. by П

and if we substitute for a (uv' - 'v)*+ &c. the value just found, the condition previously obtained may be written

(Δ+ Σ) Σ --Π* = 0.

If we recollect (Art. 321) that λ, u, v may be considered as the coordinates of a point on the reciprocal conic, the latter form may be regarded as an analytical proof of the theorem that the reciprocal of two conics which have double contact is a pair of conics also having double contact. This condition may also be put into a form more convenient for some applications, if instead of defining the lines λx+μy + vz, &c. by the coefficients λ, μ, v, &c., we do so by the coordinates of their poles with respect to S, and if we form the condition that the line P' may touch S+ P'", where P' is the polar of x'y'z', or axx' + &c. Now the polar of x'y'z' will evidently touch S when x'y'z' is on the curve; and in fact if in Σ we substitute for λ, μ, v; S, S, S, the coefficients of x, y, z in the equation of the polar, we get AS. And again two lines will be conjugate with respect to S, when their poles are conjugate; and in fact if we substitute as before for λ, μ, v in II we get AR, where R denotes the result of substituting the coordinates of either of the points x'y', "y"z", in the equation of the polar of the other. The condition that P' should touch S+ P'" then becomes (1 + S′′) S′ = R3.

387. To find the condition that the two conics

8+ (x'x + μ'y + víz)2, S+ (N′′x+μ”y + v′′z)3,

should touch each other. They will evidently touch if one of the common chords ('x+p'y + v′z) ± (X′′x + μ′′y+v"z) touch either conic. Substituting, then, in the condition of the last Article " for λ, &c., we get

(A + X') (Σ' ± 211 + Σ′′) = (Σ′ ± IT)*,

which reduced may be written in the more symmetrical form (A + Σ') (A + Σ'') = (▲ ± II)3.

The condition that S+P" and S+ P" may touch is found from this as in the last Article, and is

(1 + S′) (1 + S′′') = (1 ± R)3.

Ex. 1. To draw a conic having double contact with S and touching three given conics S + P2, S + P''?, S + P"", also having double contact with S. Let xyz be the coordinates of the pole of the chord of contact with S of the sought conic S + P2, then we have

(1 + S) (1 + 8') = (1 + P')2 ; (1 + §) (1 + S′′) = (1 + P'')2 ; (1 + S) (1 + S''') = (1 + P'"'')2

where the reader will observe that S', S", S"" are known constants, but S, P', &c. involve the coordinates of the sought point xyz. If then we write 1+ S = k2, &c., we get

kk' = 1 + P', kk" = 1+ P", kk"" = 1+ P""'.

It is to be observed that P', P", P" might each have been written with a double sign, and in taking the square roots a double sign may, of course, be given to K', k", k"". It will be found that these varieties of sign indicate that the problem admits of thirty-two solutions. The equations last written give

k (k' — k'') = P' — P" ; k (k′′ – k''') = P" — P'"' ;

whence eliminating k, we get

P' (k" − k'"') + P" (k""' − k') + P''' (k' — k'') = 0,

the equation of a line on which must lie the pole with regard to S of the chord of contact of the sought conic. This equation is evidently satisfied by the point P = P" = P"". But this point is evidently one of the radical centres (see Art. 306) of the conics S + P22, S + P"2, S+ P'''2.

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geometric interpretation of this we remark that it may be deduced from Art. 386 that the tangential equations of S + P22, S + P" are respectively

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represent points of intersection of common tangents to S+ P2, S+P", that is to x' x"

say, the coordinates of these points are ±

P' p"

, &c., and the polars of these points,

ייי

P' p"
= denote the pole, with
k' だ k"

=

with respect to S, are. It follows that respect to S, of an axis of similitude (Art. 306) of the three given conics. And the theorem we have obtained is,-the pole of the sought chord of contact lies on one of the lines joining one of the four radical centres to the pole, with regard to S, of one of the four ares of similitude. This is the extension of the theorem at the end of Art. 118.

To complete the solution, we seek for the coordinates of the point of contact of S+ P2 with S + P'2. Now the coordinates of the point of contact, which is a centre 2 x' k of similitude of the two conics, being &c., we must substitute x + x' for a, &c. in the equations kk' = 1 + P', &c., and we get

k

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k'

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where R, R' are the results of substituting x"y"z", x""'y""z"" respectively in the polar of x'y'z'. We have then

k

k (k' - k') = P' − P" +

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· (S' — R) ; k (k' — k''') = P' − P'""' + ', (S' — R′), whence eliminating k, we have

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- - - {x - 5-(-)}, P2 {k" − 1, − (k ́" — —-”)} + P'" {k” – E − ( x − §)} +

the equation of a line on which the sought point of contact must lie; and which evidently joins a radical centre to the point where P', P", P"" are respectively pro

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