2

=90- -B;

allied to each other, the doctrine 180° -B LAEB= of which is of the greatest use in physical astronomy, as well as in therefore sin A E B = sin (90 — † B) the physico-mathematical sciences, cos B, and consequently from and has been much cultivated by the above proportion AE both ancient and modern mathematicians.

CX sin B

cus.B

Again, the triangle APK gives sin A KP: sin APK, or sin A E B sin A a E, or rather cos B: sin (A + B) :: x: A K (because the external angle A a E=the two internal angles A and B). Hence A K = 2 sin (A+B). ; and there.

(A+B)

There are three methods of investigating and demonstrating the various properties of the conic sections. The first is to consider them as they are really, cut from the cone itself, which is the way adopted by all the ancients. In the second method, the properties are deduced from arbitrary deCos & B scriptions of the curves in plano; fore K E or PG= the properties belonging to these e sin B-s curves being shown to apply to those actually cut from the cone. In the third method, the chief properties are inferred from the dif ferent modifications of the general algebraic equation of lines of the second order, and the established analogies between the properties of equations and those of curves. Origin and General Equation of the Conic Sections.

Let a right cone B C D (pl. conic sections, fig. 1,) be cut by any plane AMP; required the equation of the curve MAm, which results

from the intersection.

cos B

FP; in the triangle A PF we have
To find an expression for the part
sin A FP, or sin B F G, or sin BGF,
cosB:x:: sin A: FP=
a sin A

cos & B therefore, by substituting these values of F P, P G, we have y y= sin A

ca sin B—xx sin (A+B)}

cos B
for the equation required.

Now there can occur but three cases. First. That A + B = 180°, in which case the cutting plane is parallel to the side B D (pl. conic is called a parabola, and its equasections, fig. 2). This conic section tion is

yy=

sin A X sin B

cx=

Con

sin 2 B COS B cos B cx=4 ca sin B, by (12 fig.) (be cause sin 1800, and sin Asin B), or y=2 sin B√ cx. sequently, the parabola is a curve formed by two equal and similar branches, infinitely extended, and constantly receding from each other.

If through the summit B we cause to pass a plane BCD, perpendicular to the base of the cone, and to the outting plane AMP, the intersection of these two planes will be a right line A a; and if we cut the cone parallel to its base by a plane FMG, we shall obtain a circle whose plane will be perpendicular to the triangle B CD, and whose intersection with the plane AMP will be a right line P M, perpendicular to the two right lines A a, F G. The line PM will be an ordinate common to both the circle and the section M Am. This premised, call A Pr, PM=y, AB = c, the angle A Ba =B, the angle BA a=A; the well known property of the circle gives yy FP× PG. To find analytical expressions for the lines FP and PG, draw A E parallel to CD, and P K parallel to B D, both in the plane BCD; this gives A B : sin A E B :: A E: sin B. But the (sin A + B)

Second. If A + B is less than 180°, it is easy to perceive that the plane A M P, if continued, must meet the other side BD (pl. conic sections, fig. 3). The conic section which results is called an ellipse, and is a curve formed by two equal, similar, and finite branches A Ma, Ama. Its equation is

yy=

sin A

{ COSB

ca sin B-xx

(pl. conic sec.
fig. 4.)

sin A sin (A+B-180°)

sin C sin D

Suppose the co-efficient of x equal to a constant quantity, as

a cone Bed a

ax

b

which

If we conceive equal, and opposite at the vertex, we then find y = to the right cone BCD, it is evident that the cutting plane A MP, if prolonged, will meet it, and that from their intersection will result a curve Mam; equal, similar, and opposite to the lower curve MA m; or rather these two curves which are called opposite hyperbolas, will form but one curve represented generally by same the same equation. (Pl. conic sec. fig. 4.)

is an equation to a right line; indeed is sufficiently evident, that hence we may conclude, which the hyperbola degenerates into a triangle, when co, that is, when the cutting plane passes through the vertex of the cone.

CONJUNCTION, in Astronomy, or planets in the same degree of the meeting of two or more stars the zodiac. It is true when they tude, and apparent when they have have the same latitude and longithe same latitude only. It is heliocentric when it would appear to an observer in the sun, and geocentric when it would appear to an observer on the earth. Superior when -the object is beyond the sun, and

If these sections had been made in an oblique cone, such as would have been the cone BCD (fig. 1) if the angle C had not been equal to the angle D, we should have had for the general equation sin A cr sin B. sin C sin D

y2

sin (A + B) }

or y2=

x2,

casin B

-22

sin D

inferior when between the sun and earth. The planets which are more remote from the sun than the earth, can have no inferior conjunctions, and hence they are called superior planets.

CONOID, a solid figure generated by the revolution of any conic section about its axis.

CONSEQUENTIA, a Latin term, commonly employed by astrono mers to denote the real or appa rent motion of a planet or comet, when it is moving from west to east, or according to the order of the signs; and is thus opposed to antecedentia, which denotes a con trary motion.

denoted by the leading letters of the alphabet, a, b, c, &c. to distinguish them from the variable and unknown quantities, which are represented by the final let

CONSTANT Quantities, in Algewhich are evidently equations of bra, are those whose values are the circle. This circumstance known, or which remain constantshews that in an oblique cone wely the same. These are commonly may form circular sections in two different ways; one by cutting the cone parallel to its base, the other by cutting it by a plane which makes, with one of the sides of the triangle through the axis, an an-ters, z, y, x, &c. gle equal to that which the other side of the same triangle forms with its base. This is called a subcontrary section.

In the third case, in which the

CONSTELLATION, in Astronomy, an assemblage or system of several stars, expressed and repre sented under the name and figure of some animal or other emblem.