-Let E equal the area of an ellipse; A the area of a circle, of which the radius a equals the semi-major axis of the ellipse, and let b equal the semi-minor axis. Then it has been shown that -

E: A:: b: a,

e = A b/a

The area of a circle (Art. 448) is -

A = 1/2 π dr = π2r2,

and when the radius equals a -

A = π a2 This value of A, substituted in the above equation, gives -

E = π a2 b/2

E = πab. (178)

Or: The area of an ellipse equals 3.14159 1/4 times the product of the semi-axes; or 0.7854 times the product of the axes,

459. - Ellipse: Practical Suggestions. - In order to describe the curve of an ellipse, it is essential to have the two axes; or, the major axis and the parameter; or, the major axis and the focal distance.

If the two axes are given, then with the semi-major axis for radius, from B (Fig. 299) as centre an arc may be made at F and F' the foci; and then the curve may be described by any of the various methods given at Arts. 548 to 552.

If the major axis only and the parameter are given, then (Art. 454) since -

b2 = ay, we have - -

458 Ellipse Area 427

Fig. 303.

dius and from the focal points as centres, describe arcs cutting each other at B and O (Fig. 299). The intersection of the arcs gives the limit to B O, the minor axis. With the two axes proceed as before. Points in the curve may be found by computing the length of the ordinates, and then the curve drawn by the side of a flexible rod bent to coincide with the several points..

For example, let it be required to find points in the curve of an ellipse, the axes of which are 12 and 20 feet; or

(179.)

(179.)

Or: The semi-minor axis of an ellipse equals the square root of the product of the semi-major axis into the semi-parameter. Then, having both of the axes, proceed as before.

If the major axis and the focal distance are given, or the location of the foci; then with the semi-major axis for rathe semi-axes 6 and 10 feet, or 6 x 12 = 72 inches, and 10 x 12 = 120 inches.

Fix the positions of the points NN', etc., along the semi-major axis C M (Fig. 303) at any distances apart desirable. It is better to so place them that the ordinates when drawn shall divide the curve B PM into parts approximately equal. If CM be divided into eight parts as shown, these parts measured from C will be well graded if made equal severally to the following decimals multiplied by CM. In this case CM= 120; therefore -

CN

=

120

X

0.3

=

36.

= x'

CN'

=

120

X

0.475

=

57.

= x'

CN"

=

120

X

0.625

=

75.

= x'

Etc.,

=

120

X

0.75

=

90.

= x'

I20

X

0.85

=

102.

= x'

I20

X

0.925

-

111.

= x'

I20

X

0.975

=

117.

=x'

1I20

X

1.0

=

120.

= x'

The equation of the ellipse having the origin of co-ordinates at the centre (Art. 454) is -

a2y2 = b (a* - x'2),

or, dividing by a2 -

y2 = b2/a2 (a2-x'2),

or -

458 Ellipse Area 429

or -

(180.)

(180.)

in which a and b represent the semi-axes. Substituting for these their values in this case, we have -

458 Ellipse Area 431

Now, substituting in this equation the several values of x'2 successively, the values of the corresponding ordinates will be obtained. For example, taking 36, the first value of x' as above, we have -

458 Ellipse Area 432

y - 68.684;

458 Ellipse Area 433

y = 63.359;

and so in like manner compute the others.

The ordinates for this case are as follows, viz.:

When

x'

=

0

y

=

72.0

,,

x'

=

36.

y

=

68.684

,,

x'

=

57,

y

=

63.359

,,

x'

=

75,

y

=

56.205

,,

x'

=

90,

y

=

47.624

,,

x'

=

102,

y

=

37.928

,,

x'

=

111,

y

=

27.358

,,

x'

=

117,

y

=

15.999

,,

x'

=

120,

y

=

0.0

The computation of these ordinates is accomplished easy \y by the help of a table of square roots and of logarithms.

For example, the work for one ordinate is all comprised within the following, viz.:

458 Ellipse Area 434

1202=

14400

362 =

1296

13104

-

4.1174039

Half

-

2.0587020

0.6

-

9.7781513

68.684

-

1.8368533..

The logarithm of 13104 = 4.1174039. The half of this is the logarithm of the square root of 13104. To the half logarithm add the logarithm of go; the sum is the logarithm of 68.684 found in the table (see Art. 427).