COR. If a cone standing on a circular base be cut by any plane parallel to the base, the section will be a circle. This is evident as before: we may, if we please, suppose the points A, a, the centres of the curves.

362. The sections of a cone, standing on a circular base, may be either an ellipse, hyperbola, or parabola.

A cone of the second degree is said to be right if the line joining the vertex to the centre of the circle which is taken for base be perpendicular to the plane of that circle; in which case this line is called the axis of the cone. If this line be not perpendicular to the plane of the base, the cone is said to be oblique. The investigation of the sections of an oblique cone is exactly the same as that of the sections of a right cone, but we shall treat them separately, because the figure in the latter case being more simple will be more easily understood by the learner, who may at first find some difficulty in the conception of figures in space.

N

Let a plane (OAB) be drawn through the axis of the cone OC perpendicular to the plane of the section, so that both the section MS&N and the base ASB are supposed to be perpendicular to the plane of the paper: the line RS, in which the section meets the base, is, therefore, also supposed perpendicular to the plane of the paper. Let us first suppose the line MN, in which the M section cuts the plane OAB to meet

both the sides OA, OB, as in the figure, on the same side of the vertex.

Now let a plane parallel to the base be drawn at any other point s of the section. Then we have (Euc. III. 35) the square of RS, the ordinate of the circle, AR. RB, and in like manner rsar.rb. But from, a comparison of the similar triangles ARM, arM; BRN, brN, it can at once be proved that

Therefore

AR.RB: MR.RN:: ar.rb: Mr.rN.

RS2 : rs2 :: MR.RN: Mr.rN.

Hence the section MSSN is such that the square of any ordinate

rs is to the rectangle under the parts in which it cuts the line MN in the constant ratio RS2 : MR.RN. Hence it can immediately be inferred (Art. 149) that the section is an ellipse, of which MN is the axis major, while the square of the axis minor is to MN2 in the given ratio.

RS2: MR.RN.

M

Secondly. Let MN meet one of the sides OA produced. The proof proceeds exactly as before, only that now we prove the square of the ordinate rs in a constant ratio to the rectangle Mr.rN under the parts into which it cuts the line MN produced. The learner will have no difficulty in proving that the locus will in this case be a hyperbola, consisting evidently of the two opposite branches NSS, Ms' S'.

Thirdly. Let the line MN be parallel to one of the sides. In this case, since AR=ar, and RB:rb:: RN:rN, we have the square of the ordinate rs (= ar.rb) to the abscissa N in the constant ratio

RS2 (= AR.RB) : RN.

The section is therefore a parabola.*

A

A

a

R

'S

B

363. 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

*Those who first treated of conic sections only considered the case when a right cone is cut by a plane perpendicular to a side of the cone: that is to say, when MN is perpendicular to OB. Conic sections were then divided into sections of a rightangled, acute, or obtuse-angled cone; and according to Eutochius, the commentator on Apollonius, were called parabola, ellipse, or hyperbola, according as the angle of the cone was equal to, less than, or exceeded a right angle. (See the passage cited in full, Walton's Examples, p. 428). It was Apollonius who first showed that all three sections could be made from one cone; and who, according to Pappus, gave them the names parabola, ellipse, and hyperbola, for the reason stated, p. 180. The authority of Eutochius, who was more than a century later than Pappus, may not be very great, but the name parabola was used by Archimedes, who was prior to Apollonius.

the conic section (Art. 348); 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.

364. 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 RS2: rs2, as the rectangle under the parts into 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, a 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.

365. We give formal proofs of the two following theorems, though they are evident by the principle of continuity :

I. If a circular section be cut by any plane in a line QS, the diameters conjugate to QS in that plane, and in the plane of the circle, meet QS in the same point. When qs meets the circle in real points, the diameter conjugate to it in every plane must evidently pass through its middle point ». We have therefore

T

only to examine the case where QS does not meet in real points. It was proved (Art. 361) that the diameter df which bisects chords, parallel to qs, of any circular section, will be projected into a diameter DF bisecting the parallel chords of any parallel section. The locus therefore of the middle points of all chords of the cone parallel to qs is the plane Odf. The diameter therefore, conjugate to QS in any section is the intersection of the plane Odf with the plane of that section, and must pass through the point R in which QS meets the plane Odf.

D

S

R

II. In the same case, if the diameters conjugate to QS in the circle, and in the other section, be cut into segments RD, RF; Rg, Rk; the rectangle DR. RF is to gR.Rk as the square of the diameter of the section parallel to QS is to the square of the conjugate diameter. This is evident when qs meets the circle in real points; since rs2 = dr.rf. In general, we have just proved that the lines gk, df, DF, lie in one plane passing through the vertex. The points D, d are therefore projections of g; that is to say, they lie in one right line passing through the vertex. We have therefore, by similar triangles, as in Art. 364,

dr.rf: DR.RF:: gr.rk: gR.Rk;

and since dr.rf is to gr.rk as the squares of the parallel semidiameters, DR. RF is to gR. Rk in the same ratio.

If the section gskq and the line QS be given, this theorem enables us to find DR.RF, that is to say, the square of the tangent from R to the circular section whose plane passes through QS.

366. Given any conic gskq and a line TL in its plane not cutting it, we can project it so that the conic may become a circle, and the line may be projected to infinity.

To do this, it is evidently necessary to find O the vertex of a cone standing on the given conic, and such that its sections parallel to the plane OTL shall be circles. For then any of

these parallel sections would be a projection fulfilling the conditions of the problem. Now, if TL meet the conjugate diameter in the point L, it follows from the theorem last proved that the distance OL is given: for, since the plane OTL is to meet the cone in an infinitely small circle, OL' is to gL.Lk in the ratio of the squares of two known diameters of the section. OL must also lie in the plane perpendicular to TL, since it is parallel to the diameter of a circle perpendicular to TL. And there is nothing else to limit the position of the point O, which may lie anywhere in a known circle in the plane perpendicular to TL.

367. If a sphere be inscribed in a right cone touching the plane of any section, the point of contact will be a focus of that section, and the corresponding directrix will be the intersection of the plane of the section with the plane of contact of the cone with the sphere.

N

Let spheres be both inscribed and exscribed between the cone and the plane of the section. Now, if any point P of the section be joined to the vertex, and the joining line meet the planes of contact in Dd, then we have PD=PF, since they are tangents to the same sphere, and, similarly, Pd=PF", therefore PF+PF"=Dd, which is constant. The point (R) where FF" meets AB produced, is a point on the directrix, for by the property of the circle, NFMR

A

M

B

D

is cut harmonically, therefore R is a point on the polar of F. It is not difficult to prove that the parameter of the section MPN is constant, if the distance of the plane from the vertex be constant.

COR. The locus of the vertices of all right cones, out of which a given ellipse can be cut, is a hyperbola passing through the foci of the ellipse. For the difference of MO and NO is constant, being equal to the difference between MF" and NF".*

By the help of this principle, Mr. Mulcahy showed how to derive properties of angles subtended at the focus of a conic from properties of small circles of a sphere. For example, it is known that if through any point P, on the surface of a sphere, a great circle be drawn, cutting a small circle in the points A, B, then tan AP tan BP is constant. Now, let us take a cone whose base is the small circle, and whose vertex