x, y, 1)² = 0; or by an equation of the third order, &c.; or what is the same thing, according as the equation is linear, quadric, cubic, &c.
A curve of the first order is a right line; and conversely every right line is a curve of the first order. A curve of the second order is a conic, and is also called a quadric curve; and conversely every conic is a curve of the second order or quadric curve. A curve of the third order is called a cubic; one of the fourth order a quartic; and so on.
A curve of the order m has for its equation (*
x, y, 1)m = 0; and when the coefficients of the function are arbitrary, the curve is said to be the general curve of the order m. The number of coefficients is ½(m + 1)(m + 2); but there is no loss of generality if the equation be divided by one coefficient so as to reduce the coefficient of the corresponding term to unity, hence the number of coefficients may be reckoned as ½(m + 1)(m + 2) − 1, that is, ½m(m + 3); and a curve of the order m may be made to satisfy this number of conditions; for example, to pass through ½m(m + 3) points.
It is to be remarked that an equation may break up; thus a quadric equation may be (ax + by + c)(a′x + b′y + c′) = 0, breaking up into the two equations ax + by + c = 0, a′x + b′y + c′ = 0, viz. the original equation is satisfied if either of these is satisfied. Each of these last equations represents a curve of the first order, or right line; and the original equation represents this pair of lines, viz. the pair of lines is considered as a quadric curve. But it is an improper quadric curve; and in speaking of curves of the second or any other given order, we frequently imply that the curve is a proper curve represented by an equation which does not break up.
4. Intersections of Curves.—The intersections of two curves are obtained by combining their equations; viz. the elimination from the two equations of y (or x) gives for x (or y) an equation of a certain order, say the resultant equation; and then to each value of x (or y) satisfying this equation there corresponds in general a single value of y (or x), and consequently a single point of intersection; the number of intersections is thus equal to the order of the resultant equation in x (or y).
Supposing that the two curves are of the orders m, n, respectively, then the order of the resultant equation is in general and at most = mn; in particular, if the curve of the order n is an arbitrary line (n = 1), then the order of the resultant equation is = m; and the curve of the order m meets therefore the line in m points. But the resultant equation may have all or any of its roots imaginary, and it is thus not always that there are m real intersections.
The notion of imaginary intersections, thus presenting itself, through algebra, in geometry, must be accepted in geometry—and it in fact plays an all-important part in modern geometry. As in algebra we say that an equation of the mth order has m roots, viz. we state this generally without in the first instance, or it may be without ever, distinguishing whether these are real or imaginary; so in geometry we say that a curve of the mth order is met by an arbitrary line in m points, or rather we thus, through algebra, obtain the proper geometrical definition of a curve of the mth order, as a curve which is met by an arbitrary line in m points (that is, of course, in m, and not more than m, points).