first order all the lines which, by taking x and y for the coordinates, 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 μ and cosine », we can refer the equation to that of the rectangular co-ordinates, by making y ==—, and x = +t; which will give for an equation between the perpendiculars t and u, βν น 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 whatever, 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 + B.x + xy + dx2 + ɛxy + (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 denoting 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: 8x2+6x+ a y2 + = + y + =0: and and this again may be reduced to the still more simple form y2 =ƒx2 + gx + h. Here, when the first term fr2 is affirmative, the curve expressed by the equation is a hyperbola; when fr2 is negative the curve is an ellipse; when that term is absent, the curve is a parabola. When is taken upon a diameter, the equations reduce to those already given in sect. 4 ch. i. 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 + B.x + xy + dx2 +ɛxy + {y2 + nx3 +0x22 y + 1xy2+xy3. And in like manner we regard as lines of the fourth order, those curves which are furnished by the general equation 0 = a + Bx + xy + d.x2 +exy + {y2+y.x3 + 0.x2y +6.xy2+ uy trình tưởng tượng tặng toy ; taking always x and y for rectangular co-ordinates. In the most general equation of the third order, there are 10 constant 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 now 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 x5, x^y, x3y2, x2y3, 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. 1 2 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 x 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 yariable quantities x and y, will indicate the order to which the line belongs. Thus, the curve which is denoted by this equation y-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/(a2x2), 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 = a3-ax2 a2 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 a'y + xy= 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, yax-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 appertain 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 benot decomposable into rational factors; for if it could be composed of two or of 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 yay + xy-ax, which seems to appertain to a line of the second order, if it be reduced to zero by making y2 ay xy + ax0, will be composed of the factors (y-x) (ya) = 0; it therefore comprises the the two equations y − x = 0, and y — a = 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; and the second is parallel to the axis, and drawn at a distance =a. These two lines, considered together, are comprized in the proposed equation y = ay + xy-ax. In like manner we may regard as complex this equation, yxy3 a2x2 — ̧‚ay3 + ax1y + a2xy = 0; for its factors being (y-x) (y-a) (y2-ax)= 0, instead of denoting one continued line of the fourth order, it comprizes three distinct lines, viz, two right lines, and one curve denoted by the equa. y2 — ax R M A P -- = 0. 9. We may therefore form at pleasure any complex lines whatever, which shall contain 2 or more right lines or curves. For, if the nature of each line is expressed by an equation referred to the same axis, and to the same origin of the abscissas, and after having reduced each equation to zero, we multiply them one by another, there will result a com`plex equation which at once comprizes all the lines assumed. For example, if from the centre c, with a radius CA=a, a circle be described; and further, if a right line LN be drawn through the centre C; then we may, for any assumed axis, find an equation which will at once include the circle and the right line, as though these two lines. formed only one. M Suppose there be taken for an axis the diameter AB, that forms with the right line LN an angle equal to half a right angle: having placed the origin of the abscissas in A, make the abscissa AP, and the applicate or ordinate PM = = y; we shall have for the right line, PM = CP = a x; and since the point м of the right line falls on the side of those ordinates which are reckoned negative, we have y = - a + x, or y−x + a = 0: but, for the circle, we have PM2=AP. PB, and BP = 2a x, which gives y2 =2ax x2, or y2 + x22ax=0. Multiplying these two equations together we obtain the complex equation of the third order, y3 — y2x + yx2 x3 + ay2 - 2axy + 3ax2 2a2x = 0, which represents, at once, the circle and the right line. Hence, we shall find that to the abscissa AP = x, corresponds three ordinates, namely, two for the circle, and one for the right line. Let, for example, x = a, the equation will become y3 + Zay2 — ža2y-a30; whence we first find y+a=0, and by dividing by this root, we obtain ya= 0, the two roots of which being taken and ranked with the former, give the three following values of y: VOL. III. I. V We see, therefore, that the whole is represented by one equation, as if the circle together with the right line formed only one continued curve. 10. This difference between simple and complex curves being once established, it is manifest that the lines of the second order are either continued curves, or complex lines formed of two right lines; for if the general equation have rational factors, they must be of the first order, and consequently will denote right lines. Lines of the third order will be either simple, or complex, formed either of a right line and a line of the second order, or of three right lines. In like manner, lines of the fourth order will be continued and simple, or complex, comprizing a right line and a line of the third order, or two lines of the second order, or lastly, four right lines. Complex lines of the fifth and superior orders will be susceptible of an analogous combination, and of a similar enumeration. Hence it follows, that any order whatever of lines may comprize, at once, all the lines of inferior order, that is to say, that they may contain a complex line of any inferior orders with one or more right lines, or with lines of the second, third, &c, order; so that if we sum the numbers of each order, appertaining to the simple lines, there will result the number indicating the order of the complex line. Def. 9. That is called an hyperbolic leg, or branch of a curve, which approaches constantly to some asymptote; and that a parabolic one which has no asymptote. ART. 11. All the legs of curves of the second and higher kinds, as well as of the first, infinitely drawn out, will be of either the hyperbolic or the parabolic kind: and these legs are best known from the tangents. For if the point of contact be at an infinite distance, the tangent of a hyperbolic leg will coincide with the asymptote, and the tangent of a parabolic leg will recede in infinitum, will vanish and be no where found. Therefore the asymptote of any leg is found by seeking the tangent to that leg at a point infinitely distant: and the course, or way of an infinite leg, is found by seeking the position of any right line which is parallel to the tangent where the point of contact goes off in infinitum: for this right line is directed the same way with the infinite leg. Sir Isaac Newton's Reduction of all Lines of the Third Order, to four Cases of Equations; with the Enumeration of those lines. CASE I. |