368. It is evident that the projections of the tangents at the points A, B of the circle are the tangents at the points M, N of the conic section (Art. 362); now in the case of the parabola the point M and the tangent at it go off to infinity; we are therefore again led to the conclusion that every parabola has one tangent altogether at an infinite distance.

369. Let the cone now be supposed oblique. The plane of the paper is a plane drawn through the line OC, perpendicular to the plane of the circle AQSB. Now let

the section meet the base in any line QS, draw a diameter LK bisecting QS, and let the section meet the plane OLK in the line MN, then the proof proceeds exactly as before; we have the square of the ordinate RS equal to the rectangle LR. RK; if we conceive a plane, as before, drawn parallel to the base (which, however, is left out of the figure in order to avoid rendering it too complicated), we have the square

M

of any other ordinate, rs, equal to the corresponding rectangle lr.rk; and we then prove by the similar triangles KRM, krM; LRN, IN, in the plane OLK, exactly as in the case of the right cone, that RS: rs2, as the rectangle under the parts in which each ordinate divides MN, and that therefore the section is a conic of which MN is the diameter bisecting QS, and which is an ellipse when MN meets both the lines OL, OK on the same side of the vertex, an hyperbola when it meets them on different sides of the vertex, and a parabola when it is parallel to either.

In the proof just given QS is supposed to intersect the circle in real points; if it did not, we have only to take, instead of the circle AB, any other parallel circle ab, which does meet the section in real points, and the proof will proceed as before.

370. If a circular section be cut by any plane in a line RS, the rectangle DR. RF of the segments of the diameter of the circle conjugate to QS is to the rectangle gR. Rk under the segments of the diameter of the section conjugate to QS as the square of the diameter of the section parallel to QS is to the square of the conjugate diameter gk.

This has been proved in the last Article, in the case where

QS meets the circle in real points,

=

T

S R

since rs2 dr.rf. Now, if the plane meet any other parallel plane in a line QS which does not meet the curve: First, we say that the diameters conjugate to QS with regard to the circle, and with regard to the other section, will meet QS in the same point R; for, by Art. 366, the diameter df, bisecting chords of any circular section parallel to qs, will be projected into a diameter bisecting the parallel chords of any parallel section. The middle points, therefore, of all chords parallel to qs, must lie in the plane Odf, and, consequently, the diameter conjugate to QS, in the section gqks, must be the line gk, in which it is met by the plane Odf. DF, therefore, and gk, intersect in the point R, where QS meets the plane Odf.

Now, since we have proved that the lines gk, df, DF, lie in one plane passing through the vertex, the points D, d, are projections of g, that is, they lie in one right line passing through the vertex; we have, therefore, by similar triangles, as in Art. 367, dr.rf: DR.RF::gr.rk.gR.Rk; and, since dr.rf:gr.rk, as the squares of the parallel semidiameters, DR. RF:gR. Rk in the same ratio.

371. If a plane be drawn through the vertex parallel to the circular base meeting the section gqks in TL, it follows, as a particular case of the preceding, that gL.Lk: OL in the ratio of the squares of the parallel diameters of the section. Hence we see that, given any conic section and a line, TL, in its plane, it is an indeterminate problem to find O the vertex of a cone such that the section of it, by any plane parallel to OTL, should be a circle. For, draw the diameter of the section conjugate to TL, then the distance of L from the vertex of the cone is determined by the present Article; also OL must lie in the plane perpendicular to TL, since it is parallel to the diameter of a circle perpendicular to TL; O may, therefore, be any point of a certain circle in a plane perpendicular to TL.

Hence, Given any conic section, and any line TL in its plane not cutting it, we can project it, so that the conic section may become a circle; and the line may be projected to infinity, for we have only to take any point O, such that the plane OTL may be parallel to the planes of circular section, and then any plane parallel to OTL will be a plane of projection fulfilling the required conditions.

372. Given any conic section and a point in its plane, we can project it into a circle, of which the projection of that point is the centre, for we have only to project it so that the projection of the polar of the given point may pass to infinity (Art. 157).

Or again, Any two conic sections may be projected so as both to become circles, for we have only to project one of them into a circle so as that any of its chords of intersection with the other shall pass to infinity, and then, by Art. 259, the projection of the second conic passing through the same points at infinity as the circle must be a circle also.

Any two conics which have double contact with each other may be projected into concentric circles.

For we have only to project one of them into a circle so that its chord of contact with the other may pass to infinity (Art. 259). Strictly speaking, all these projections have only been shown to be possible when the line projected to infinity does not meet the conic in real points; but it will be found in practice that this is a limitation which it is unnecessary to attend to, and that a projective proposition once proved true for any state of a figure may become unmeaning, but will never become false, when certain lines in that figure have become imaginary. Thus, for example, although the method of projecting into concentric circles only directly proves properties of conics having double contact, whose chord of contact is imaginary, we shall not think it necessary to seek for an independent proof of the same properties in the case where the chord of contact is real.

373. We shall now give some examples of the method of deriving properties of conics from those of the circle, or from other more particular properties of conics.

Ex. 1. "A line through any point is cut harmonically by the curve and the polar of

that point." This property and its reciprocal are projective properties (Art. 364), and both being true for the circle, are true for every conic. Hence all the properties of the circle depending on the theory of poles and polars are true for all the conic sections.

Ex. 2. The anharmonic properties of the points and tangents of a conic are projective properties, which, when proved for the circle, as in Art. 313, are proved for all the conics. Hence, every property of the circle which results from either of its anharmonic properties is true also for all the conic sections.

Ex. 3. Carnot's theorem (Art. 314), that if a conic meet the sides of a triangle,

Ab. Ab'. Bc. Be'. Ca. Ca' Ac. Ac'. Ba. Ba'. Cb. Cb',

=

is a projective property which need only be proved in the case of the circle, in which case it is evidently true, since Ab. Ab' Ac. Ac', &c.

=

The theorem is evidently true, and can be proved in like manner for any polygon.

Ex. 4. From Carnot's theorem, thus proved, could be deduced the properties of Art. 151, by supposing the point C at an infinite distance; we then have

The first theorem is obviously true, since the four lengths are constant. The second may be considered as an extension of the anharmonic property of the tangents of a conic. In like manner, the theorems (in Art. 278) with regard to anharmonic ratios in conics having double contact are immediately proved by projecting the conics into concentric circles.

Ex. 7. We mentioned already, that it was sufficient to prove Pascal's theorem for the case of a circle, but by the help of Art. 362 we may still further simplify our figure, for we may suppose the line joining the intersection of AB, DE, to that of BC, EF, to pass off to infinity; and it is only necessary to prove that, if a hexagon be inscribed in a circle having the side AB parallel to DE, and BC to EF, then CD will be parallel to AF; but the truth of this can be shown from elementary considerations.

Ex. 8. A triangle is inscribed in any conic, two of whose sides pass through fixed points, to find the envelope of the third (p. 229). Let the line joining the fixed points be projected to infinity, and at the same time the conic into a circle, and this property

becomes,- -"A triangle is inscribed in a circle, two of whose sides are parallel to fixed lines, to find the envelope of the third." But this envelope is a concentric circle, since the vertical angle of the triangle is given; hence, in the general case, the envelope is a conic touching the given conic in two points on the line joining the two given points.

Ex. 9. To investigate the projective properties of a quadrilateral inscribed in a conic. Let the conic be projected into a circle, and the quadrilateral into a parallelogram (Art. 365). Now the intersection of the diagonals of a parallelogram inscribed in a circle is the centre of the circle; hence the intersection of the diagonals of a quadrilateral inscribed in a conic is the pole of the line joining the intersections of the opposite sides. Again, if tangents to the circle be drawn at the vertices of this parallelogram, the diagonals of the quadrilateral so formed will also pass through the centre, bisecting the angles between the first diagonals; hence, "the diagonals of the inscribed and corresponding circumscribing quadrilateral pass through a point, and form an harmonic pencil."

Ex. 10. Given four points on a conic, the locus of its centre is a conic through the middle points of the sides of the given quadrilateral.

Ex. 11. The locus of the point where parallel chords of a circle are cut in a given ratio is an ellipse having double contact with the circle. (Art. 166.)

Given four points on a conic, the locus of the pole of any fixed line is a conic passing through the fourth harmonic to the point in which this line meets each side of the given quadrilateral.

If through a fixed point O a line be drawn meeting the conic in A, B, and on it a point P be taken, such that {OABP} may be constant, the locus of P is a conic having double contact with the given conic.

374. We may project several properties relating to foci by the help of the definition of a focus given, page 233.

Ex. 1. The locus of the centre of a circle touching two given circles is a hyperbola, having the centres of the given circles for foci.

If a conic be described through two fixed points, and touching two conics which also pass through those points, the locus of the pole of the line joining those points is a conic inscribed in the quadrilateral formed by joining the two given points to the poles of the same line with regard to the given conics.

We give this example as worth the learner's study, because it illustrates the different principles that all circles pass through two fixed points at infinity (Art. 259); that the centre is the pole of the line joining them (Art. 157); that a focus is the intersection of tangents passing through these fixed points (Art. 282); and that we are safe in extending our conclusion from imaginary to real points (Art. 372).

Ex. 2. Given the focus and two points of a conic section, the intersection of tangents at those points will be on a fixed line. (Art. 196.)

Ex. 3. Given a focus and two tangents to a conic, the locus of the other focus is a

Given two tangents, and two points on a conic, the locus of the intersection of tangents at those points is a right line.

Given four tangents and a fixed point on each of two of them, the locus of the