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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
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.
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:
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.
12. All the lines of the first, third, fifth, and seventh order, or of any odd order, have at least two legs or sides proceeding on ad infinitum, and towards contrary parts. And all lines of the third order have two such legs or branches running out contrary ways, and towards which no other of their infinite legs (except in the Cartesian parabola) tend. If the legs are of the hyperbolic kind, let GAs be their asymptote; and to it
let the parallel CBC be drawn, terminated (if possible) at both ends at the curve. Let this parallel be bisected in x, and then will the locus of that point x be the conical or common hyperbola xa, one of whose asymptotes is as. Let its other asymptote be AB. Then the equation by which the relation between the ordinate Bcy, and the abscissa AB = x, is determined, will always be of this form: viz,
xy2 + ey = ax3 + bx2 + cx + d . . . (I.) Here the coefficients e, a, b, c, d, denote given quantities, affected with their signs + and of which terms any one may be wanting, provided the figure through their defect does not become transformed into a conic section. The conical hyperbola xa may coincide with its asymptotes, that is, the point x may come to be in the line AB; and then the term +ey will be wanting.
13. But if the right line CBC cannot be terminated both ways at the curve, but will come to it only in one point; then draw any line in a given position which shall cut the asymptote as in a ; as also any other right line, as BC, parallel to
the asymptote, and meeting the curve in the point c; then the equation, by which the relation between the ordinate BC and the abscissa AB is determined, will always assume this form: viz. ry = ax3 + bx2 + cx + d . . . (II.)
14. If the opposite legs be of the parabolic kind, draw the right line CBC, terminated at both ends (if possible) at the curve, and running according to the course of the legs; which line bisect in B: then shall the locus of в be a right line. Let that right line be AB, terminated at any given point, as A: then the equation, by which the relation between the ordinate BC and the abscissa AB is determined, will always be of this form: y2 = a.r3 + bx2 + c.x + d . . . . (III.)
15. If the right line CBC meet the curve only in one point, and therefore cannot be terminated at the curve at both ends; let the point where it comes to the curve be c, and let that right line at the point B, fall on any other right line given in position, as AB, and terminated at any given point, as A. Then will the equation expressing the relation between BC and AB, assume this form :
y = ax3 + bx2 + cx + d . . . . (IV.)
16. In the first case, or that of equation 1, if the term at be affirmative, the figure will be a triple hyperbola with six hyperbolic legs, which will run on infinitely by the thre asymptotes, of which none are parallel, two legs towards eac! asymptote, and towards contrary parts; and these asymptote if the term br2 be not wanting in the equation, will mutual intersect each other in 3 points, forming thereby the triang Dad. But if the term br2 be wanting, they will all conver to the same point. This kind of hyperbola is called redun ant, because it exceeds the conic hyperbola in the number its hyperbolic legs.
In every redundant hyperbola, if neither the term ey wanting, nor b2 4ac = aea, the curve will have no meter; but if either of those occur separately, it will h only one diameter; and three, if they both happen. St diameter will always pass through the intersection of two the asymptotes, and bisect all right lines which are termina each way by those asymptotes, and which are parallel to third asymptote.
17. If the redundant hyperbola have no diameter, let four roots or values of r in the equation àxa + bx3 + C dr +462 = 0, be sought; and suppose them to be AP,