476 
CURVE. 
[785 
as at once the locus of the centre of curvature and the envelope of the normal. It 
may be added that the given curve is one of a series of curves, each cutting the 
several normals at right angles. Any one of these is a “ parallel ” of the given curve; 
and it can be obtained as the envelope of a circle of constant radius having its centre 
on the given curve. We have in like manner, as derivatives of a given curve, the 
caustic, catacaustic, or diacaustic, as the case may be, and the secondary caustic, or 
curve cutting at right angles the reflected or refracted rays. 
We have in much that precedes disregarded, or at least been indifferent to, reality; 
it is only thus that the conception of a curve of the mth order, as one which is 
met by every right line in m points, is arrived at; and the curve itself, and the line 
which cuts it, although both are tacitly assumed to be real, may perfectly well be 
imaginary. For real figures we have the general theorem that imaginary intersections, &c., 
present themselves in conjugate pairs: hence, in particular, that a curve of an even 
order is met by a line in an even number (which may be = 0) of points ; a curve 
of an odd order in an odd number of points, hence in one point at least; it will be seen 
further on that the theorem may be generalized in a remarkable manner. Again, when 
there is in question only one pair of points or lines, these, if coincident, must be real; 
thus, a line meets a cubic curve in three points, one of them real, the other two real 
or imaginary; but if two of the intersections coincide they must be real, and we have 
a line cutting a cubic in one real point and touching it in another real point. It 
may be remarked that this is a limit separating the two cases where the intersec 
tions are all real, and where they are one real, two imaginary. 
Considering always real curves, we obtain the notion of a branch; any portion 
capable of description by the continuous motion of a point is a branch; and a curve 
consists of one or more branches. Thus the curve of the first order or right line 
consists of one branch; but in curves of the second order, or conics, the ellipse and 
the parabola consist each of one branch, the hyperbola of two branches. A branch 
is either re-entrant, or it extends both ways to infinity, and in this case, we may 
regard it as consisting of two legs {crura, Newton), each extending one way to infinity, 
but without any definite separation. The branch, whether re-entrant or infinite, may 
have a cusp or cusps, or it may cut itself or another branch, thus having or giving 
rise to crunodes; an acnode is a branch by itself,—it may be considered as an 
indefinitely small re-entrant branch. A branch may have inflexions and double tangents, 
or there may be double tangents which touch two distinct branches; there are also 
double tangents with imaginary points of contact, which are thus lines having no visible 
connexion with the curve. A re-entrant branch not cutting itself may be everywhere 
convex, and it is then properly said to be an oval; but the term oval may be used 
more generally for any re-entrant branch not cutting itself; and we may thus speak 
of a once indented, twice indented oval, &c., or even of a cuspidate oval. Other 
descriptive names for ovals and re-entrant branches cutting themselves may be used 
when required; thus, in the last-mentioned case a simple form is that of a figure of 
eight; such a form may break up into two ovals, or into a doubly indented oval or 
hour-glass. A form which presents itself is when two ovals, one inside the other, 
unite, so as to give rise to a crunode—in default of a better name this may be called,