785] 
CURVE. 
463 
A curve of the second order is a conic, or as it is also called a quadric; and 
conversely every conic, or quadric, is a curve of the second order. 
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 on has for its equation (*]£#, y, l) 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 ^-(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 \ (m+1) (to + 2) — 1, that is, |-m(m + 3); and a curve of the order m may be 
made to satisfy this number of conditions; for example, to pass through \ on (m 4- 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 + d = 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. 
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 rn 
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 on, points). 
The theorem of the m intersections has been stated in regard to an arbitrary 
line; in fact, for particular lines the resultant equation may be or appear to be of