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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; 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.
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 every different value of a there are n values of y, it will commonly happen that the ordinate will cut the curve in n or in n 2, n-4, &c, points, according as the equation has n, or n 2, n 4, &c, possible roots. It is not however to be inferred, that a right line will cut a curve of n dimensions, in n, n 4, &c, points, only; for if this were the case, a line of the 2d order, a conic section for instance, could, only be cut by a right line in two points ;-but this is manifestly incorrect, for though a conic parabola will be cut in two points by a right line oblique to the axis, yet a right line parallel to the axis can only cut the curve in one point.
29. It is true in general, that lines of the n order cannot be cut by a right line in more than n points; but it does not hence follow, that any right line whatever will cut in n points every line of that order; it may happen that the number of intersections is n − 1, n 2, n 3, &c, ton - n. The number of intersections that any right line whatever makes with a given curve line, cannot therefore determine the order to which a curve line appertains. For, as Euler remarks, if the number of intersections ben, it does not follow that the curve belongs to the n order, but it may be referred to some superior order; indeed it may happen that the curve is not algebraic, but transcendental. This case excepted, however, Euler contends that we may always affirm positively that a curve line which is cut by a right line in n points, cannot belong to an order of lines inferior to n. Thus, when a right line cuts a curve in 4 points, it is certain that the curve does not belong to either the second or third order of lines; but whether it be referred to the fourth, or a superior order, or whether it be transcendental, is not to be decided by analysis.
30. Dr. Waring has carried this enquiry a step further than Euler, and has demonstrated that there are curves of any number of odd orders, that cut a right line in 2, 4, 6, &c, points only; and of any number of even orders that cut a right line in 3, 5, 7, &c, points only; whence this author likewise infers, that the order of the curve cannot be announced from the number of points in which it cuts a right line. See his Proprietates Algebraicarum Curvarum.
31. Every geometrical curve being continued, either returns into itself, or goes on to an infinite distance. And if any plane curve has two infinite branches or legs, they join one another either at a finite, or at an infinite distance.
32. In any curve, if tangents be drawn to all points of the curve; and if they always cut the abscissa at a finite distance from its origin; that curve has an asymptote, otherwise not..
33. A line of any order may have as many asymptotes as it has dimensions, and no more.
34. An asymptote may intersect the curve in so many points abating two, as the equation of the curve has dimensions. Thus, in a conic section, which is the second order of lines, the asymptote does not cut the curve at all; in the third order it can only cut it in one point; in the fourth order, in two points; and so on.
35. If a curve have as many asymptotes, as it has dimensions, and a right line be drawn to cut them all, the parts of that measured from the asymptotes to the curve, will together be equal to the parts measured in the same direction, from the curve to the asymptotes.
36. If a curve of n dimensions have n asymptotes, then the content of the n abscissas will be to the content of the n ordinates, in the same ratio in the curve and asymptotes; the sum of their n subnormals, to ordinates perpendicular to their abscissas, will be equal to the curve and the asymptotes; and they will have the same central and diametral curves.
37. If two curves of n and m dimensions have a common asymptote; or the terms of the equations to the curves of the greatest dimensions have a common divisor; then the curves cannot intersect each other in n × m points, possible or impossible. If the two curves have a common general centre, and intersect each other in n x m points, then the sum of the affirmative abscissas, &c, to those points, will be equal to the sum of the negative; and the sum of the n subnormals to a curve which has a general centre, will be proportional to the distance from that centre.
38. Lines of the third, fifth, seventh, &c order, or any odd number, have, as before remarked, at least two infinite legs or branches, running contrary ways; while in lines of the second, fourth, sixth, or any even number of dimensions, the figure may return into itself, and be contained within certain limits.
39. If the right lines AP, PM, forming a given angle APM, cut a geometrical line of any order in as many points as it has dimensions, the product of the segments of the first terminated by P and the curve, will always be to the product of the segments of the latter, terminated by the same point and the curve, in an invariable ratio.
40. With respect to double, triple, quadruple, and other multiple points, or the points of intersection of 2, 3, 4, or more branches of a curve, their nature and number may be estimated by means of the following principles. 1. A curve of the n order is determinate when it is subjected to pass through
the number (~+1) · (~ + 2)
· 1 points. 2. A curve of the order cannot intersect a curve of the m order in more than min points.
Hence it follows that a curve of the second order, for example, can always pass through 5 given points (not in the same right line), and cannot meet a curve of the m order in more than mn points; and it is impossible that a curve of the m order should have 5 points whose degrees of multiplicity make together more than 2m points. Thus, a line of the fourth order cannot have four double points; because the line of the second order which would pass through these four double points, and through a fifth simple point of the curve of the fourth dimension, would meet 9 times; which is impossible, since there can only be an intersection 2 x 4 or 8 times.
For the same reason, a curve line of the fifth cannot, with one triple point, have more than three double points: and in a similar manner we may reason for curves of higher orders.
Again, from the known proposition, that we can always make a line of the third order pass through nine points, and that a curve of that order cannot meet a curve of the m order in more than 3m points, we may conclude that a curve of the m order cannot have nine points, the degrees of multiplicity of which make together a number greater than 3m. Thus, a line of the fifth order cannot have more than 6 double points; a line of the 6th order, which cannot have more than one quadruple point, cannot have with that quadruple point more than 6 double points; nor with two triple points more than 5 double points; nor even with one triple point more than 7 double points. Analogous conclusions obtain with respect to a line of the fourth order, which we may cause to pass through 14 points, and which can only meet a curve of the order in 4m points, and so on.
41. The properties of curves of a superior order, agree, under certain modifications, with those of all inferior orders. For though some line or lines become evanescent, and others become infinite, some coincide, others become equal; some points coincide, and others are removed to an infinite distance; yet, under these circumstances, the general properties still hold good with regard to the remaining quantities; so that whatever is demonstrated generally of any order, holds true in the inferior orders: and, on the contrary, there is hardly any property of the inferior orders, but there is some similar to it, in the superior ones.
For, as in the conic sections, if two parallel lines are drawn