1

CON

190

CON

The line is passing through the focus perpendicular to thedirectrix, is the axis.

The point I, where the axis cuts the curve, is the vertex.

Any line parallel to the axis, terminated at one extremityby the curve, and on the concave side of it, is called, adiameter.

Any line parallel to a tangent at the limited end of a dia-meter, is called a double ordinate to that diameter.

The limited part of a diameter, contained by the curveand a double ordinate, is called the abscissa of that doubleordinate.

Figures 4, 5, 6. An abscissa, the ordinate, and the diameterbeing given, to describe the ellipsis or hyperbola.

Let A b be the diameter, a c the abscissa, and c d the ordi-nate. Draw a e parallel to c d, and d e parallel to C a. InDC take any number of points, r, g, h, and divide de in thesame proportion at f g, h. Draw bfi, b g k, b ii l ; likewisef i a, g k A, h l a, and through the points d, i, k, l, a, drawa curve. In the same manner may the curve for the oppositeordinate be drawn.

When the extremities of the diameter are on differentsides of the ordinate, the curve is an ellipsis; but when theextremities of the diameter are on the same side of the ordi-nate, the curve is an hyperbola. When the diameter, a b, isof infinite length, the ordinates, F x, G k, h l, will be parallel,then the curve is a parabola. Therefore, in Figure 4, thelines drawn from the points f, g, h, parallel to the transverseaxis, or abscissa, a b, instead of being drawn to the point b,as in Figures 1, 2, 3, make the only difference.

It is hardly possible to conceive more convenient or easiermodes of description than these; their correctness may beproved by showing that their common properties are similarto those demonstrated of conic sections.

Figures 7, 8 , 9.—Let A b be the diameter, c d the ordinate,and a c the abscissa, as before. In CD take any point, g,and divide d e by g, in the same ratio as d c is by the point,G ; draw g k a, b g k, Figure 7, and ekg, Figure 8 ; then,because of the similar triangles, B N it and b c g, b N : b c : :N K : C G; and also, because of the similar triangles A n k andA M G , A x : A M or <7 E : : N K : M G or c D. By constructionwe have g e :de or ca : :cg :Cd; and therefore by mul-tiplication we have bn + na : bc + ca :: n k 2 : c d 2 ,which property is known to be that of the ellipsis andhyperbola.

Corollary. — Since, in the parabola, b n and b c are of infi-nite length, and may therefore be said to be equal, b n andBC may therefore be expunged from the first two terms of theanalogy in the above general property; then we shall have

N A : 0 A : : NK 2 : CD 2 .

Or the truth of the operation may be shown by a particulardemonstration for the parabola thus: See Figure 9.

Because of the similar triangles ask and am g , an:AM : : N K : M g or C d ; by construction we have a m : a c : :C G or N K : C D; and consequently, by multiplication, a n :a c : : n k 2 : C d 2 , which is the property of the parabola.

Figures 7, 8, 9 .—In a conic section, are given the abscissa,a c, an ordinate, C d, and a point, k, in the curve : to deter-mine the species, and thence to describe the curve.

Draw e g D parallel to a C, and A e parallel to CD; throughthe points a and k draw ak g, cutting ED at gr; make dg:g C :: d g : g E, and through the points k and G draw k g bor g k b, which, if not parallel to a C, produce it until itmeet a c or C a in b ; then A b will be a diameter. In thiscase, the curve is an ellipsis or hyperbola. It is an ellipsiswhen the extremities of the diameter are on different sides ofthe ordinate, as in Figure 5 ; but when the extremities of thediameter are on the same side of the ordinate, the curve is an

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hyperbola. If k g be parallel to A c, the curve is a P a t „; n e4

A diameter, A b, and an ordinate, c d, being thus ascer ,

the curve will be described as in Figures 1, 2, 3. Gtnticulars relating to these curves will be found under tcles Ellipsis, Hyperbola , and Parabola. . , ap ed

CONICAL ROOF, a roof whose exterior surface is slike a cone.

CONICS. See Conic Section.

CONISTRA, the pit of a theatre. j, 0 la,

CONJUGATE DIAMETERS, of an ellipsis or h fP e e2£tr e-any two diameters that are parallel to tangents at themities of each other. . f lh e

CONOID, (from the Greek, KuroeiSric, partaking afigure of a cone), a figure generated by the revoluti ^conic section round one of its axes. There are t * iree , g pat®'conoids, viz., the elliptical, the hyperbolical, and t ... s0 idibolical; which are sometimes otherwise denominated, e "

or spheroid, hyperboloid, and paraboloid.

Now because the solid is generated by the revo

,lutin ' 1

al=o

the section of a cone upon its axis, the axis will then ‘that of the solid. In this case, since, in the generatin' 1solid, every point of the curve will describe a circle,tion of the solid parallel to the base will be a circle-

every

theP 1

see"

In" 6

or

If a conoid be cut by a plane meeting the base, or j S)of the base produced, the section will be either an c Pan hyperbola, or a parabola. . is

Every section of an ellipsoid oblique to its a ’ellipsis ; and if a paraboloid or hyperboloid be cut 1>J^ th emeeting the plane of the base, produced on theoutsi^^joid;

A'i

;Il'

In the pa'^Tn-il 1

figure, the section will also be an ellipsis, ju* ^ it j 0 pif the cutting plane be parallel to the axis, the sec ^ c ,be an equal parabola. In the hyperboloid, if the so ^ ^peby a plane parallel to a section of the cone, made 3 ^ tl' apassing through the point where the asympte ® ^ ^generating section meets the axis of the solid pro j a pesection will be an hyperbola : but if the cutting^ ^ pig'^parallel to the plane in which is the asymptote, 8,1 .y b»angles with the generating section, the section

yip' «■

and the ellipsis : the paraboloid, three sections; t fo"

parabola

Thus the ellipsoid has only two sections,

ellipsis : the paraboloid, three section®; ^circle, the ellipsis, and the parabola : the hyperb° t >sections, viz., the circle, the ellipsis, the hyperbo tJ -ipou’parabola: and the cone itself has five sections, viz.; ^ a

triangle is a section peculiar to the cone alone; the j yP 1(J pt>

‘ 4 ,-cle '

the circle, the ellipsis, the hyperbola, and the p ar8

to the cone and hyperboloid ; the parabola, to

parabolical and hyperbolical conoids; and the - ^- but also m

cirt' ie .j, J

ellipsis are common not only to the cone, ,

the three conoids. e9; tb° a ~ Vj

All parallel sections of conoids are of similar Salit may seem singular that this should be a gen ^

when it is considered that, in a cone, a sectm' 1 ^ the r<J ( ||Eapex, or point, is a triangle, and a section para er ally> ,|f the property existed g p 0 ' 1

“ the cone S “V''

, • niira^ 0 *’ n

this P< u e a j,etions.e

lui fe

when it is considered that, in a cone, a sectio' 1 ,,

section

an hyperbola ; so thattriangular and hyperbolic sections ofought also to be similar. To reconcileconsider, that in all hyperbolical parallel s»

sect! 01 *

(d> I

7 J r t , g QQH

the asymptotes make equal angles, and tne ^e,are nearer to that passing through the apex of ^g c 1a greater degree of curvature at the vertex o ^ fig 11 ' *^1/

o - —O- — - — i

than those which are more remote, thougnsimilar. Farther, if the legs of the hyper

bola be

' 1 *crht li ,]< '' '"'j per

extended, they will be infinitely near a straI =L e cut'" 1, pa 1 ' 1will fall in with the asymptotes nearly, andtion will bear no sensible magnitude, compare p/P®which is comparatively straight, as the leg 3 0

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