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to the plane VDF, the three sections, A1Mami, Q1Ri, NMOm, are all perpendicular to the plane VDF; therefore their common sections, li, Mm, are perpendicular to the plane VDF, and to the lines Aa, QR, NO; but, because the plane VDF passes through the axis of the cone, it will divide all the circles parallel to the base into two equal parts; therefore QR and NO are the diameters of the circles Q1Ri, NMOm; and, because the chords li, Mm, are perpendiculars to the diameters, QR, NO, they will be bisected; let H be the point of bisection in 1i, and P the point of bisection in Mm.

Let CA = Ca = a, CP=x, PM=y, CH=2, HI = 7, PN = t, PO = u, HQ = v, and HR=w.

Then APCA + CP=a+x, aP=Ca- CP=a-x,

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Wherefore, eliminating t, u, v, w, by multiplying the given equations, the result will be y2(a+x) (a—x) = y2(a−x) (a+x), or, by actual multiplication, r2(a2 — x2) = y2(a2 — x2).

249. COROLLARY 1.-Hence, every chord perpendicular to the axis major is bisected by the axis major.

250. COROLLARY 2.-Hence the tangents at the extremity of the axis major are perpendicular to the axis major.

DEFINITIONS RELATIVE TO THE ELLIPSE, CONTINUED.

251. A straight line drawn through the centre, perpendicularly to the axis major, and terminated by the curve, is called the axis minor, or conjugate axis. 252. A third proportional to the axis major and minor, is called the parameter, or the latus rectum of the axes.

Thus a and b being the semi-transverse and semi-conjugate axes, 2a:2b::2b:p, the parameter; therefore, ap=26', or if f=1p, we shall have af b', therefore f ==

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253. That point in the axis, cut by an ordinate, which is equal to half the parameter, is called the focus.

In the figure here annexed, Bb, drawn through C, is the semi-axis minor; and, if Ll be a third proportional to Aa, Bb, then Ll is the parameter, and the point F, where it cuts Aa, is the focus.

254. Any line drawn through the centre, and terminated at each extremity by the curve, is called a diameter.

255. A diameter, which is parallel to a tangent at one extremity of another diameter, is called a conjugate diameter to that other diameter.

256. A straight line, parallel to a tangent, at the extremity of any diameter, terminated at one extremity by that diameter and the curve at the other, is called an ordinate to that diameter.

257. The portion of a diameter between the centre and an ordinate, is called the abscissa of that ordinate, or of that diameter.

M

In the figure here annexed, the straight line, Aa, drawn through the centre, C, is a diameter; and, if ST be a tangent at A, and the diameter Bb be drawn parallel to ST, the diameter is called the conjugate diameter of Aa; and PM, parallel to ST or Bb, is an ordinate to the diameter Aa; and the distance CP, on the diameter Aa, is called the abscissa.

ELLIPSE. THEOREM 2.

258. The square of the axis major is to that of the axis minor as the rectangle contained by the two parts of the axis major, from the ordinate to each vertex, to the square of the ordinate.

M

From the preceding theorem, y2 (a2 — x2)=y2 (a2 — ≈2). Now let b represent the semi-axis minor, and let the ordinate y become b, then will its abscissa, ≈, become zero, and, consequently, y2 (a2x2)=y2 (a2x2) will become b2 (a2 — x2) =a'y2; whence a2 : b2 : ; a2 —x2 : y2.

259. COROLLARY 1.-Hence every ellipse has two focii at an equal distance from the centre; because y2=(a2x2).

260. COROLLARY 2.-Hence the tangent at either vertex of the curve is parallel to the ordinates; and, consequently, perpendicular to the axis major.

ELLIPSE. THEOREM 3.

261. The square of the axis minor is to that of the axis major as the rectangle contained by the two parts of the axis minor, from the ordinate to the extremity of the axis minor, to the square of the ordinate.

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262. COROLLARY 1.-Hence the tangent at each extremity of the axis minor is parallel to the ordinates; and, consequently, parallel to the axis major.

263. COROLLARY 2.-Hence the axis major and minor are reciprocally conjugate diameters.

264. COROLLARY 3.-If a circle be described on either axis of an ellipse, an ordinate of the circle will be to the corresponding ordinate of the ellipse as the axis of this ordinate is to the other axis.

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Therefore, multiplying these two equations, we have b'ya'y', and ex

tracting the root by=ay; wherefore a b: 7 y.

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265. COROLLARY 4.-Hence any two corresponding ordinates of the circle and ellipse are in the same constant ratio of the two axes.

ELLIPSE. THEOREM 4.

266. The square of the distance of the focus from the centre is equal to the difference of the squares of the semi-axes.

CF2 AC2-BC2

=

Let the ordinate FG, which passes through the focus, be denoted by f, and CF, the distance of the focus from the centre, by ɛ.

A

G

Then, by the equation of co-ordinates....a3 y2 = b2 (a2 — x2) and, since (252) AC: BC:: BC: FG, .. a2ƒ2=b1. Now, in the first of these two equations, when the abscissa x becomes ε, the ordinate y will become f, and, consequently, a2ƒ2 = b2 (a2 — ε2). Whence b1 = b2 (a2 — ε2) or b2-a-2; and, by transposition, e2 – a2 — b2.

267. COROLLARY 1.-Hence, because af=b', therefore afa2- ¿3.

268. COROLLARY 2.—Hence b2 = a2 — e2 = a2 — c2 a2=a2 (1 — c2).

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269. COROLLARY 3.-Because a2y2=b2 (a2 — x2) and that b2=a2 (1 − c2). Therefore, by substitution, there will arise y2 = (1 — c2) (a2 — x2) = a2 — x2 − c2 a2+c2x2.

270. COROLLARY 4.-The semi-conjugate axis CB is a mean proportional between AF, FB, or between Af, fB, the distances of either focus from the two vertices, for b2 = a2 — e2 = (a + ε ) ( a − ε).

a

ELLIPSE.—THEOREM 5.

271. The sum of two lines drawn from the focii, to meet any point in the curve, is equal to the transverse axis.

B

M

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By Geom. (160).... FM2 = PM2 + FP2, R2 = y2 + c2a2 +2cax+x2,

y2=a2-c2a2 + c2x2 — x2.

and, by theorem 4, cor. 3, Wherefore, eliminating y, by adding these equations together, we have R2 a2+2cax + c2x2.

the equations.. Then, extracting the roots of each side of this equation, we have R = a + cx. In the same manner will be found ra-cx; therefore R + r = 2a. 272. COROLLARY 1.-Hence Ra+x, and r = a-x, for e=ca or c=

273. COROLLARY 2.-Because R=a+x, we shall have aRa2 +ɛx; and, by transposition, a (R- a)=ex, hence R-a : x ¦ ¦ ɛ : a.

R

M

274. COROLLARY 3.--Since the ratio of to a, or that CF to CA is constant, if we produce aA to R, and find the point R by making CF : CA :: CA : CR, and draw RX perpendicular to aR, and MX parallel to aR, and let CR=d; then will R-a:x:: a: d; wherefore Rd-ad=ax, or, by transposition, Rd=ad+axa (d+x). Therefore d+x: R::d: a; wherefore MX is to MF always in the constant ratio of CR to CA; and hence we have another method of constructing the ellipse.

N.B. The line RX is called the directrix. Several writers make this the fundamental principle from which all the other properties emanate.

275. COROLLARY 4.-Hence CR: CA :: aR: aF; because, when the point M comes to a, the line MX or PR will become aR, and the radius vector, FM, will become Fa.

276. COROLLARY 5.-Hence CA: CF:: aR: aF.

ELLIPSE. THEOREM 6.

277. The tangents from the corresponding points in the curves of a circle and ellipse, made by the prolongation of an ordinate of the ellipse, will meet the axis major produced in the same point T.

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