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 AB = reduces to the simple ratio - when DC BC

(Art. 56) = BC 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 OR'RR” 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" 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 66 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 O. 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)

gx+fy + c

0.

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

becomes

=

m

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

(The reader must be careful, in this and the following P examples, to take the points of the

order on both sides of the equation.

pencil in the same

Thus, on the left

hand 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

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

TK2 TC.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 fixed points, A, B, on a hyperbola or parabola, be joined to any variable point 0,

Cbi

Q Q.

and the joining lines meet a fixed parallel to an asymptote (if the curve be a hyperbola), or a diameter (if the curve be a parabola), in a, b, then the ratio Ca: Cb will be constant.

Ex. 5. If the same ratio be estimated on any other parallel line, lines inflected from any three fixed points to a variable point, on a hyperbola or parabola, cut a fixed parallel to an asymptote or diameter, so that ab: ac is constant.

Ex. 6. It follows from Ex. 4, that if the lines joining A, B to any fourth point O' meet C∞o in a', b', we must have

ab ас
=
a'b' a'C'

Now let us suppose the point C to be also at an infinite distance, the line C∞ becomes an asymptote, the ratio ab: a'b' becomes one of equality, and lines joining two fixed points to any variable point on the hyperbola intercept on either asymptote a constant portion (Art. 199, Ex. 1).

Let these ratios be estimated on C∞; then if the tangents at A, B, cut C∞ in a, b, and the chord of contact AB in K, we have

c K

B

(observing the caution in Ex. 1). Or, if any paralle to an asymptote of a hyperbola, or a diameter of a parabola, cut two tangents and their chord of contact, the intercept from the curve to the chord is a geometric mean between the intercepts from the curve to the tangents. Or, conversely, if a line ab, parallel to a given one, meet the sides of a triangle in the points a, b, K, and there be taken on it a point C such that CK2 Ca. Cb, the locus of C will be a parabola, if Cb be parallel to the bisector of the base of the triangle (Art. 211), but otherwise a hyperbola, to an asymptote of which ab is parallel.

=

Ex. 8. Let two of the fixed points be at infinity,

{∞.AB∞∞'} = {∞'. AB ∞ ∞ '};

the lines ∞ ∞ ∞∞', are the two asymptotes, while ∞ ∞o' is altogether at infinity. Let these ratios be estimated on the diameter OA; let this line meet the parallels to the asymptotes B∞, B∞',

OA Oa' in a and a'; then the ratios become = Oa OA'

Or,

parallels to the asymptotes through any point on a hyperbola cut any semi-diameter, so that it is a mean proportional between the segments on it from the centre.

Hence, conversely, if through a fixed point O a line be drawn cutting two fixed lines, Ba, Ba', and a point A taken on it so that OA is a mean between Oa, Oa', the locus of A is a hyperbola, of which O is the centre, and Ba, Ba', parallel to the asymptotes.

{∞ .AB ∞ ∞ '} = {∞'. AB ∞ ∞ '}.

Ex. 9.

Let the segments be measured on the asymptotes, and we have

(0 being Oa'

the centre), or the rectangle under parallels to the asymptotes through any point on the curve is constant (we invert the second ratio for the reason given in Ex. 1).

327. We next examine some particular cases of the anharmonic property of the tangents to a conic (Art. 275).

P

R

Q

Ex. 1. This property assumes a very simple form, if the curve be a parabola, for one tangent to a parabola is always at an infinite distance (Art. 254). Hence three fixed tangents to a parabola cut any fourth in the points A, B, C, so that AB: AC is always constant. If the variable tangents coincide in turn with each

T

of the given tangents, we obtain the theorem,

F

Ex. 2. Let two of the four tangents to an ellipse or hyperbola be parallel to each other, and let the variable tangent coincide alter

nately with each of the parallel tangents. In the

first case the ratio is

Hence the rectangle Ab. Db' is constant.

It may be deduced from the anharmonic pro

perty of the points of a conic, that if the lines joining any point on the curve O to A, D, meet the parallel tangents in the points b, b', then the rectangle Ab.Db' will be constant.

328. We now proceed to give some examples of problems easily solved by the help of the anharmonic properties of conics.

Ex. 1. To prove Mac Laurin's method of generating conic sections (p. 248), viz.— To find the locus of the vertex V of a triangle whose sides pass through the points A, B, C, and whose base angles move on the fixed lines Oa, Ob.

Let us suppose four such triangles drawn, then since the pencil {C.aa'a"a""} is the same pencil as {C.bb'b''b'''}, we have

{aa'a"a""} = {bb′b"b"},

and, therefore,

{A. aa'a"a""} = {B.bb'b''b'''};

or, from the nature of the question,

{A. VV'V"V""} = {B. VV'V"V""};

and therefore A, B, V, V', V", V'"' lie on the same conic section. Now if the first three triangles be fixed, it is evident that the locus of V"" is the conic section passing through ABVV'V".

"

A

V'

Vm

Or the reasoning may be stated thus: The systems of lines through A, and through B, being both homographic with the system through C, are homographic with each other; and therefore (Art. 297) the locus of the intersection of correspond