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

By Geometry, (prop. 62,) .. FM2=PM2+FP2, R2=y2+x2+2acx+c3a2. And, by theorem 4, cor. 3, (314) y2=a2-x2+c2x2-c2 a2, wherefore, eliminating y, by adding these equations together, we have the equation R2=c2x2+2acx+a2.

then, extracting the roots of each side of this equation, we have R=cx+a =x+a. In like manner will be found r=cx-a=x-a; therefore R—r=2a.

HYPERBOLA.-THEOREM 6.

316. The line bisecting the angle, at any point in the curve formed by the two lines drawn from that point to each focus, is a tangent.

T a

The tangent MT at M will bisect the angle FMƒ. In MF take MK = Mƒ, and in MT take any point L; join fL, and let it meet the curve in Q; join also KL, FL, FQ. Then, by hypothesis, the angle KML=ƒML, KM =ƒM, and LM is common, the base LK is Lf; but the difference of any two sides of a triangle is less than the third; therefore FL-LK, or FL-Lf, is less than FK, or Ff, or FM -ƒM, or FQ-ƒQ. HencefL is greater than ƒQ ; for, since FL-fL is less than FQ-fQ, if fL were less than fQ, FL+LQ

=

* It is shown by every writer of Elementary Geometry, that the sum of every two sides of a triangle are greater than the third. Let a, b, c, be the three sides of a triangle; then, a+bc, a+c-b, b+ca: therefore, by transposition, arc-b, a-b-c, b—c-a, b—a—c, c—b—a, c-a-b. N.B. signifies greater than.

would be less than FQ; which is impossible. Therefore every point, L, in MT, except M, is without the curve of the hyperbola; and MT touches it at M.

HYPERBOLA.—THEOREM 7.

317. In the line of the axis major, the rectangle contained by the distance between the centre and the intersection of the tangent, and the distance between the centre and the ordinate, is equal to the

....

and by Geometry, (theorem 57,) · r (ε +u) = R(ε—u). Multiplying these three equations together, we have ux=a2.

HYPERBOLA.-THEOREM 8.

318. The semi-transverse axis is a mean proportional between the two distances in the line of the transverse axis; the one from the centre to the ordinate, and the other from the centre to the intersection of the tangent. For, by the preceding proposition, uxa2, therefore x: a :: a: u. (See figure, theorem 7.)

Whence

Or, if the subtangent PTs, then will CTCP-PT=x-s. (x− s)x=a2 or x2-sxa2; expressed as in the same proposition of the ellipse.

HYPERBOLA.- THEOREM 9.

319. If there be any tangent, and four perpendiculars to the line of axis, contained between the tangent and the line of axis, the rectangle under two, which passes through the vertices, will be equal to the rectangle of the third, which passes through the centre, and the fourth which is the ordinate.

Rovacsi-ime sift ai o

Therefore, by transposition,

a2+sx-x2=0; add sx-x2 to each side of this equation, and a2x2+2sx — s2 = sx -- s2, or a2 — (x2 – 2sx+s2)=s(x− s); and, since the difference of two squares is equal to the rectangle of the sum and difference of their roots, (a+x− s) (a−x+8)=s(x − s).

Therefore, eliminating AT, aT, PT, CT, by multiplying these equations and aG × AI=CH × PM.

HYPERBOLA.-PROBLEM.

320. Given the transverse axis of an hyperbola and an ordinate, to find the conjugate axis and assymtotes, (which are two straight lines, such as, as, if produced indefinitely with the curve, will never meet each other,) and thence to describe the curve itself.

Let Aa, (fig. 1, pl. V,) be the transverse axis, and let PM be an ordinate. Make PD equal to AP. Then on aD describe the semi-circle aND. Produce PM to N. Draw AR perpendicular to CD, and make AR equal to CA. Join NR, and produce NR and DA, if necessary, to meet each other in S; and draw MS, cutting AR in Q. Produce QA to T, and make AT equal to AQ. Then QT will be the conjugate axis, or AQ, AT, will each be the semi-conjugate axis. Through the points C, T, draw JH; and through the points C, Q, draw IK: then JH and IK are the assymtotes, by which the curve may be described.

Because CP=x, and CA=a, AP=PD=x-a, and aP equal to x+a; therefore PN is a mean proportional between x+a and x-a, or between aP and PD; but, in the hyperbola, the transverse axis is to the conjugate as the mean proportion between x+a, and x-a to the ordinate y or PM; but AR is divided in Q, in the same ratio as PN is in N. Therefore PN: PM :: AR: AQ. That is, as the mean proportional between x+a, and x-a is to the ordinate PM, so is the semi-transverse axis to the semi-conjugate axis.

OF THE PARABOLA.

DEFINITIONS RELATIVE TO THE PARABOLA.

321. That portion of the primary line which is within the curve, and which is terminated at one extremity by the vertex, is called the axis.

322. A straight line, drawn perpendicularly to the axis, between it and the curve, is called an ordinate.

PARABOLA.-THEOREM 1.

323. The squares of the ordinates of the axis are to each other as their distances from the vertex.

Let VRQ be a plane, passing through the axis of the cone, perpendicular to the cutting plane of the section AMII'M'; and let AH be their common section; then AH will be the axis. Let QIRI' be a section of the cone parallel to the base :

Then, because the base RIQI' is perpendicular to the plane VRQ, the two sections AMII'M', OMNM', are perpendicular to the plane VRQ; therefore their common sections, MM', II', are also perpendicular to VRQ and to the lines AH, RQ, ON; but, because the plane VRQ passes through HE AFWOLD the axis of the cone, it will divide every circle pa

rallel to the base into two equal parts; therefore ON is a diameter of the circle OMNM'; and, because the chords MM', II', are perpendicular to the diameters RQ, ON, they will be bisected; let H be the point of bisection in II', and P the point of bisection in MM'.

324. Let AP=x, PM=y, AH=%, HI=7, HQ=v, HR=OP=w, and PN=t.

By similar triangles, APN, AHQ......0%=t

and, by the circle,

QIR......y = vw

By multiplying these equations, the result will be xyzy. Therefore

x : 2 :: y2 : y2.

325. COROLLARY 1.-Hence =

326. COROLLARY 2.-Hence, because whatever may be the values of x, x, y, y, therefore or 2 is a constant quantity: hence, putting =p; therefore y2=px.

PARABOLA.DEFINITIONS, CONTINUED.

327. The constant quantity p is called the parameter or latus rectum.

328. COROLLARY.-Hence the parameter is a third proportional to the distance of the ordinate from the vertex, and the ordinate itself; for 2x: 2y: 2y: p, or px=2y2, that is, y2=1px.

THE THREE CURVES OF THE CONIC SECTIONS.-PROBLEM.

329. The vertical section of a right cone being given, and the position of the axis of a conic section, to describe that section.

Let AVB, (fig. 2, pl. V,) be the section of a cone through its axis; let ig be the line of the axis, and let it cut the section AVB at h, and the opposite side BV, produced, at g. On gh describe the semi-circle hqsg. Draw Vp parallel to AB, cutting the axis in p. Bisect hg in r, and draw pq, rs, perpendicular to hg. Make pw equal to pV; then, with the transverse axis, hg, and the ordinate, pw, describe the ellipse hwtg, cutting rs at t; then rt is the semi-conjugate axis.

In fig. 3, pl. V, draw the line aA, for the transverse axis, equal to gh, fig. 5; and bisect Aa in C, the centre. Through A draw DE, perpendicular to Aa;