with regard to x2 + y2 + z2 = 0. Then the polar with regard to this of any point on the reciprocal curve will touch the given curve. But the equation of the polar is xx' + yy' + 'zz' 0; and expressing (Art. 154) the condition that this line should touch the given conic, the equation of the reciprocal is found to be

(a'a” – b2) x2 + (a'a − b'2) y2 + (aa′ – b'′′2) z2

=

+ 2 (b′b′′ – ab) yz + 2 (b′′b – a'b') zx + 2 (bb′ – a′′b′′) xy = 0. We have seen (Art. 296) that the coefficients in this equation are

dv dv
da' da'

equal to &c. We shall denote these coefficients by A, A', A”, B, B ́, B". It is easy to deduce from this equation the properties which we have already obtained geometrically, such as, that if the curve be a parabola, the origin will be a point on the reciprocal curve, &c.

Ex. 1. To find the equation of the reciprocal of the reciprocal of a given conic. This must evidently represent the given curve itself. The equation is

(A' A′′ – B2) x2 + &c. = 0;

and writing for A', &c., their values, this is found equal to the given equation multiplied by v. In like manner the discriminant of the reciprocal is found = √2.

Ex. 2. To find the reciprocal of a system of conics which pass through four points. The equation of any conic of the system being S + kS' = 0, the equation of the reciprocal is found by writing a + kA for a, a' + kA' for a', &c., in the equation of the reciprocal. It is easy to see that the result will contain k in the second degree. We may write it Σ + k¥ + k2Σ'= 0, where Σ and ' are the reciprocals of S and S', while • = (a ́A" + a′′A' – 26B) x2 + (a′′A + aA” — 2b'B') y2 + (aA' + a'A − 2b′′B′) z2

+ 2 (b′B′′ + b′′B′ – aB − bA) yz + 2 (b′′B +bB” — a′B' — b'A') zx +2 (bB'+b'B-a′′B′′ – b′′A") xy.

But the form of the equation This, then, is the equation of the

Now, since the original system of conics passes through four fixed points, the reciprocal system always touches four fixed right lines. shows that the reciprocal always touches 42' = p2. four lines which are common tangents to Σ, E', and the other conics of the reciprocal system. But the form of 4ΣΣ' = p2, the equation to these four lines, touched by them, and that passes through the points of contact. passes through the four points where is touched by the common tangents. Hence the eight points of contact of common tangents to the two conics E, Σ', all lie on the same conic .

1=

Ex. 3. To find the equation of the common tangents to S and S'.

shows that Σ is In like manner,

The system reciprocal to the system of conics which have the same common tangents will pass through four fixed points, and will be Σ + kΣ' = 0. Forming, then, the reciprocal of this latter system, we find VS + kF + k2V'S'= 0, where F is what becomes when the coefficients A, A', &c. of the reciprocal are written for A, A', &c. The equation, then, of the common tangents will be F2 = 4VV'SS'.

Ex. 4. To find the envelope of a system of confocal conics.

(Art. 319) (a2 - k2) x2 + (b2 — k2) y2 = 1; and as this denotes a system of conics through the four points of intersection of (a2x2 + b2y2 − 1) and (x2 + y2), it follows that the system of confocal conics touches four fixed right lines. Arranging their equation,

(b2x2 + a2y2 — a2b2) + k2 (a2 + b2 − x2 — y2) — k1 — 0,

they always touch

=

(a2 + b2 − x2 − y2)2 + 4 (b2x2 + a2y2 − a2b2) = 0,

which will be found to be equivalent to

{y2 + (x − c)a} {y2 + (x + c)2} = 0,

a result in accordance with Art. 282.

Ex. 5. The equation of the pair of tangents from any point x'y'z' to S is found by substituting yz' — zy', zx' — xz', xy' — yx' for x, y, z in the equation of the reciprocal

curve.

Any point on either tangent through x'y'z' evidently possesses the property that the line joining it to x'y'z' touches the curve. In order, then, to find the equation of the pair of tangents, we have only to express (Art. 154) the condition that the line joining two points x (yz” — y ̋z) + y (z'x” — z′′x') + z (xy” — x”y') = 0

should touch the curve, and to consider then "y"z" as variable. And remembering (Art. 321) that the coefficients are the same in the condition that a line should touch the curve, and in the equation of the reciprocal curve, the truth of the theorem is manifest. As we have already (Art. 150) obtained the equation of the pair of tangents in another form, it follows (as may easily be verified) that

(ax2 + a'y2 + &c.) (ax'3 + a'y'2 + &c.) -- (axx' + a'yy' + &c.)3 = A (yz' — zy')2 + A' (zx' − xz')2 + &c.

In like manner,

(Ax2 + &c.) (Ax'a + &c.) − (Axx' + &c.)2 = ▼ {a (yz' — zy)2 + &c. }.

Ex. 6. To verify that, if two conics have double contact with each other, their reciprocals have double contact with each other (Art. 294).

The reciprocal of S + (lx + my + nz)2 is (Art. 297) Σ + {a (mz − ny)2 + &c.}. But since (Ex. 5)

The reciprocal is

{v + (Al2 + &c.)} Σ - (Alx + &c.)2 = 0,

a conic evidently having double contact with Σ.

322. Given the reciprocal of a curve with regard to the origin of co-ordinates, to find the equation of its reciprocal with regard to any point (xy).

If the perpendicular from the origin on the tangent be P, the perpendicular from any other point is (Art. 27)

Pa'cos-y'sin 0,

and, therefore, the polar equation of the locus is

we must, therefore, substitute, in the equation of the given reci

procal,

k2x
xx + yy + k2

for x, and

k2y
xx' + yy + k2

for y.

The effect of this substitution may be very simply written as follows: Let the equation of the reciprocal with regard to the origin be Un + Un-1 + Un-29 &c. (see Art. 271),

then the reciprocal with regard to any point is

a curve of the same degree as the given reciprocal.

323. Before quitting the subject of reciprocal polars, we wish to mention a class of theorems, for the transformation of which M. Chasles has proposed to take as the auxiliary conic a parabola instead of a circle. We proved (Art. 216) that the intercept made on the axis of the parabola between any two lines is equal to the intercept between perpendiculars let fall on the axis from the poles of these lines. This principle, then, enables us readily to transform theorems which relate to the magnitude of lines measured parallel to a fixed line. We shall give one or two specimens of the use of this method, premising that to two tangents parallel to the axis of the auxiliary parabola correspond the two points at infinity on the reciprocal curve, and that, consequently, the curve will be a hyperbola or ellipse, according as these tangents are real or imaginary. The reciprocal will be a parabola if the axis pass through a point at infinity on the original curve.

66

Any variable tangent to a conic intercepts portions on two parallel tangents whose rectangle is constant."

To the two points of contact of parallel tangents answer the asymptotes of the reciprocal hyperbola, and to the intersections of those parallel tangents with any other tangent answer parallels

to the asymptotes through any point; and we obtain, in the first instance, that the asymptotes and parallels to them through any point on the curve intercept portions on any fixed line whose rectangle is constant. But this is plainly equivalent to the theorem: "The rectangle under parallels drawn to the asymptotes from any point on the curve is constant."

Chords drawn from two fixed points of a hyperbola to a variable third point, intercept a constant length on the asymptote.

If any tangent to a parabola meet two fixed tangents, perpendiculars from its extremities on the tangent at the vertex will intercept a constant length on that line.

This method of parabolic polars is plainly much more limited in its application than the method of circular polars, whose resources in transforming theorems of magnitude M. Chasles has possibly underrated.

HARMONIC AND ANHARMONIC PROPERTIES OF CONICS.

*

324. The harmonic and anharmonic properties of conic sections admit of so many applications in the theory of these curves, that we think it not unprofitable to spend a little time in pointing out to the student the number of particular theorems either directly included in the general enunciations of these properties, or which may be inferred from them without much difficulty.

The cases which we shall most frequently consider are, when one of the four points of the right line, whose anharmonic ratio we are examining, is at an infinite distance. The anharmonic ratio of four points, A, B, C, D, being in general

AB.CD

if

AD.BC'

D be at an infinite distance, the ratio is ultimately = 1, and

the anharmonic ratio becomes simply

CD
AD
AB

If the line be cut

harmonically, its anharmonic ratio = 1, and if D be at an infinite distance AC is bisected. The reader is supposed to be acquainted with the geometric investigation of these and the other fundamental theorems connected with anharmonic section.

The discovery of the anharmonic properties of conics is due to M. Chasles, from the notes to whose History of Geometry the following pages have been developed.

325. We shall commence with the theorem (Art. 147): "If any line through a point O meet a conic in the points R′, R”, and the polar of O in R, the line ORRR" is cut harmonically."

First. Let R" be at an infinite distance; then the line OR must be bisected at R'; that is, if through a fixed point a line be drawn parallel to an asymptote of an hyperbola, or to a diameter of a parabola, the portion of this line between the fixed point and its polar will be bisected by the curve (Art. 216).

Secondly. Let R be at an infinite distance, and R'R" must be bisected at O; that is, if through any point a chord be drawn parallel to the polar of that point, it will be bisected at the point.

If the polar of O be at infinity, every chord through that point meets the polar at infinity, and is therefore bisected at O. Hence this point is the centre, or the centre may be considered as a point whose polar is at infinity (p. 139).

Thirdly. Let the fixed point itself be at an infinite distance, then all the lines through it will be parallel, and will be bisected on the polar of the fixed point. Hence every diameter of a conic may be considered as the polar of the point at infinity in which its ordinates are supposed to intersect (p. 241).

This also follows from the equation of the polar of a point (Art. 144),

Now, if x'y' be a point at infinity on the line my = nx, we must

and 'infinite, and the equation of the polar becomes

m(2Ax + By + D) + n (2Cy + Bx + E) = 0,

a diameter conjugate to my = nx (Art. 139).

326. We may, in like manner, make particular deductions from the theorem (Art. 149), that the two tangents through any point, any other line through the point, and the line to the pole of this last line, form an harmonic pencil.

Thus, if one of the lines through the point be a diameter, the other will be parallel to its conjugate, and since the polar of any point on a diameter is parallel to its conjugate, we learn that the