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fame manner may any number of points be found, and these being joined, will give a representation of the curve, which will be more or lefs juft, according to the number of points found, and the accuracy of the feveral operations employed.
By the fame methods the locus of any other equation is to be traced: Thus, by varying the former equation, the figure of its locus will be varied. If bo, then the points A and E coincide, the nodus vanishes, and A is called a Cufpis.
If b is negative, then E is to the right of A, which will now be a punctum conjugatum. The rest of the curve will be between E and C, and CD becomes an afymptote. If a=0, then-xy2=x3—bx2, or y2=bx -x2, which is an equation to the circle of which 6AE is the diameter.
H. General Properties of Curves from their.
The general properties of equations lead to the general affections of curve lines. For
x bestill reImaining
A ftraight line may meet a curve in as many points as there are units in the dimention of its equation; for fo many roots may that equation have. An afymptote may cut a curve line in as many points excepting two as it has dimenfions, and no more. The fame may be obferved of the tangent.
Impoffible roots enter an equation by pairs, therefore the interfection of the ordinate and curve muft vanish by pairs.
The curves of which the number expreffing the order is odd, must have at least two infinite arcs; for the abfcifs may be fo affumed, that, for every value of it, either pofitive or negative, there must be at least one value of y, &c.
The properties of the coefficients of the terms of equations mentioned Part II. Ch. 1. furnish a great number of the curious and universal properties of curve lines. For example, the fecond term of an equation is the fum of the roots with the figns changed, and if the second term is wanting, the pofitive and negative roots must be equal. From this it is easy to demonftrate, "That,
if each of two parallel ftraight lines meet a curve line in as many points as it has dimenfions, and if a ftraight line cut these two parallels, fo that the fum of the fegments of each on one fide be equal to the fum of the fegments on the other, this ftraight line will cut any other line parallel to thefe in the fame manner." Analagous properties, with many other confequences from them, may be deduced from the compofition of the coefficients of the other
Many properties of a particular order of curves may be inferred from the properties of equations of that order.. Thus, if ftraight line cut a curve of the third order in three points, and if another straight line be drawn, making a given angle with the former, and cutting the curve also in three points, the parallelopiped by the segments of one of these lines between its intersection with the other, and the points where it meets the curve, will be to the parallelopiped by the like segments of the other line in a given ratio." This depends upon the compofition of the abfolute be extended to curves of
term, and may
III. The Subdivifion of Curves.
As lines are divided into orders from the dimenfions of their equations, in like manner, from the varieties of the equations of any order, may different Genera and Species of that order be distinguished, and from the peculiar properties of these varieties, may the affections of the particular curves be discovered.
For this purpose a complete general equation is affumed of that order, and all the varieties in the terms and coefficients which can affect the figure of the locus are enumerated.
It was formerly obferved, that the equations belonging to any one curve, may be of various forms, according to the pofition of the bafe, and the angle which the ordinate makes with it, though they be all of the fame order, and have alfo certain properties, which diftinguish them from the other equations of that order.
The locus of fimple equations is a straight line. There are three fpecies of lines of the fecond order, which are easily shown to be
the conic fections, reckoning the circle and ellipfe to be one. Seventy-eight fpecies
have been numbered of the third order: And, as the fuperior orders become too numerous to be particularly reckoned, it is ufual only to divide them into certain general claffes.
A complete arrangement of the curves of any order, would furnish canons, by which the species of a curve whofe equation is of that order might be found.
IV. Of the place of Curves defined from other principles, in the Algebraical System.
If a curve line be defined from the fection of a folid, or from any rule different from what has been here fuppofed, an equation to it may be derived, by which its order and fpecies in the algebraical system may be found. And, for this purpose, any bafe and any angle of the co-ordinates may be affumed, from which the equation may