Az2+ By D - Ca2.... (1,)

a® b2 z2 + a3 c2 y2 + b2 c2 x2 = a2 b2 c2 .... (2.) If any two of the coefficients be equal, for example, those of r and y', then the equation becomes

b2 z2 + c2 (x2 + y2) = a2 c2,

which is the equation to an ellipsoid of revolution about the axis of c. In like manner, if a = c, or b = c, the resulting equation will be that to an ellipsoid of revolution about the axis of b, or of a. If a = b = c, the equation will be

z2 + y2 + x2 = a2,

Az2 + Cr2 = DB ẞ.... (2,) By+CxD A .... (3,)

belong to the sections made by planes parallel respectively to the planes of x z and x y.

These sections therefore are ellipses, which become imaginary when

(113.) Secondly. We have hitherto supposed D to be negative, let it now be considered positive, then A z2 + B y2+ C x3 = − D,

which is impossible; therefore the surface is in this case imaginary.

(114.) Thirdly. Let D= 0,

... A z2 + By + C x2 = 0, which is the equation to a point.

Hence, the first species of surfaces that have a centre is an ellipsoid, which in particular cases becomes an ellipsoid of revolution, a sphere, a point, and an imaginary surface.

(115.) II. When A and B are positive and C negative, the equation becomes

A2+ByCx2 + D = 0.

First. Let D be negative.

It appears, therefore, that the sections of the surface made by planes parallel to y z are in all cases ellipses; the two remaining sections are hyperbolas. Hence the surface is continuous, and is called the

sects the three axes.

Referring to the last article,

In the first equation, let

The principal sections of the surface are 1. An ellipse, whose equation is

which is the equation to the elliptic or hyperbolic paraboloid, according as the upper or lower sign is used.

For farther information on the subject of Curves and of Surfaces, the reader is referred to the following works:

Annales de Mathématiques; Biot's Essai de Géométrie Analytique, sixth edition; Bourdon's Application de l'Algèbre à la Géométrie ; Boucharlat's Théorie des Courbes et des Surfaces du Second Ordre; Correspondance sur l'Ecole Polytechnique; Cramer's Introduction à l'Analyse des Lignes Courbes Algebriques; Euler's Introductio in Analysin Infinitorum, tom. ii.; Garnier's Géométrie Analytique; Hamilton's Principles of

x2 yo 22 2 x + ± =

a2 62 c2

a

que; Lacroix's Traité Elémentaire de Trigonométrie Rectiligne et Spherique, et d'Application de l'Algèbre à la Géométrie; Lardner's Algebraic Geometry; Maclaurin's Algebra; Monge's Application de l'Algèbre à la Géométrie; Peacock's Examples on the Differential and Integral Calculus; Poullet-Delisle's Application de l'Algèbre à la Géométrie; Reynaud's Traité d'Application d'Algèbre à la Géométrie.

Conic

THE principal properties of Lines of the Second Sections. Order, or, as they are more generally termed, of Conic Sections, may be derived with great facility from their equations, which have been already obtained in the Article entitled Analytical Geometry. But with a view of rendering the following Treatise independent of any previous investigation, we propose to deduce those equations from a general definition first assumed by Boscovich, and subsequently adopted by other writers of celebrity, as the basis of geometrical systems of

Fig. 1.

Fig 2.

Conic Sections.

(1.) Definition. A Conic Section is the locus of a point, whose distances from a fixed point and a straight line given in position, are to each other in a constant ratio.

Thus, let S be a fixed point, Kk a straight line given in position, P any point; join P, S, and let fall the perpendicular PQ upon Kk; then, if P be al ways taken to PQ in the same constant ratio, the locus of P will be a conic section.

The fixed point S is called the focus, and the straight line K k, given in position, the directrix.

(2.) The particular species of the conic section will depend upon the constant ratio of PS: PQ, which may be either a ratio of equality, or of lesser or greater inequality.

1. Let

PSPQ.

therefore the curve passes through the origin A. Let x be supposed to have any positive value. Then, for each assumed value of x, there are two equal values of y, with contrary signs; as a increases, the values of y increase; and when is taken indefinitely great, the values of y will also become indefinitely great.

Let x be now supposed to have any negative value. Then the values of y being in this case imaginary, it is plain that no part of the curve can lie to the left of A.

The parabola consists, therefore, of two infinite branches A Z, A z situated to the right of the point A, and symmetrically placed with respect to the straight line A X.

The point A is called the vertex, and the line A X the aris, of the parabola.

Cor. 1. The parabola can have but one focus, and one directrix.

Cor. 2. To find the value of S L, the ordinate pass

ing through the focus.

At the point S, x = A S = m,

Fig 3.

Conic

The double ordinate L passing through the focus, Sections. is called the principal parameter; or latus rectum, of the parabola.

Cor. 3. Hence, if P be any point in the parabola,
PMLI. AM;

that is, the square of the ordinate is equal to the latus
rectum multiplied into the corresponding abscissa.

(5.) To find the intersection of a straight line with a parabola.

Let the equation to the proposed straight line be y = ax +ẞ.... (1.)

Then the coordinates of the point or points of intersection with the parabola will be determined by combining this equation with the equation

This quadratic gives two values of y, which, substituted in (1), furnish two corresponding values of x; therefore the coordinates required may be determined.

When the two roots of the quadratic are equal, the points of section coincide, and the straight line Pp will then touch the parabola; and when the two roots are imaginary, the straight line P p falls entirely without the parabola.

Hence it appears, that a straight line cannot cut a parabola in more than two points.

That part of the straight line contained within the parabola, is called a chord; when it passes through the focus, it is then called the focal chord.

(6.) To find the equation to a straight line that touches a parabola in a given point.

Let x', y' be the coordinates of the given point, and x", y" those of any other point in the parabola near the first.

Then the equation to the straight line drawn through .these two points, and cutting the parabola, is

or

In the equation y y' 2 m (x + x).

Let

that is,

y = 0, as at T, then x + x = 0,
x = -x',
AT=AM;

the negative sign merely implying that AT must be measured in the contrary direction to A M.

Cor. 1. Hence MT 2 M A.

=

Def. The line M T intercepted between the foot of the ordinate, and the point where the tangent meets the axis, is called the subtangent.

It appears, therefore, that the subtangent is equal to twice the abscissa.

Cor. 2. Hence is derived a simple method of drawing a tangent to a parabola at a given point.

=

Let P be the given point, and A M, M P its coordinates; in M A produced take AT AM, join TP, then T P touches the parabola in P.

Def. The straight line which is drawn from the point of contact at right angles to the tangent, is called the normal.

(8.) To find the equation to the normal.

Let TP touch the parabola in P, and from this point draw Pg at right angles to PT. Then, since Pg is at right angles to PT, whose equation is

the equation to Pg will be

2 m

(9.) To find the intersection of the normal with the axis.

When the normal cuts the axis, as at G, then y = 0,

that is,

AGAM, or M G = 2 m.

Def. The line M G, incercepted between the foot of the ordinate and the point where the normal cuts the axis, is called the subnormal.

Hence it appears, that the subnormal is equal to half the latus rectum.

We have considered the normal P g as an indefinite line, but it is customary to give that name to the straight line PG intercepted between the point of contact and the point in which Pg cuts the axis.

Fig. 4.