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same manner may any number of points be found, and these being joined, will give a representation of the curve, which will be more or less just, according to the number of points found, and the accuracy of the several operations employed.

By the same 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 b=o, 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 conjuga-
tum. The rest of the curve will be between
E and C, and CD becomes an asymptote.
If a=o, then – xy2 =x2-bx?, or yż=bx

bouillsea --, which is an equation to the circle of Basining

megative, which b=.AE is the diameter.

II. General Properties of Curves from their

Equations.

The general properties of equations lead to the general affections of curve lines. For example, Hh

A

more.

A ftraight line may meet a' curve in 26 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 dimensions, and no

The fame may be observed of the tangent.

Impossible roots enter an equation by pairs, therefore the interseciion of the ordinate and curve must vanish by pairs.

The curves of which the number exprefsing the order is odd, must have at least two infinite arcs ; for the absciss may be fo assumed, that, for

every

value of it, either po sitive or negative, there must be at least one value of

J',

&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 second term of an equation is the sum of the roots with the signs changed, and if the second term is wanting, the pofitive and negative roots must be equal. From this it is easy to demonstrate, That,

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if each of two parallel straight lines meet a curve line in as many points as it has dimensions, and if a straight line cut these two parallels, so that the sum of the segments of each on one side be equal to the fum of the segments on the other, this straight line will cut any other line parallel to thefe in the same manner.

r.” Analagous properties, with many other consequences from them, may be deduced from the composition of the coefficients of the other

terms.

Many properties of a particular order of curves may be inferred from the properties of equations of that order.. Thus, “ if a straight 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 composition of the absolute

may be extended to curves of

any
order.

III.

term, and

III. The Subdivision of Curves.

As lines are divided into orders from the dimentions 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 assumed 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 observed, that the equaticns belonging to any one curve, may be of various forms, according to the position of the base, and the angle which the ordinaie makes with it, though they be all of the same order, and have also certain properties, which distinguish them from the other equations of that order.

The locus of simple equations is a straight line. There are three species of lines of the second order, which are easily shewn to be

the

the conic sections, reckoning the circle and ellipse to be one. Seventy-eight species have been numbered of the third order : And, as the superior orders become too numerous to be particularly reckoned, it is usual only to divide them into certain general classes.

A complete arrangement of the curves of any order, would furnish canons, by which the species of a curve whose equation is of that order might be found.

IV. Of the place of Curvés defined from other

principles, in the Algebraical System.

If a curve line be defined from the section of a solid, or from any rule different from what has been here supposed, an equation to it may be derived, by which its order and species in the algebraical system may be found. And, for this purpose, any base and any angle of the co-ordinates may be assumed, from which the equation may

be

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