And by addition we have P, + P2+

1=3

Σ + Σ. + ... Σ. - ( + + ...", - 2p) π

−

...

n2

2 w r2.

COR. 4. If the polygons P1, P2, ... P, together entirely and exactly cover the sphere, we shall have

4 w r2 =

− (n1 + n2 + ... n, - 2p) * . 2 w r2.

2 π

EXERCISES ON THE CONE, SPHERE, AND CYLINDER.

1. Equal circular sections are equally distant from the centre of the sphere; and those which are equally distant are equal.

2. The greatest section of the sphere is made by a plane through the centre; and that section which is nearer to the centre is greater than the more remote. And conversely.

2

3. If two spheres touch each other, the line joining their centres will pass through the point of contact.

. 4. Two spheres cannot touch one another in more than one point.

5. Through any point on the surface of a cone one tangent plane, and only one, can be drawn: and through any point without the surface, two.

6. If a sphere and right cone touch in two points, they touch in an entire circle.

7. Through any point on the surface of a right cone, innumerable tangent spheres may be described: but only one of them can touch it in more points than one, and this touches in a circle whose centre is in the axis of the cone.

8. If a plane cut a right cone and be not parallel to a tangent plane, then two spheres can be found (one on each side of the plane) which shall touch the plane in points and the cone in circles.

9. If two unequal spheres lie each wholly without the other, two right cones can be described which shall be tangential to the spheres; if the spheres cut each other, only one such cone can be described; and if one sphere lie wholly within the other, no such cone cau exist.

10. Planes may be described to touch three spheres, cach of which lies wholly without the other two:-how many? Also, investigate the circumstances of any of the spheres cutting or lying within another.

11. Four unequal spheres lie each wholly without the others, and common cones are described to envelope each two: the vertices of these cones are situated in threes upon straight lines, and in sixes in four planes.

CHAPTER IX.

THE CONIC SECTIONS.

THE first volume of this course contains a short discussion on the conic sections by coordinates. So far as the investigation of properties is concerned, the coordinate method is by far the most simple and direct. The elements of the subject are then limited to the equations of the curves, and the equations of their tangents, normals, and diameters. In establishing that method, however, a geometrical property of the conic sections is assumed without proof. It is proposed, therefore, in this chapter, to show how those curves are formed on the surface of a cone, and thence to deduce by geometry some of their most important properties. This course will then contain the elements of the conic sections, both by geometry and analysis.

DEFINITIONS.

1. A conic section is the figure formed by the intersection of a plane with the surface of a right cone.* The cone is supposed to admit of indefinite extension, both below the circular base and beyond the vertex, so that the complete conical system is composed of two opposite cones, or two opposite sheets of the same cone.

2. A plane drawn through the vertex of the cone parallel to the cutting or sectional plane is called the directing plane.

3. The section receives different names, according to the position of the directing plane. When this plane touches the cone along one of its edges, the section made by the cutting plane is called the parabola. When the directing plane cuts both sheets of the cone, the figure is called the hyperbola. In this case there are two parts of the curve, which are often called opposite hyperbolas. In the investigation of properties, however, those parts are treated only as two branches of the same curve. When the directing plane cuts one sheet only of the complete cone, the figure is called the ellipse.†

4. The plane which passes through the axis of the cone, and is perpendicular to the sectional plane, is called the transverse plane.

5. The intersection of the transverse and sectional planes is called the transverse axis of the conic section. When this line meets the section in two points (as it evidently does in the ellipse and hyperbola), the portion of it intercepted between these points is called the transverse diameter; and when the transverse axis meets the section in one point

The sections made by a plane with the oblique cone, the ellipsoid, and some other surfaces, are also conic sections; but it will be sufficient for the objects of this course to consider only the sections made with the right cone.

† As varieties of these curves, the circle is a case of the ellipse, and is formed when the cutting plane is parallel to the base of the cone (Pls. i. 24). The point is also another case when the cutting and directing planes coincide; also when the directing plane of the hyperbolic section coincides with the sectional plane, the section becomes two straight lines.

only (as in the parabola), the portion of it within the curve is called the principal diameter or axis of the parabola.

6. The middle of the transverse diameter of the ellipse or hyperbola is called the centre of the curve, and a line drawn through the centre at right angles to that diameter is the conjugate diameter.

7. Any line drawn through the centre of a conic section, and limited by the curve, is called a diameter of the conic section. In the case of the parabola, the centre is infinitely distant, and therefore all its diameters are parallel to the principal diameter.

8. The points of section of the diameters with the curve are called the vertices of those diameters.

9. A tangent to a conic section is a straight line which meets the curve, but being produced, does not cut it.

10. If a tangent and a diameter of a conic section pass through the same point, they are said to be conjugate to one another; and any lines drawn parallel to these are said to be conjugate to one another. Of two lines drawn in this way, if both pass through the centre of a conic section, they are said to be conjugate diameters; and if one be a diameter, and the other a chord of a conic section, the chord is called an ordinate of that diameter. Also the part of the diameter between the vertex and ordinate is called the abscissa; and the abscissa and ordinate, when spoken of together, are called coordinates of the point or points. 11. The subtangent is that portion of the principal diameter of a conic section intercepted between the tangent and the ordinate drawn from the same point in the curve.

12. The normal is that portion of the perpendicular drawn to the tangent from the point of contact, which is intercepted between that point and the transverse axis; and the subnormal is the portion of the transverse axis intercepted between the normal and the ordinate at the same point.

13. The parameter, or (as it is sometimes called) the latus rectum of any diameter, is the third proportional to that diameter and its conjugate in the ellipse and hyperbola, and to any abscissa and its ordinate in the parabola.

[This is not a mere arbitrary definition, as might be supposed, but a term suggested by a property of the curve. Its demonstration is given in a subsequent part. By the term parameter in general is meant, some constant quantity of the curve. In the case of the circle, for instance, the radius may be considered as the parameter].

14. The focus of a conic section is that point in the transverse diameter at which the double ordinate is equal to the parameter of the transverse diameter; and the distance between the centre and focus is called the eccentricity.

15. If a third proportional (estimated from the centre in the ellipse and hyperbola upon the transverse axis) be taken to the eccentricity, and the semi-transverse diameter, then the perpendicular to the transverse axis drawn through the extremity of the third proportional is called the directrix. In the parabola, the directrix is at the same distance from the vertex that the focus is,

[It will be seen, when we come to the investigation of properties, that the directrix has a remarkable relation to each curve. For instance, the distance of any point in the curve from this line has a constant ratio to the distance of the same point from the focus. This property of the

conic sections is made the foundation of the coordinate method given in the first volume. The constant ratio has been named the determining ratio.]

THE PARABOLA.

PROPOSITION I.

In the parabola, the abscissas are to one another as the squares of their ordinates.

Let MVN be the transverse plane, ¿AI the parabolic sectional plane, AH the transverse axis, KGL, MIN two sections of the cone perpendicular to its axis. Also let KL, MN be the intersections of KGL, MIN with the transverse plane, and GFg, IHi those with the sectional plane. Then because KGL, MIN are sections perpendicular to the axis of the right cone, they are circles (Pls. 1. 24); and since the transverse plane passes through the axis of the cone, it passes through the centres of these circles, and therefore the lines KL, MN are diameters of the circles KGL, MIN.

K

H

L

G

N

Again, because the plane MVN passes through the axis of the cone, it is perpendicular to the base, and each of its parallel sections KGL, MIN (Pls. II. 16). It is also perpendicular to the sectional plane IAi (Def. 4); hence the two planes KGL, IAi being each perpendicular to the plane MVN, their common section Gg is perpendicular to the same plane (Pls. II. 18), and consequently to every line in it, as AH, KL. In a similar way it is shown that Ii is perpendicular

to AH and MN.

Also, since KL is a diameter of the circle KGL, and Gg is perpendicular to this diameter, Gg is bisected in F. Similarly Ii is bisected in H.

Now by the similar triangles FAL, HAN, and the equal lines KF, MH (AH being parallel to VM by Def. 3), we have

AF: AH:: FL: HN:: KF.FL:HM.HN. But by a property of the circle (Euc. 111. 35),

KF.FL

=

FG2, and MH. HN = HI2; wherefore AF: AH:: FG2: HI2.

And as AF, AH are obviously the abscissas, and FG, IH the ordinates of the points G and I in the curve, the property is consequently established.

COR. 1. Let B and P be any two points in the parabola PAP', and BF, PM their semi-ordinates; then by the proposition,

AF: AM::BF2 : PM2,

or AF: BF2 :: AM: PM2.

Hence, if we denote the third proportional to AF and BF, or to

AM and PM, by p, it will be obvious that p will be constant for every point in the curve; and therefore we have at any point Q,

NQ2=p. AN;

that is, if we call p the parameter of the principal diameter, the rectangle under the abscissa of any point in the parabola and the parameter of the axis is equal to the square of the semiordinate of that point.

This is one of the properties to which reference is made in Def. 13.

A

M

N

=

8

X

p; then by

COR. 2. Let AF be of such magnitude that BF Cor. 1, AF tp. Whence (Def. 14), the abscissa of the ordinate which passes through the focus of a parabola is equal to one-fourth of the parameter of the axis.

COR. 3. Let EX be the directrix of the parabola PAP'; then (Def. 15), FA = AE = (Cor. 2) p. Wherefore, the distance between the focus and directrix of a parabola is equal to one-half the parameter of the axis.

COR. 4. Let QQ' be any other ordinate to the axis of the parabola, and PC a line parallel to the axis, meeting the curve in P and the ordinate QQ'in C. Then by the proposition, p. AN - p. AM = NQ2 — MP2 = (NQ+ MP) (NQ – MP) = Q'C.QC; or p. PC = Q'C. QC. Hence, if a double ordinate of the parabola be divided into two segments by a line parallel to the axis, the rectangle under the two segments is equal to the rectangle under the dividing line and the parameter of the axis, the dividing line being limited by the ordinate and curve.

PROPOSITION II.

In the parabola, the line drawn from the focus to any point in the curve, is equal to the perpendicular drawn from the same point to the directrix.

(See last Figure.)

Let P be any point in the curve, PX a perpendicular on the directrix, and F the focus, of a parabola; then PF = PX. For PM being a perpendicular on the axis AM, we have (Fuc. 11. 4),

PX2 = EM2 = EF2 + FM2 + 2EF.FM
=2EF (EF + FM) + FM2.

But by Def. 15, and Prop. I., Cor. 3, EF

= p 2 AF, and also (Prop. 1.), p AM = MP2. MP2. Whence PX2 = p (AF + FM) + FM2 pAM+FM2

=

=

PM2 +FM2 = PF2;
or PX = PF.

COR. A line drawn from the vertex A perpendicular to the axis, is a tangent at the vertex.

For any point in this perpendicular is evidently at a greater distance from the focus F than A is, and therefore it is at a greater distance from the focus than from the directrix. Consequently by the proposition, this perpendicular does not meet the curve again; and hence (Def. 9) it is a tangent to it.