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Let then a denote a constant quantity which I suppose=5, and let there be drawn an indefinite right line BD, on which I take AD-10=2 a; and which I divide into ten equal

HD parts AP, PP, &c.

Let A be B

ANP PP the origin of the abscissæ, BD their axis, AD the direction of the positive abscissæ, then AB will be that of the negative, if the curve required has any.

M To the point A let there be drawn the indefinite perpendicular Ex which I take as the axis of the ordinates, and of which I consider AE as the positive part. Lastly, let AP=r, and PM=y.

By the equation itself y=+V2 ar-ex, from which it is clear that when x=0, we have y=0; therefore the curve has the point A in common with the line of the abscissæ. If we make x=1, y becomes + 3; if x=2, y becomes + 4, &c.; so that the corresponding values of x and

y X=0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 y=0,+3, +4,+21, +24, +5,+24, + 21, +4, +3, +0 These values of y determine the lengths of

determine the lengths of so many ordinates PM, whose extremities M are points of the curve required; and since these values are at once both positive and negative, it is evident that drawing through the point A two equal branches, one of which passes through the points M, above the axis of the abscissæ, and the other through the corresponding points M', M' below, we shall have the curve required.

With respect to the description, it will be the more exact in proportion as we multiply the divisions of the line AD. In this manner we may describe any curve by referring each of its points M to two lines BD, EF, given in position; for if we compleat the parellelogram APMN, of which we know the two sides AP or NM, and I'M, the intersection of these last two lines will determine the point M of the curve. Thie parallelogram is called the parallelogram of the co-ordinates.

The values of y increase here from 0 to 5, and then decrease in the same proportion to zero again. Hence we may conclude 19, that there is one ordinate PM greater than any other ; this is called the maximum of the ordinates. (The investigation of the maxima and minima is one of the most curious analytical subjects; we shall hereafter give examples of it.)

Qly. We may also conclude, that the curve whose equation is yy=2 a1—1x, is a curve returning into itself, and compleally including a space. It does not extend beyond the point A ; for there its

abscissæ becoming negative, the values of y would be imaginary, which indicates that none of its branches can extend on the other side of the origin of the abscissæ. Let us now investigate some of its properties.

5. From C, the middle of the line AD, I draw the right sines CM, and thereby form as many right-angled triangles CPM, in which CM2= PM+ CP? = y2 +a+ - 2 ax +xx; therefore since y=2 ax-xx, we shall always have CM-a; and consequently all the points M of the curve are at an equal distance from the centre C, a distinguishing property of the circumference of the circle.

Again the equation yy=2 ax-.xr, gives x:y::y: 22-1, or AP : PM::PM: PD; therefore every perpendicular PM is a mean proportional between the two segments of the diameter AD, another property of the circle.

Draw the chord AM, and we shall have AMP=2 ax; therefore #:AM::AM : 2a; this shews that in the required curve any chords drawn from the point A to any point M, are mean proportionals between the diameter AD, and the corresponding segment AP; this also agrees with the circle.

If we draw the chord MD, we shall have AM2+ MD-4 a'-AD", a property of the right-angled triangle. Therefore all the angles AMD are right-angles, as they ought to be in the circle.

Inscribing the quadrilateral figure AMDM', we shall find that AMX M'D+AM'X MD = AD X MM. And in the same manner we may discover the remaining properties.

6. Let it now be required to describe a curve, supposing the equation of its co-ordinates to be y'=ar.

We see at once that this curve must cut E, the axis of the abscissæ at their origin,

M

N since supposing x=0, we also have y=0. It is also evident that it must have two equal branches, one positive, the other negative. These branches will extend ad

B A

P D infinitum, receding from their axis, as we assign greater values to x. But these values must all be positive, otherwise the ordinates would become imaginary. For these reasons, the curve will have the form N MAM.

7. Again suppose yy=xx-ad. It is clear that if the curve to which

E this equation belongs, cuts the axis of the abscissæ, or even

10 only touches it in some points, we shall determine them by supposing y=0. On this sup- BP А) S

D position we have x = + a. Therefore taking in an indefi. nite right line BD a point A for

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the origin of the abscissæ, and two parts AS, As equal to the given quantity a, the curve must pass through the points S, s, which are called its vertices, or summits.

To find the direction of its branches, let AD be the side of the positive abscissæ ; we shall have y=+V (x-a)=+(x+a) (t-a), this gives two branches, the one SM, the other SM', whose course extend indefinitely, provided x is greater than a. If it were less, the values of y would be imaginary; hence the curve does not pass on the other side of the point S, so long as the abscissæ are positive.

Suppose that we take them negative, the equation will become y==(-x+a)(-x-a). Here as long as z is less than a, the values of y will be imaginary. Hence no part of the curve will lie betwcen the points A and s.

If x=2, we find as above, y=0; if x > a, then y has two real values, one positive, the other negative ; and these values constantly increasing, the curve will have two new branches, opposite to, but equal to the two first. The axis of the abscissæ is BD, that of the ordinates, EF ; assigning values to x, we shall determine those of y or PM, and the parallelogram of the co-ordinates will give the points M, m, &c. through which the required curve must pass. We shall soon have occasion to examine the properties of these two curves. 8. Let us now investigate the curve whose equation is

bx2+x2

y =

P

Assume BD for the axis of the abscissæ, AD-a for the direction of the positive abscissæ, AB-6 for the direction of those that

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H are negative; the point A for their origin, EF for the axis of

M the ordinates; and we have

M
E

F y=+- (+1); this gives I",

in y=0, when x=0; the curve must therefore pass through the point A.

2ly. For each value of x, there are two values of y. Hence there are both positive and negative ordinates. It now remains to determine the points at which they cease to be real.

Bly. Take a positive, but less than a or AD, and we have for y two values PM, PM', which continue to increase till x=ra, when

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they become infinite ; for then y=+*+/+*); supporting there

fore as usual, that zero expresses an infinitely small quantity, or th

0 = -. line GH, without limit, that it may meet the two branches of the

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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 6. 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 xs b renders the values of y imaginary. 7ly. Make y=0; on the supposition of x being negative and =,

.bwe have yy=xx ( )=0; whence # (b-x)=0, which gives

a + x I=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 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'=x c) becomes yʻ= ; this equation belongs to an ancient curve called the cissoid, of which we shall presently treat. 11. Besides these multiple points,

I there are also points of inflexion, or of contrary flexure, and points of re

É flexion. The first are those where the curve, after having its convexity turned in one direction, begins_to

D turn it in the opposite sense. For example, the curve MAM', whose equation is yo=ax, has a point of

M inflexion at Å. 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 เรthe case in the curve m A m'; its equation is yr=ar?

12. If the equation of the coordinates is of the first degree, it always belongs to a right line, and for this reason, right lines are called lines of the first order or species.

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If in the equation of the co-ordinates, there occur only ‘yy, or At, 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

Ist. 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.

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