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straight lines. The object of the few following pages is to prove such properties of these curves as shall enable us to give rules for their construction. The complete investigation of their properties, which forms a distinct and very interesting branch of mathematics, is not here intended.

DEFINITION 1.-Let dD (Fig. 60) be a given fixed straight line, called the directrix; Sa given fixed point, called the focus; P a moveable point. Let Pd be perpendicular to the given straight line; then if P move in such a manner that SP bears a contant ratio to Pd, it traces out one of the curves called conic sections.

D

A S

Fig. Co.

DEFINITION 2.-Suppose SP: Pd::e: 1, then ife 1,. the curve is an ellipse; if e = 1, the curve is a parabola; if e 7 1, the curve is a hyperbola.

THE ELLIPSE.

Let Dd, S and P (Fig. 61) be the same as in the foregoing definitions; through S draw DAa perpendicular to Dd. Then if A and a are so taken that SA: AD: :e: 1, and Sa: aD::e: 1, then A and a are points in the ellipse; also the line Aa is the transverse diameter or major axis; D bisect Aa in C, then C is the centre of the ellipse; through C draw BCb perpendicular to Aa, then Bb is called to conjugate to the transverse diameter, or the minor axis; take H, so that aH AS, or so that CS CH, then H and S are called the foci (singular focus).

We have already

(aD-DA), and aH

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1. To show that in the ellipse CS=e, A

seen that AS=e. AD, also that Sa = e. aD .. aS — SA — e AS... SHe. Aa, or SC=e. AC.

N.B. e is called the eccentricity of the ellipse.

(2.) To show that SB AC.

e AD + e AC

=

=

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Manifestly SB = e. DC
AS SC AC. Q. E. D.
COR.-Hence, BC2= SB2 - SC2 = AC2 — SC2 = AC2 (1 − e2).

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Q.E.D.

=

AC-e

-e CN and SN SC- CN

= e AC.

.. (AC -e CN)2= (SC · CN)2+ PN2 and SC

.. AC2

2 e AC, CN + e2 CN2e2 AC2-2 e AC CN + CN2 + PN2.

.. AC2 (1 e2) = CN2 (1 — e2) + PN2.

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PN2
AC2 (1 — e2)

Q.E.D.

Now AC (1-e) = CB2.

BC2 AC

COR. With centre C and radius CA, describe a circle Apa; produce NP to meet the circumference in p; join Cp, which is plainly equal to AC.

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Cp = CA?.

=

PN2 CN2
+
CA2 CA2

PN2 pN2

CB2
PN pN

=

CA2
CB

= 1.

CA.

This result, and the previous one, we have already had occasion to use in the article on Mensuration. (See p. 376, &c.)

(5.) To show that HP = AC + e CN.

For HP2 PN2 + NH2 = PN2 + (CN + CS)2 = PN2 + (CN + e CA)2.

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Or, SP + PH = the major axis of the Ellipse.

On this property of the Ellipse the first and third practical methods of construction depends.

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(6.) If EFGH (Fig. 62) is a rectangular parallelogram, and DC and AB are the lines joining the bisections of the opposite sides, divide GB into n, equal parts, and also OB into n, equal parts. Let Gp, contain p of the parts into which GP is divided, and let Op contain p of the equal parts into which OB is divided, join Dp, and Cp, and produce it to meet Dp, in P; then P is a point in the Ellipse, whose major and minor are AB and

H

OD.

For, drawn PN parallel to DO and .. perpendicular to AB,

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axes

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Hence by article (4), P is a point in an ellipse, whose major and minor axes are OB

and OD respectively.

On this principle the second practical method is founded,

(7). If AB, CD (Fig. 63) are two lines at right angles to each other, and bisecting each other in the point O, then if Pab is a straight line, so placed that Pb = OC

and Pa OA, then the point P is in an

=

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Fig. 63.

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Hence (art. 4) P is a point in the specified ellipse. Q. E. D.

This is the principle of the Trammel described in the fourth practical method.

To describe an Ellipse, the transverse (AB) and conjugate (CD) diameters being given.

1ST METHOD.-Draw AB and CD, bisecting each other at right angles, in the centre

O (Fig. 64); with the centre D or C and radius OA or OB, describe arcs cutting AB in F and F, which points will be the foci of the ellipse; take any other point P in the transverse diameter AB, and with the centres F, F' and radii equal to AP and PB, respectively, describe arcs intersecting in the point E, which will be a point in the curve. By taking several points in the transverse diameter, and proceeding in a similar manner, as many points in the

D

C

Fig. 64.

curve may be found as may be required, through which the ellipse may be drawn.

B

2

F

D

3B

2ND METHOD.-Through the extremities of the given diameters draw the rectangle EFGH (Fig. 65); divide AE, AF and BH, BG into any number of equal parts (say 3); also divide AO and OB into the same number of equal parts; join D and the points of division in AF and BG, also C and those in AE, and BH. Again, from D draw lines through the points of division in AO and OB, intersecting the lines drawn from C to the points in AE and BH; and in the same manner from C draw lines through the points of division in AO and OB, intersecting the lines drawn from D to the points in AF and BG; the intersections thus derived are points in the curve through which the required ellipse may be drawn.

E

C

Fig. 65.

H

3RD METHOD.-An ellipse may easily be drawn by means of a thread, in the following manner :—

D.

B

Find the foci F and F' (Fig. 66) as in the first method; take a piece of thread equal in length to the distance betwen the foci added to the transverse diameter, and fasten the two ends; having fixed two pins in the foci, pass the thread round them and stretch it, place a pencil in the angle made by the thread when stretched, and commencing at one end (D) of the conjugate diameter, the thread being in the position FDF" (as shown in dotted lines), move the pencil point round, keeping the thread always stretched until the point again meets at D, the ellipse will be drawn.

A

F

C

Fig. 66.

4TH METHOD.-An instrument called a trammel is sometimes used for describing an ellipse, and is generally formed of two pieces of wood fixed at right angles to each other, and having a groove running through the centre of each, the groove being made a little wider at the bottom than at the top, in order to keep the guides b, c, (Fig. 67) attached to the bar, abc, in the groove; the bar, abc, should have a pencil fixed at the end a, the two guides, b and c, being moveable or sliding along the bar, but when set ready for use, secured by small screws in the proper positions. The method of using the trammel is as follows:

1st. Set the grooved pieces of wood on the given diameters AB and CD of the ellipse (as in the figure); the centre, where the two pieces cross, being on the centre O of the ellipse.

2nd. Make the distance from the pencil at a to the first sliding guide at b equal to

b

D

C

half the conjugate diameter, (or ab equal

Fig. 67.

to OD or OC); and from the same point a to the second sliding guide at c equal to half the transverse diameter (or ac equal to OA or OB).

3rd. Move the end a round, commencing at the point D, allowing the guides to move freely in the grooves, and the pencil will be found to describe the ellipse required.

The OVAL although not an ellipse, so nearly resembles it, that very frequently, from the simplicity of drawing an oval, it is used in place of an ellipse, and is formed of parts of circles by the following methods:

1ST METHOD.-Divide any line AB (Fig. 68) representing the transverse diameter into three equal parts in the points CD. With the centres C and D and the equal radii CA and DB, describe two circles, at the points E and F, where they intersect, as centres, and the equal radii AD or BC, describe parts of other two circles joining the circumferences of the two first circles (the exact points of junction of the two circles may be got by joining the centres of both circles, as shown by the dotted lines), which will complete the oval required.

E

Fig. 68.

B

2ND METHOD.It is often required to draw a flat or long-shaped oval, in which case divide the line AB (Fig. 69) into four equal parts in the points C, D, G, and with the centres C and G and radii equal to CA and GB, describe two circles; also with the same centres and radii equal to any two parts of the line AB, such as AD, describe arcs intersecting at E and F, with which, as centres and radii equal to three parts, of AB, such as AG, describe portions of other two circles joining the circumferences of the first circles, which will complete the oval required.

3RD METHOD.-To draw an egg-shaped oval.

Bisect the diameter AB in the point D (Fig. 70), with which, as a centre, and D'A or DB as a radius, describe a circle cutting the diameter EF in the point G: with the

E

H

Fig. 69.

Fig. 70.

centres A and B and radii equal to AB describe two arcs, AH and BH'; join AG and BG, and produce them until they cut the arcs AH and BH' in the points H and H', and with the centre G and radius GH, describe a part of a circle touching the two latter drawn arcs, which will complete the oval.

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