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9. Lines which in this manner approach nearer and nearer to tho branches of a curve, and yet can never meet it, are called asympoles.

4ly. If x> a, y becomes imaginary. Hence the curve cannot extend beyond GH.

5ly. If x is negative, y has two values, provided that x be less than b. The curve has therefore two negative branches.

6ly. If x=b, we have y=0; consequently the curve must pass through the point B. But it cannot descend lower, since x> b renders the values of y imaginary. 7ly. Make y=0; on the supposition of x being negative and=,

6we have yy=XX

= ) =0; whence x (6x)=0, which gives

atx x=0, x=0, x=b. Consequently the curve will pass once through the point B, and twice through the point A, where it forms a node

10. When several branches of the same curve pass through the same point, that point is called in general a multiple point ; and in

a particular cases a double, triple, &c. point, when two, three, &c. branches concur in that same point. Algebra teaches us to discern these points, and to ascertain the number of the concurring branches.

8ly. If b=0, the node vanishes, and the equation y=r (6+x) becomes yo= b

x3)

this equation belongs to an ancient curve called the cissoid, of which we shall presently treat. 11. Besides these multiple points,

E there are also points of inflexion, or

É of contrary flexure, and points of re

Ć flexion. The first are those where the curve, after having its convexity turned in one direction, begins to B

D turn it in the opposite sense. For

А) example, the curve MAM', whose equation is y:=ax, has a point of

P

M inflexion at A. Points of reflexion, otherwise

'T called cusps, are those where two branches of the same curve touch, without passing on the other side of the point of contact. Such is

Q the case in the curve m A m'; its equation is y:=ar?

12. If the equation of the coordinates is of the first degree, it always belongs to a right line, and

é

P for this reason, right lines are called lines of the first order or species.

; ar

B

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m

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A

If in the equation of the co-ordinates, there occur only yy, or It, or xy; the lines which they represent, are called lines of the second order.

When this equation is of the third degree, the lines which result are of the third order, &c. And as lines of the second order, are the simplest curves, they are also called curves of the first order ; so that lines of the third order are curves of the second, and so forth. Of the first species or order there is only the right line. There are four lines of the second order ; 72 of the third, (as may be seen in Newton's Enumeratio Linearum tertii Ordinis, and in the works of more recent Geometers, as Maclaurin, Emerson, Euler, Cramer, &c.; and there are a still greater number of the fourth order.

13. But in this division of lines into different orders, it must be observed, that we comprehend geometrical curves only. This term is applied to those curves which have for their abscissæ and ordinates right lines, whose relation can be determined geometrically. Thus a curve having for its abscissæ circular arcs, or right lines equal to sines, would not be a geometrical curve. Such a curve would be one of those called mechanical or transcendental. The former sort are also called algebraical curves.

In the examination of a curve, the principal object of the analysis is

lot. To find the equation when the curve is given; or to trace the curve, when the equation is given.

2ly. To determine its tangent.
3ly. To ascertain its curvature in any given point.
4ly. To find its greatest or least ordinates.

5ly. To obtain its exact quadrature, if susceptible of it; or at least its approximate quadrature.

6ly. To find its rectification; that is to determine the length of a right line equal to any one of its arcs.

Algebra is undoubtedly competent to all these investigations ; but fluxions, or as it is also termed the differential and integral calculus, is much more expeditious.

ORIGIN AND GENERAL EQUATION OF

THE CONIC SECTIONS.

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14. Let a right cone BCD be cut by any plane AMP; required the equation of the curve MAm, which results from the intersection. If through the summit Bwe

B cause to pass a plane BCD, perpendicular to the base of the cone, and to the cutting plane AMP, the intersection of these two planes will be a right line Aa; and if we

I cut the cone parallel to its base by a plane FMG, we shall obtain a circle whose plane will be perpen

E

K dicular to the triangle BCD, and whose intersection with the plane (

D AMP will be a right line PM, perpendicular to the two right lines Aa, FG. The line PM will be an ordinate common to both the circle and the section MAm.

This premised, call AP=, PM=y, AB=c, the angle ABa =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 AE parallel to Có, and PK parallel to BD, both in the plane BCD; this gives AB : sin AEB::AE : sin B. But the L AEBN

180°-B

= 90 - B; therefore sin AEB = sin

2 (90—1B)=cos B, and consequently from the above proportion

cx sin B

cos B Again the triangle APK gives sin AKP : sin APK, or sin AEB : sin AaE, or rather cos B: sin (A+B) :: *: AK (because the external angle AaE = the two internal angles A and B). Hence AK

x sin (A+B)

; and therefore KE or

cos B c sin B-I sin (A+B)

cos B To find an expression for the part FP; in the triangle APF we have sin AFP, or sin BFG, or sin BGF, or...cosB :::: sin A:

AE =

PG=

FP =

{

că sin B--x sin (A+B)}

2

P

-CI

2

2

x sin A

3

therefore by substituting these values of FP, PG, cos B

sin A we have yy=

A+B cos? | B for the equation required. 15. Now there can occur but

B three cases, Io. That A+B=180°, in which case the cutting plane is parallel to the side BD. This conic section is called a parabola, and its equation is

m2 sin A x sin B

sin ?B yy = cos? | B

M

cos ? B CI=4 cx sin ?} B, by (12 Fig.) (because sin 18 0=0, and sin A= sin B), or y=+ 2 sin ļ B cx. Consequently the parabola is a curve formed by two equal and similar branches, infinitely extended, and constantly receding from each other.

16. II°. If A+B is less than 180°. it is easy to perceive that

B the plane AMP, if continued, must meet the other side BD. The conic section which results is called an Ellipse, and is a curve formed by two equal, similar, and finite branches AMa, Ama. Its equation is sin AS

-1 }

K 17. IIIS. If A+B is more than

C

D 180°, the section is called an hyperbola, and its equation is

sin A yy=

cos B If we conceive a cone Bed equal, and opposite at the vertex to the right cone BCD, it is evident that the

d cutting plane AMP, if prolonged,

1 will meet it, and that from their inter

mí section will result a curve M'am' equal, similar, and opposite to the lower curve MAm; or rather these

(В two curves which are called opposite

А hyperbolas, will form but one same curve represented generally by the same equation.

B

M

yy=

com IB {cx sin B-xx(sinA+B)}

{ cx sin B+zx sin (A+B—1600)}

2

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a

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{ cx sin B~-sin (A+B)}

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2

18. Instead of supposing these sections to be made in a right cone, we might have supposed them to have been made in an obe lique cone, such as would have been the cone BCD (see figure to 14) if the angle C had not been equal to the angle D. In this case we should have found for the general equation

sin A yy =

sin C sin D This equation, in common with the foregoing one, belongs to an ellipse, a parabola, or an hyperbola, according as the sum of the angles A and B is less, equal to, or greater than 180o.

In the first case it expresses a circle, whenever the angle A is equal to either of the angles C or D; for then we have one or other of the following equations

cx sin B y =

or....yo =

cx sin B sin C

sin D which are evidently equations of the circle". This circumstance shews that in an oblique cone we may form circular sections in two different manners; 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 angle equal to that which the other side of the same triangle forms with its base. This is called a sube contrary section.

In the third case, in which the equation expresses an hyperbola, we may suppose c=0, and then we have

sin A sin (A+B-1800) y'=

sin C sin D Suppose the co-efficient of pl equal to a constant quantity, as a?

we then find G

which is an equation to a right line; hence we may conclude, which indeed is sufficiently evident, that the hy. perbola degenerates into a triangle, when c=0, that is, when the cutting plane passes through the vertex of the cone.

To reduce the equations of the sections of an oblique cone to those of a right cone, we need only observe that sin C sin Dcos? | B.

Such is the origin of the three conic sections ; we now proceed to investigate their principal properties, and to simplify this analysis, we shall suppose these curves described on a plane.

a

y=7

This will appear by comparing these equations with the equation of the circle in article 4, which is yy=2 ax-xx; for we have only to make the constant co-efficient

csin B c sin B

-2a and the equations become iden

or

sin C tical.

sin D

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