points may be determined, and a fair curve drawn through them is the conic required.

It is obvious that the ratio of Nn to XN is the same as the ratio of DE to XD, and therefore the ratio of FP to PM is the same as the ratio of DE to XD which was made equal to the given eccentricity. The line XE will touch the conic at a point R obtained by drawing FR perpendicular to the axis to meet XE at R.

These

In Fig. 75, three different cases are shown. In the ellipse the eccentricity is 1:3, in the parabola the eccentricity is of course 1:1, and in the hyperbola the eccentricity is 3: 1. eccentricities make the angle DXE equal to 30° for the ellipse, 45° for the parabola, and 60° for the hyperbola.

It is obviously unnecessary to draw the line XE in the case of the parabola. The radius for the arc through P may be taken at once from XN in this case, but it is instructive to notice that the construction given applies to all the conics.

If lines FY and FY, be drawn making 45° with the axis and

In

meeting XE at Y and Y1, then perpendiculars from Y and Y, to the axis determine the points A and A, where the conic cuts the axis. the case of the parabola there is only one such point, or to put it in another way, the point A, for the parabola is at an infinite distance from A.

It will be seen that the ellipse is a closed curve with two vertices, and that the hyperbola has two separate branches each with its own

vertex.

If A1F1, measured to the left of A,, be made equal to AF, and if A,X, measured to the right of A, be made equal to AX, and if X,M, be drawn perpendicular to the axis, then the ellipse may be constructed from the focus F, and directrix X,M,, both to the right of the figure, in the same way as from the focus F and directrix XM, using the same eccentricity. Also the hyperbola may be constructed from the focus F, and directrix X, M1, both to the left of the figure, in the same way as from the focus F and directrix XM, using the same eccentricity.

Referring to Fig. 75, it will be seen that as the eccentricity increases or diminishes the angle DXE increases or diminishes, and therefore as the eccentricity of the ellipse increases and approaches to unity the ellipse approaches to the parabola, and as the eccentricity of the hyperbola diminishes and approaches to unity the hyperbola also approaches to the parabola. A parabola is therefore the limiting form of an ellipse or an hyperbola. Remembering this fact, many of the properties of the parabola may be deduced at once from those of the ellipse or hyperbola. For example, the tangent to an hyperbola at any point P on the curve bisects the angle between the focal distances FP and FP. Now in the parabola the focus F, is at an infinite distance from the focus F, therefore FP is parallel to the axis, and the tangent to a parabola at any point P on the curve bisects the angle between the focal distance FP and the perpendicular PM on the directrix.

35. Conics defined with Reference to the Cone.-In Figs. 76, 77, and 78, vtu is the projection of a right circular cone on a plane parallel to its axis, vs being the projection of the axis of the cone. In each Fig. xx, represents a plane which cuts the cone and is perpendicular to the plane of projection.

If xx, cuts vt and vu below the vertex of the cone (Fig. 76), the section is an ellipse. If xx, cuts vt above and vu below the vertex (Fig. 78), the section is an hyperbola. If xx, is parallel to vt (Fig. 77), the section is a parabola.

In each Fig. the true shape of the section is shown, and is obtained by the rules of solid geometry. The axis XX, of the true shape of the section is drawn parallel to xx, and any point P on the curve is determined as follows:-Through any point n within the projection of the cone and on x.r, draw nN perpendicular to x, meeting XX, at N. Through a draw a line perpendicular to us and terminated by vt and vu. On this line as diameter describe a semicircle. Through n draw np parallel to vs to meet the semicircle at p. On the line nN make NP equal to np. P is a point on the true shape of the section. By repeating this construction any number of points may be obtained,

and a fair curve drawn through them is the curve required. The theory of the above construction will be understood after Art. 226, Chap. XVIII, has been studied.

To determine the positions of the directrices and foci of the conics, draw spheres inscribed in the cone and touching the plane of section. These spheres are represented by the circles whose centres are at 8 and 8, on the projection of the axis of the cone. These spheres will touch the cone in circles whose projections are the chords of contact of the circles which are the projections of the spheres and the lines vt and vu. The planes of these circles of contact intersect the plane of section in lines which are the directrices of the sections, and the projections of these lines are the points x and x. (In the case of the parabola the

D

point x, is at an infinite distance from x.) Hence if perpendiculars be drawn from x and x, to XX, the directrices XM and X,M, of the true shape of the section are obtained.

The inscribed spheres touch the plane of section at the foci of the section, and the projections of these points are at f and f. Hence perpendiculars from ƒ and f, to xx to meet XX, determine the foci F and F, of the true shape of the section.

A reference to the sketch shown in Fig. 79 will perhaps make the meaning of the foregoing statements a little clearer.

As in Art. 34, it is instructive to observe that the parabola is the limiting form of an ellipse or an hyperbola, as the plane of section

FIG. 79.

(Figs. 76 and 78) is turned round so as to come nearer and nearer to a position (Fig. 77) in which it is parallel to vt. Also, when the hyperbolic section is taken through the vertex of the cone, the hyperbola becomes two straight lines, and when the plane which gives the elliptic section is turned round so as to be perpendicular to the axis of the cone the ellipse becomes a circle. Again, if the plane of the parabolic section be moved parallel to itself nearer and nearer to vt, the ultimate form of the parabola will be a straight line. Studying Figs. 76, 77, and 78 still further, it will be seen that when the hyperbola becomes two straight lines, the directrices will coincide, and the foci will coincide at the point where the axis of the conic cuts the directrix, and where the two straight lines which form the hyperbola intersect. Again, when the parabola becomes a straight line, that line will be the axis of the conic, and the focus will be on the directrix. Lastly, when the ellipse becomes a circle the foci will coincide at the centre of the circle, and the directrices will move off to infinity.

36. Additional Definitions relating to Conics.-A perpendicular PN (Fig. 80) from a point P on a conic to the axis is called the ordinate of the point P, and if PN be produced to cut the conic again at P', the line PP' is called a double ordinate of the conic.

RFR', the double ordinate through the focus, is called the latus rectum of the conic.

In the ellipse and hyperbola the point which is midway between A and A, the points where the conic cuts the axis, is called the centre of the conic, and the ellipse and hyperbola are called central conics.

A straight line joining two points on a conic is called a chord of the conic.

R

N

A

FIG. 80.

37. General Properties of Conics.-Following from the general definition of a conic (Art. 33), there are many properties possessed by all conics which may be demonstrated. A few of the more important of these general properties will now be given.

(1) If a straight line cuts the directrix at D and the conic at P and Q (Figs. 81 and 82), then, F

being the focus, the straight line DF will bisect, either the exterior or the interior angle between PF and QF. Only in the case of the hyperbola, and only when P and Q are on different branches of the curve (Fig. 82) is it the interior angle PFQ which is bisected by DF.

If the straight line FP'

FIG. 81.

FIG. 82.

be drawn bisecting the angle PFQ (Fig. 81) and meeting the conic at I", then the angle DFP' is evidently a right angle, and if the line DPQ be turned about D so as to make P and Q approach nearer and nearer to one another, P' will always lie between P and Q, and in the limit when P and Q coincide they will coincide at P' and a straight line through D and P' will be a tangent to the conic at P'. Hence the next general property of conics.

(2) The portion of a tangent intercepted between its point of contact and the directrix subtends a right angle at the focus.

(3) Tangents at the extremities of a focal chord intersect on the directrix. This follows at once from

the preceding property. It is evident, conversely, that if tangents be drawn to a conic from a point on the directrix, the chord of contact passes through the focus.

(4) If PFP and QFQ (Fig. 83) be two focal chords, the straight lines P'Q and Q'P intersect at a point on the directrix; also the straight lines PQ and Q'P' intersect at a point on the directrix, and the

FIG. 83.

portion of the directrix DD' between the points of intersection subtends a right angle at the focus. The above follows very easily from the first property given in this article. (5) Tangents TP and TQ (Fig. 84) from any

point T subtend equal angles at the focus F.

(6) If the normal at P (Fig. 84) meet the axis at G, the ratio of FG to FP is equal to the eccentricity of the conic.

(7) PL (Fig. 84) the projection of PG on FP is equal to the semi-latus rectum.

A

FIG. 84.