equal to the semiparameter.-Hence the ellipse and hyperbola have each two foci, but the parabola only one. The foci, or burning points, were so called, because all rays are united or reflected into one of them, which proceed from the other focus, and are reflected from the curve.

E'

E

M

M

18. The directrix is a line drawn perpendicular to the axis of a conic section, through an assignable point in the prolongation of that axis; such that lines drawn from that directrix parallel to the axis to meet the curve, shall be to lines drawn from the points of intersection to the focus, in a constant ratio

B

T

A

F

P C

for the same curve. Thus, if E M: MF:: E' M': M' F, then TE' E is the directrix. The curve will be a parabola, an ellipse, or an hyperbola, according as F M is equal to, less than, or greater than, M E.

19. An asymptote is a right line towards which a certain curve line approaches continually nearer and nearer, yet so as never to meet, except both be produced indefinitely. The hyperbola has two asymptotes.

2. If A в, a b, be the two axes, centre, F, f, the foci, P any point in the curve, P D, an ordinate: also, if A B = 2 t, ab 2 c C D = x', A D = x, D P = V, FPZ, angle P F D = o, √ (ta — c3)

=

(3)

E

D

K

H

L

The first of these is the equation of the curve when the abscissæ are reckoned from the extremity a of the transverse axis the second is the equation when the abscissæ are reckoned from the centre c and the third is called the polar equation, and is principally used in the investigations of astronomy.

Ex. Suppose A B = 20, a b = 12, A D = 4. Required the numeral value of P D.

=

Or, taking Equa. II. where c D = x' = 10- - 4 6. and t and c as before we have

also FP+ Pƒ= A B (5); and ƒ ƒ” = A B3 ·

4. Let Tκ be a tangent to the ellipse at any point к, and let T be the point where that tangent meets the prolongation of the axis: let also F H, fh be perpendiculas from the foci, F, f, upon the tangent, and let GH FH

then

G

H

T

A

F D

as before.

(4) a ba

(8) (9)

LFPT = LfPK.... (7) CD: CA: CA: CT. H and h fall in circumf. of circle whose diam. is A B .. If m be in the middle of P D, then Am produced will meet the two tangents т K, в K, in their point of intersection ¤ (10.)

If D the foot of the ordinate pass through F the focus, then the point, T, of intersection of the tangent and the prolongation of the axis will be the point r of the directrix (Def. 18) .... (11.)

f P

(13)

5. If an ordinate be drawn to any diameter of an ellipse, then will the rectangle of the abscissæ be to the square of the ordinate in a given ratio..... (14)

6. All the parallelograms that may be circumscribed about an ellipse are equal to one another: and every such parallelogram is equal to the rectangle of the two axes.. . . (15)

7. The sum of the squares of every pair of conjugate diameters is equal to the same constant quantity; viz. the sum of the squares of the two axes. . . ... (16)

8. Def. The radius of curvature of a conic section or other curve is the radius of that circle which is precisely of the same curvature as the curve itself, at any assigned point, or the radius of the circle which fits the curve, and coincides with it, at a small distance on each side of the point of contact. The circle itself is called the osculatory circle, or the equicurve circle; and if the curve be of incessantly varying curvature, each point has a distinct equicurve circle, the radius of which is perpendicular to the tangent at the point of contact.

9. Let p c be the radius of curvature at any point p in an ellipse or hyperbola whose major axis is a в, minor axis a b, and

The radius of curvature is greatest at the extremities of the

The radius of curvature is least at the extremities of the

10. PROB. To construct an ellipse whose two axes are given.

2

Find the distance F f, from the value of Ff, given in equa. 6, or from Ff= ✔A B a b3. Then, let a fine thread, FPF, in length=rf+A B, be put round two pins fixed at the points F, f: then, if a pencil be put within the cord, and the

whole become tightened so as to make three right lines F p,

Pf, fr, the point p may be carried on, the cord slipping round the fixed pins F, f, so as to describe and complete the ellipse A Pрвb A.

H

Otherwise.-Let there be provided three rulers, of which the two F I, ƒ H, are of the same length as the transverse axis A B, and the third H 1, equal in length to Ff, the focal distance. Then connecting these rulers so as to move freely about F f, and about HI, their intersection P will always be in the curve of the ellipse: so that, if there be slits running along the two rulers, and the apparatus turned freely about the foci, a pencil put through the slits at their point of intersection will describe the

curve.

A

F

** There are various other methods, as by the elliptic compasses, the trammels, &c. But the first of the above methods is as accurate and easy as can well be desired.

11. PROB. To find the two axes of any proposed ellipse.

Draw any two parallel lines across the ellipse, as м L, F K : bisect them in the points 1 and D, through which draw the right line NIDO, and bisect it in c.

From o

as a centre, with any adequate radius, describe an arch of a circle to cut the ellipse in the points G, H. Join G, H, and parallel to the line G H draw

A

a F

G

M

C

B

K

through c the minor axis a b; perpendicular to which through c draw A B, it will be the major axis.

12. PROB. From any given point out of an ellipse to draw a tangent to it.

8

R

M

Let T be the given point, through it and the centre c draw the diameter A B; and parallel to it any line HI terminated by the curve. Bisect HI in o : and co produced will be the conjugate to A B.Draw any line т s=T B, and make TR Draw R A, and parallel to it s p cutting A B in P. Through P, draw Pм parallel to c D, and join T м, it will be the tangent required.

=

T C.

B

H

D

SECTION II.-Properties of the Hyperbola.

1. If, in the annexed diagram, the conjugate hyperbolas whose vertices are D, B, are cut from the two opposite right cones whose common summit is N v, and в C, D E, be the diameters of the two circular bases of the two cones; then D B, K N, being the axes, and s, s', the foci, we shall have

If these three properties be compared with the corresponding ones for the ellipse, they will be found to agree, with the simple difference of the signs and of the connecting quantities in the first property. This at once indicates a general analogy between the properties of the two

curves.

2. Hence, putting a CC Bt, a c = cb= c, CF = d, AD = x, C D = x', DP =y, angle P F D = 9, Z = F P, we have y2

=

=

=

c2

t2

for the three most useful forms of the equation to the hyperbola, agreeing with those to the hyperbola, except in the signs.

P

3. And hence it follows, taking this and the preceding marginal figures to correspond with those in arts. 3 and 4 Ellipse; that the properties indicated by the parenthetical figures (4), (5), (6), (7), (8), (9), (11), (12), (13), (14), (15), and (16), hold in the hyperbola; simply changing+toin (5), to + in (6),

H

C TB

D

circumscribed to inscribed between the four hyperbolas in (15), and sum to difference in (16). Those properties, therefore, need not be here repeated.