SECTION SECOND.

The Ellipse.

Def. 1. An ellipse is a plane curve described by the intersection of two radii, varying in such manner as to preserve in sum the same constant quantity, while they revolve about two fixed points as centres.

PROPOSITION I.

a+u

C

I

F

To find the Equation of the Ellipse. Let P be any point of the curve, formed by the intersection of the variable radii a +u, au, which are equal in sum to a constant quantity, 2a, and which revolve about two fixed points, F, F', distant from each other by the line 2c. Take the middle point O between F, F', for the origin of rectangular coördinates, the line passing through F, F', being the axis of X. There results,

Fig. 43.

a2

• x2 = a2 + u2 = y2 + c2 + x2,

(200)

a2 a2 + c2x2 = a2y2 + a2c2 + a2x2;

a2y2+(a2 — c2)x2 = a2(a2 — c2),

the equation of the ellipse.

Now, it is obvious that, if the curve cuts the axis of x, y for that point will be reduced to nothing; therefore, if we make y = 0, and denote by x, what x becomes for this value of y, we have (200) a2 • 02 + (a2 — c2)x3_o= a2(a2 — c2);

Xy_o=±a; hence

Cor. 1. The ellipse cuts the axis of abscissas at equal (201) distances on the right and left of the origin, which x'

0, the curve cuts the axis of y, but this condition

Cor. 2. The ellipse cuts the axis of ordinates at equal (202) distances above and below the origin, which distance, denoted by

Def. 2. The line MON = 2a, is denominated the Major Axis or the Conjugate Diameter, and passes through the Foci, F, F'; the line N POQ=2b, is the Minor Axis or the Transverse Diameter, being perpendicular to the former. [b can never > than a.]

If in (200) we substitute b2 for (a2 — c2), the equation of the ellipse becomes

y2 x2

a2y2+b2x2 = a2b2, or

b2

+3=1,

(203)

where the constants are the semimajor and semiminor axes. Cor. 3. The ellipse is symmetrical in reference to both (204) axes; since,

For every value of x, whether + or —,

and, for every value of y, whether + y or -y,

22), two equal values

Fig. 435.

-;

Cor. 4. The major axis bisects all chords parallel to the (205) minor, and the minor axis bisects all chords parallel to the major. Cor. 5. The origin bisects all chords drawn through it, (206)

Foci, plural of focus, fire-place.

and is, consequently, the Centre of the Ellipse; T therefore these chords are Diameters.

Fig. 437.

Cor. 6. Diameters, equally inclined to the major axis, (207) are equal; and the converse.

Cor. 7. Any ordinate of the ellipse is to the correspond- (208) ing ordinate of the circle, described on the major axis, as the semiminor axis is to the semimajor,

For let y, Y, be corresponding ordinates of the ellipse and circle; we have (203)

Cor. 8. The circle, described on the major axis, circum- (209) scribes the ellipse. Hence,

Cor. 9. The angle embraced by chords, drawn from any (210) point of the ellipse to the extremities of the major axis,

is obtuse.

[How?]

Fig. 439.

Cor. 10. Any abscissa of the ellipse is to the correspond- (211) ing abscissa of the circle described on the minor axis, as the semimajor axis to the semiminor.

Cor. 11. The circle described on the minor axis is in- (212) scribed in the ellipse.

Cor. 12. The angle embraced by chords drawn from any (213) point of the ellipse to the extremities of the minor axis, is acute.

Def. 3. The double ordinate drawn through the focus, is denominated the Parameter of the major axis, and sometimes the Latus Rectum.

To find the parameter we have only to make x = c in (200); whence there results

PARAMETER AND ECCENTRICITY.

117

or

a2y_c+(a2 - c2)c2 = a2(a2 — c2), Parameter = 2yx = c

α

=2.

a2 - c2 262 4b2 (2b)2

=

=

(26)2;

a 2a 2a

2a 2b 2b: Parameter; i. e,

Cor. 13. The Parameter is a third proportional to the (214) major and minor axes.

Putting the parameter = p, and substituting in (203) we get

y2 = 1 (a2 — x2),

P 2a

(215)

for the equation of the ellipse, in terms of the parameter and semimajor axis.

Def. 4. We call e the Eccentricity because it expresses the ratio of the distance, c, of the focus from the centre to the semimajor axis, and thus determines the form of the ellipse, as round or flat. When the eccentricity is O, the ellipse becomes a circle. Equation (216) is that of the ellipse, referred to its eccentricity and semimajor axis, and is convenient in astronomy.

=

It is sometimes desirable to have the equation of the ellipse, when the left hand extremity of the major axis is made the origin of abscissas. In order to this, we have only to substitute x a instead of ≈ in (203), as the new x exceeds the old by a, y remaining the same; which done, there results,

The origin might be transported to any other point, either in the curve or elsewhere, by changing the value of y as well as that

of x.

Scholium. It is easy to show that any equation referred to rectangular coördinates, and of the form

py2+qx2 = r;

is the equation of an ellipse; for we have

A tangent to the Ellipse makes equal angles with the (219) lines joining the foci and the point of tangency.

Let P be any point of the ellipse, through which the line P'PR is drawn so as to make the angles FPR, F'PP', equal; then will P'PR be tangent at the point P. For, producing F'P to Q, and making PQPF, we have

F'P'P'Q> F'P+ PQ = F'P+PF.

Fig. 44.

Now, if P'PR be not a tangent, let the second point, in which it cuts the curve, be P', which we are at liberty to suppose, since P' may be any point of P'PR; then the definition of the ellipse gives

F'P'P'FF'P+ PF, which is less than F'P'+P'Q;

.. PFP'Q, and .. / PPF < P'PQ, or FPR > QPR = F'PP', which is contrary to the hypothesis; hence, so long as the FPR = F'PP', the line P'PR cannot cut the ellipse, and is tangent to it. Q. E. D.

Scholium. It is on this account that F, F', are called the Foci of the ellipse; since, from the principle of light and heat, that the angle of reflection is equal to the angle of incidence, if the curve were a polished metallic hoop, and a flame placed at F, the rays, reflected from all points of the ellipse, would pass through F'.

Cor. 1. The tangents drawn through opposite extremities (220) of any diameter, are parallel.

[See (206), (99).]

Fig. 442.