Take OT' in FO produced =4. OF.

Bisect TN in M.

Take Q in OY such that MQ = MN=MT.

Fold through Q so that QP may be at right angles to OK

Let P be the point where QP meets the ordinate of N.

Then P is a point on the curve.

239. The subnormals= 2 OF and FP=FG = FT'. These properties suggest the following construction.

Take N any point on the axis.

On the side of N remote from the vertex take NG = 2OF.

Fold NP perpendicular to OG and find P in NP such that FP = FG.

Then P is a point on the curve.

A circle can be described with F as center and FG, FP and FT' as radii.

The double ordinate of the circle is also the double ordinate of the parabola, i. e., P describes a parabola as N moves along the axis.

240. Take any point N' between O and F(Fig. 69). Fold RN'P at right angles to OF.

Take R so that OP=OF.

Fold RN perpendicular to OP, N being on the axis.

Fold NP perpendicular to the axis. Now, in OX take OT=ON'. Take P' in RN so that FP = FT. Fold through P F cutting NP in P Then P and P' are points on the curve.

Fig. 69.

241. N and N' coincide when PFP is the latus rectum.

As N' recedes from F to O, N moves forward from F to infinity.

At the same time, T moves toward O, and T'(OT'= ON) moves in the opposite direction toward infinity.

242. To find the area of a parabola bounded by the axis and an ordinate.

Complete the rectangle ONPK. Let OK be divided into n equal portions of which suppose Om to contain r and mn to be the (r + 1)th. Draw mp, nq at right angles to OK meeting the curve in p, q, and pn' at right angles to nq. The curvilinear area OPK is the limit of the sum of the series of rectangles constructed as mn' on the portions corresponding to mn.

But pn :NK=pm . mn : PK- OK, and, by the properties of the parabola, pm : PK= Om2 : OK2 = r2 : n2 and mn : OK= 1 : n. ... pm .mn:PK. OK= r2 : n3.

... pn = r2 / n3 XNK.

Hence the sum of the series of rectangles

= 12 +22 + 32..... + (n-1)2 / n3 X NK

= (n-1)n(2n-1) / 1.2.3.n3 X NK

= 2n3 - 3n2 + n / 1.2.3.n3 X NK

= NK

= 1/3 of NK in the limit, i. e., when n is ∞. ... The curvilinear area OPK= 1/3 of NK, and the parabolic area OPN=2/3 ofNK.

243. The same line of proof applies when any diameter and an ordinate are taken as the boundaries of the parabolic area.

Section III. - The Ellipse

244. An ellipse is the curve traced by a point which moves in a plane in such a manner that its distance from a given point is in a constant ratio of less inequality to its distance from a given straight line.

Let F be the focus, OY the directrix, and XX' the perpendicular to O Y through F. Let FA : AO be the constant ratio, FA being less than AO. A is a point on the curve called the vertex.

Fig. 70.

As in § 116, find A' in XX' such that

FA':A'O = FA : AO.

Then A' is another point on the curve, being a second vertex.

Double the line A A' on itself and obtain its middle point C, called the center, and mark F' and O' corresponding to F and O. Fold through O' so that O' Y' may be at right angles to XX'. Then F' is the second focus and O'Y' the second directrix.

By folding A A', obtain the perpendicular through C.

FA .AO = FA':A'0

= FA + FA' : AO + A'O = AA': 00' = CA : CO.

Take points B and B' in the perpendicular through C and on opposite sides of it, such that FB and FB' are each equal to CA. Then B and B' are points on the curve.

AA' is called the major axis, and BB' the minor axis.

245. To find other points on the curve, take any point E in the directrix, and fold through E and A, and through E and A'. Fold again through E and F and mark the point P where FA' cuts FA produced. Fold through PF and P on EA'. Then P and P are points on the curve.

Fold through P and P so that KPL and K'L'P' are perpendicular to the directrix, K and K' being on the directrix and L and Z' on EL.

FL bisects the angle A'FP,

... LFP=PLF and FP = PL.

FP:PK=PL : PK

= FA : AO. And

FP:PK'=P'L' : PK'

= FA'.A'0

= FA :A0.

If EO = FO, FP is at right angles to FO, and FP = FP'. PP' is the latus rectum.

246. When a number of points on the left half of the curve are found, corresponding points on the other half can be marked by doubling the paper on the minor axis and pricking through them.

247. An ellipse may also be defined as follows : If a point P move in such a manner that PN2

: AN.NA' is a constant ratio, PN being the distance of P from the line joining two fixed points A, A', and N being between A and A', the locus of P is an ellipse of which AA' is an axis.

248. In the circle, PN2 = AN.NA'.

In the ellipse PN2 : AN.NA' is a constant ratio.

This ratio may be less or greater than unity. In the former case APA' is obtuse, and the curve lies within the auxiliary circle described on AA' as diameter. In the latter case,APA' is acute and the curve is outside the circle. In the first case AA' is the major, and in the second it is the minor axis.