CASE I.

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

E

X

T WP

B

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.

CASE II.

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

02

the

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.)

CASE III.

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.)

CASE IV.

....

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,

20. When, in equa. I, the terms ar3, br, are wanting, or when that equation becomes xy + ey = c.x + d, it expresses a figure consisting of three hyperbolas opposite to one another, one lying between the parallel asymptotes, and the other two without each of these curves having three asymptotes, one of which is the first and

principal ordinate, the other two parallel to the abscissa, and equally distant from it; as in the annexed figure of Newton's 60th species. Otherwise the said equation expresses two opposite circumscribed hyperbolas, and an anguineal hyperbola between the asymptotes. Under this class there are 4 species, called

by Newton Hyperbolisma of an hyperbola. By hyperbolismæ of a figure he means to signify when the ordinate comes out, by dividing the rectangle under the ordinate of a given conic section and a given right line, by the common abscissa.

21. When the term cr2 is negative, the figure expressed by the equation xy2+ey - cx+d, is either a serpentine hyperbola, having only one asymptote, being the principal ordinate; or else it is a conchoidal figure. Under this class there are 3 species, called Hyperbolism of an ellipse.

22. When the term cr is absent, the equa. xy2 + ey = d, expresses two hyperbolas, lying, not in the opposite angles of the asymptotes (as in the conic hyperbola), but in the adjacent angles. Here there are only 2 species, one consisting of an inscribed and an ambigeneal hyperbola, the other of two inscribed hyperbolas. These two species are called the Hyperbolisma of a parabola.

23. In the second case of equations, or that of equation II, there is but one figure; which has four infinite legs. Of these, two are hyperbolic about one asymptote, tending towards contrary parts, and two converging parabolic legs, making with the former nearly the figure of a trident, the familiar name given to this species. This is the Cartesian parabola, by which equations of 6 dimensions are sometimes constructed: it is the 66th species of Newton's enumeration. 24. The third case of equations, or

equa. III, expresses a figure having two parabolic legs running out contrary ways: of these there are 5 different species, called diverging or bell-form parabolas; of which 2 have ovals, 1 is nodate, 1

punctate, and I cuspidate. The figure shows Newton's 67th

species

species; in which the oval must always be so small that no right line which cuts it twice can cut the parabolic curve c more than once.

25. In the case to which equa. IV refers, there is but one species. It expresses the cubical parabola with contrary legs. This curve may easily be described mechanically by means of a square and an equilateral hyperbola. Its most simple property is, that RM (parallel to AQ) always varies as QN3 - QR3.

N

26. Thus according to Newton there are 72 species of lines of the third order. But Mr. Stirling discovered four more species of redundant hyperbolas; and Mr. Stone two more species of deficient hyperbolas, expressed by the equation xy2 = bx2 + cx + d: i. e. in the case when ba2+cx+d=0, has two unequal negative roots, and in that where the equation has two equal negative roots. So that there are at least 78 different species of lines of the third order. Indeed Euler, who classes all the varieties of lines of the third order under 16 general species, affirms that they comprehend more than 80 varieties; of which the preceding enumeration necessarily comprizes nearly the whole.

27. Lines of the fourth order are divided by Euler into 146 classes; and these comprize more than 5000 varieties: they all flow from the different relations of the quantities in the 10 general equations subjoined.

1. y4 +fx2y2+ gxy3 + hx2y + iy2+hxy+ly }

2. y4 +ƒxy3 +gx2y + hxy2 + ixy+ky

3. x2y2+fy3 +gx2y+hy3 +ky

4. x2y2+ƒŸ3 +gy3 +hxy + iy

5. y3 +fxy +gx3y + hy

6. y3 +fxy2 +gxy + hy

7. y4 + ex3y +fxy3 +gxy2+ hy2+ ixy + ky

8, x3y +exys +fï3y +gy2 +hxy+iy

9. xsy eyз +fxy2 +gxy + hy 10. 13y ey3 +fy2 +gxy + hy

axt + bx3 + cx2 + dx + e.

•= ax3 + bx2 + cx + d.

28. Lines of the fifth and higher orders, of necessity become still more numerous; and present too many varieties to admit of any classification, at least in moderate compass. Instead, therefore, of dwelling upon these; we shall give a concise sketch of the most curious and important properties of curve lines in general, as they have been deduced from a contemplation of the nature and mutual relation of the roots of the equations representing those curves. Thus a curve being called of n dimensions, or a line of the nth order when its representative equation rises to a dimensions; then since

for