The effect of this substitution may be very simply written as follows: Let the equation of the reciprocal with regard to the origin be

where u denotes the terms of the nth degree, &c., then the reciprocal with regard to any point is

'xx' + yy′+ k2\*

'xx′ + yy' + k2 • + &c. = 0,

a curve of the game degree as the given reciprocal.

321. To find the reciprocal with respect to x2 + y2 - k3 of the conic given by the general equation.

We find the locus of a point whose polar xx+yy' - k' shall touch the given conic by writing x', y', - k2 for λ, μ, v in the tangential equation (Art. 151). The reciprocal is therefore

Ax+2Hxy+By' - 2 Gkx-2Fky + Ck' = 0.

Thus, if the curve be a parabola, C or ab - h=0, and the reciprocal passes through the origin. We can, in like manner, verify by this equation other properties proved already geometrically. If we had, for symmetry, written - 2*, and looked for the reciprocal with regard to the curve x2 + y2 + z2 = 0, the polar would have been xx + yy′+zz', and the equation of the reciprocal would have been got by writing x, y, z for λ, μ, v in the tangential equation. In like manner, the condition that λx + μy + vz may touch any curve, may be considered as the equation of its reciprocal with regard to x* + y2 + z2.

A tangential equation of the nth degree always represents a curve of the nth class; since if we suppose λx + μy + vz to pass through a fixed point, and therefore have λx' + μy' + vz′ = 0; eliminating between this equation and the given tangential equation, we have an equation of the nth degree to determine λ:μ; and therefore n tangents can be drawn through the given point.

322. 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. 211) 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.

"Any variable tangent to a conic intercepts on two parallel tangents, portions 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 on any fixed line portions 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. (Art. 199, Ex. 1).

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 very limited in its application.

CHAPTER XVI.

HARMONIC AND ANHARMONIC PROPERTIES OF CONICS.*

323. 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 AD

(Art. 56)

=

÷ reduces to the simple ratio BC DO

AB

--

BC

when

D is at an infinite distance, since then AD ultimately = - - DC. If the line be cut harmonically, its anharmonic ratio = -1; and if D be at an infinite distance AB = BC, and 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.

324. We commence with the theorem (Art. 146): "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. 211).

The fundamental property of anharmonic pencils was given by Pappus, Math. Coll. VII. 129. The name "anharmonic 99 was given by Chasles in his History of Geometry, from the notes to which the following pages have been developed. Further details will be found in his Traité de Géométrie Supérieure; and in his recently published Treatise on Conics. The anharmonic relation, however, had been studied by Möbius in his Barycentric Calculus, 1827, under the name of "Doppelschnittsverhältniss." Later writers use the name "Doppelverhältniss."

Secondly. Let R be at an infinite distance, and R'R" must be bisected at 0; 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 0. Hence this point is the centre, or the centre may be considered as a point whose polar is at infinity (Art. 154).

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.

This also follows from the equation of the polar of a point (Art. 145)

(ax +hy+g) + (hx + by +ƒ) 2 + gx +fy + c

= 0.

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

make 2

becomes

n

=

ገቢ

and x infinite, and the equation of the polar

m (ax+hy+g) +n (hx + by +ƒ)=0,

a diameter conjugate to my = nx (Art. 141).

325. Again, it was proved (Art. 146) that the two tangents through any point, any other line through the point, and the line to the pole of this last line, form a harmonic pencil.

If now 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 portion between the tangents of any line drawn parallel to the polar of the point is bisected by the diameter through it.

Again, let the point be the centre, the two tangents will be the asymptotes. Hence the asymptotes, together with any pair of conjugate diameters, form a harmonic pencil, and the portion of any tangent intercepted between the asymptotes is bisected by the curve (Art. 196).

326. The anharmonic property of the points of a conic (Art. 259) gives rise to a much greater variety of particular theorems. For, the four points on the curve may be any whatever, and

either one or two of them may be at an infinite distance; the fifth point 0, to which the pencil is drawn, may be also either at an infinite distance, or may coincide with one of the four points, in which latter case one of the legs of the pencil will be the tangent at that point; then, again, we may measure the anharmonic ratio of the pencil by the segments on any line drawn across it, which we may, if we please, draw parallel to one of the legs of the pencil, so as to reduce the anharmonic ratio to a simple ratio.

The following examples being intended as a practical exercise to the student in developing the consequences of this theorem, we shall merely state the points whence the pencil is drawn, the line on which the ratio is measured, and the resulting theorem, recommending to the reader a closer examination of the manner in which each theorem is inferred from the general principle.

We use the abbreviation {0.ABCD} to denote the anharmonic ratio of the pencil OA, OB, OC, OD.

Let these ratios be estimated by the segments on the line CD; let the tangents at A, B meet CD in the points T, T', and let the chord

that is, if any chord CD meet two tangents in T, T, and their chord of contact in K,

KC.KT.TD = KD.TK. T'C.

(The reader must be careful, in this and the following P examples, to take the points of the pencil in the same order on both sides of the equation. Thus, on the lefthand side of this equation we took K second, because it answers to the leg OB of the pencil; on the right hand we take K first, because it answers to the leg OA).

Ex. 2. Let T and T' coincide, then

KC. TD= == KD. TC,

or, any chord through the intersection of two tangents is cut harmonically by the chord of contact.

Ex. 3. Let T be at an infinite distance, or the secant CD drawn parallel to PT', and it will be found that the ratio will reduce to

TKTC.TD.

Ex. 4. Let one of the points be at an infinite distance, then {0. ABC} is constant. Let this ratio be estimated on the line Co. Let the lines AO, BO cut C∞ Ca

in a, b; then the ratio of the pencil will reduce to ; and we learn, that if two Cb

fixed points, A, B, on a hyperbola or parabola, be joined to any variable point 0,

Q Q.