ON THE NATURE AND PROPERTIES OF CURVES, AND THE CONSTRUCTION OF EQUATIONS.

SECTION I.

Nature and Propertics of Curves.

DEF. 1. A curve is a line whose several parts proceed in different directions, and are successively posited towards different points in space, which also may be cut by one right line in two or more points.

If all the points in the curve may be included in one plane, the curve is called a plane curve; but if they cannot all be comprised in one plane, then is the curve one of double cur

vature.

Since the word direction implies straight lines, and in strict. ness no part of a curve is a right line, some geometers prefer defining curves otherwise: thus, in a straight line, to be called the line of the abscissas, from a certain point let a line arbi. trarily taken be called the abscissa, and denoted (commonly) by at the several points corresponding to the different values of x, let straight lines be continually drawn, making a certain angle with the line of the abscissas: these straight lines being regulated in length according to a certain law or equation, are called ordinates; and the line or figure in which their extremities are continually found is, in general, a curve line. This definition, however, is not free from objection; for a right line may be denoted by an equation between its abscissas and ordinates, such as y=ax+b.

Curves are distinguished into algebraical or geometrical, and transcendental or mechanical.

Def. 2. Algebraical or geometrical curves, are those in which the relations of the abscissas to the ordinates can be denoted by a common algebraical expression: such, for example, as the equations to the conic sections, given at page 536, &c. vol. i.

Def. 3. Transcendental or mechanical curves, are such as cannot be so defined or expressed by a pure algebraical equation; or when they are expressed by an equation, having one

of its terms a variable quantity, or a curve line. Thus, y = log x, y = a. sin x, y = A. cos x, y = A*, are equations to transcendental curves; and the latter in particular is an equation to an exponential curve.

Def. 4. Curves that turn round a fixed point or centre, gradually receding from it, are called spiral or radial curves.

Def. 5. Family or tribe of curves, is an assemblage of several curves of different kinds, all defined by the same equation of an indeterminate degree; but differently accord. ing to the diversity of their kind. For example, suppose an equation of an indeterminate degree, a"--1 x=y" : if m = 2, then will ax = y2; if m = = 3, then will a2x: = y3; if m = = 4, then is a'r = y; &c.: all which curves are said to be of the same family or tribe.

Def. 6. The axis of a figure is a right line passing through the centre of a curve, when it has one if it bisects the ordinates, it is called a diameter.

Def. 7. An asymptote is a right line which continually approaches towards a curve, but never can touch it, unless the curve could be extended to an infinite distance.

Def. 8. An abscissa and an ordinate, whether right or oblique, are, when spoken of together, frequently termed co-ordinates.

ART. 1. The most convenient mode of classing algebraical curves, is according to the orders or dimensions of the equa. tions which express the relation between the co-ordinates. For then the equation for the same curve, remaining always of the same order so long as each of the assumed systems of co-ordinates is supposed to retain constantly the same inclination of ordinate to abscissa, while referred to different points of the curve, however the axis and the origin of the abscissas, or even the inclination of the co-ordinates in different systems, may vary; the same curve will never be ranked under dif ferent orders, according to this method. If therefore we take, for a distinctive character, the number of dimensions which the co-ordinates, whether rectangular or oblique, form in the equation, we shall not disturb the order of the classes, by changing the axis and the origin of the abscissas, or by varying the inclination of the co-ordinates.

2. As algebraists call orders of different kinds of equations, those which constitute the greater or less number of dimensions, they distinguish by the same name the different kinds of resulting lines. Consequently the general equation of the first order being 0 = a + Bx + ry; we may refer to the first order all the lines which, by taking x and y for the co

ordinates, whether rectangular or oblique, give rise to this equation. But this equation comprises the right line alone, which is the most simple of all lines; and since, for this reason, the name of curve does not properly apply to the first order, we do not usually distinguish the different orders by the name of curve lines, but simply by the generic term of lines: hence the first order of lines does not comprehend any curves, but solely the right line.

As for the rest, it is indifferent whether the co-ordinates are perpendicular or not; for if the ordinates make with the axis an angle whose sine is p and consine v, we can refer the equation to that of the rectangular co-ordinates, by making and x = +; which will give for an equation

y

μ

ชน

μ

Thus it follows evidently, that the signification of the equation is not limited by supposing the ordinates to be rightly applied and it will be the same with equations of superior orders, which will not be less general though the co-ordinates are perpendicular. Hence, since the determination of the inclination of the ordinates applied to the axis, takes nothing from the generality of a general equation of any order what ever, we put no restriction on its signification by supposing the co-ordinates rectangular; and the equation will be of the same order whether the co-ordinates be rectangular or oblique.

3. All the lines of the second order will be comprised in the general equation

0 = a + Bx + yy + dx2 + ɛry +(y2; that is to say, we may class among lines of the second order all the curve lines which this equation expresses, x and y de noting the rectangular co-ordinates. These curve lines are therefore the most simple of all, since there are no curves in the first order of lines; it is for this reason that some writers call them curves of the first order. But the curves included in this equation are better known under the name of CONIC SECTIONS, because they all result from sections of the cone. The different kinds of these lines are the ellipse, the circle, or ellipse with equal axes, the parabola, and the hyperbola; the properties of all which may be deduced with facility from the preceding general equation. Or this equation may be transformed into the subjoined one:

and this again may be reduced to the still more simple form y3 = ƒx2 + gr + h.

Here, when the first term fr2 is affirmative, the curve ex. pressed by the equation is a hyperbola; when fr is negative, the curve is an ellipse: when that term is absent, the curve is a parabola. When r is taken upon a diameter, the equations reduce to those already given in sect. 4, ch. 1.

The mode of effecting these transformations is omitted for the sake of brevity. This section contains a summary, not an investigation of properties: the latter would require many volumes, instead of a section.

4. Under lines of the third order, or curves of the second, are classed all those which may be expressed by the equation 0=a+Be + ry + dx2 + ɛry + (y2 + nx3 + 8x3y+ıxy1+xy3.. And in like manner we regard as lines of the fourth order, those curves which are furnished by the general equation 0 = a + B x + 7y + dx2 + sxy + (y2 + ne2 + Ox2y + uy2+ xử từ t phương về giờ tặng tay;

taking always x and y for rectangular co-ordinates. In the most general equation of the third order, there are 10 con. stant quantities, and in that of the fourth order 15, which may be determined at pleasure; whence it results that the kinds of lines of the third order, and much more those of the fourth order, are considerably more numerous than those of the second.

5. It will not be easy to conceive, from what has gone before, what are the curve lines that appertain to the fifth, sixth, seventh, or any higher order; but as it is necessary to add to the general equation of the fourth order, the terms

x3, x'y, x3y3, x'y', xy1, y3,

with their respective constant coefficients, to have the general equation comprising all the lines of the fifth order, this latter will be composed of 21 terms: and the general equation comprehending all the lines of the sixth order, will have 28 terms; and so on, conformably to the law of the triangular numbers. Thus, the most general equation for lines of the order n, will (n + 1). (n -- 2) contain terms, and as many constant letters,

which may be determined at pleasure.

6. Since the order of the proposed equation between the co-ordinates makes known that of the curve line; whenever we have given an algebraic equation between the co-ordinates z and y, or t and u, we know at once to what order it is necessary to refer the curve represented by that equation. If the equation be irrational, it must be freed from radicals, and if there be fractions, they must be made to disappear; this done, the greatest number of dimensions formed by the va

riable quantities x and y, will indicate the order to which the line belongs. Thus, the curve which is denoted by this equation y2 — ax = 0, will be of the second order of lines, or of the first order of curves; while the curve represented by the equation y2 = x √ (a2 — x2), will be of the third order (that is, the fourth order of lines), because the equation is of the fourth order when freed from radicals; and the line which is indicated by the equation y = will be of the third order, or of the second order of curves, because the equation, when the fraction is made to disappear, becomes a3y + x'y a3 — ax2, where the term 'y contains three dimensions.

7. It is possible that one and the same equation may give different curves, according as the applicates or ordinates fall upon the axis perpendicularly or under a given obliquity. For instance, this equation, y2=ax-x2, gives a circle, when the co-ordinates are supposed perpendicular; but when the co-ordinates are oblique, the curve represented by the same equation will be an ellipse. Yet all these different curves ap. pertain to the same order, because the transformation of rectangular into oblique co-ordinates and the contrary, does not affect the order of the curve, or of its equation. Hence, though the magnitude of the angles which the ordinates form with the axis, neither augments nor diminishes the generality of the equation, which expresses the lines of each order; yet, a particular equation being given, the curve which it expresses can only be determined when the angle between the co-ordinates is determined also.

8. That a curve line may relate properly to the order indicated by the equation, it is requisite that this equation be not decomposable into rational factors; for if it could be composed of two or more such factors, it would then comprehend as many equations, each of which would generate a particular line, and the re-union of these lines would be all that the equation proposed could represent. Those equations, then, which may be decomposed into such factors, do not comprise one continued curve, but several at once, each of which may be expressed by a particular equation; and such combinations of separate curves are denoted by the term complex curves.

Thus, the equation y2 = ay + xy-ax, which seems to appertain to a line of the second order, if it be reduced to zero by making y2 — ay—xy + ax = 0, will be composed of the factors (y — x) (y — a) =0; it therefore comprises the two equations y r = = 0, and ya = 0, both of which belong to the right line: the first forms with the axis at the origin of the abscissas an angle equal to half a right angle;