785] 
CURVE. 
477 
after the curve of that name, a limaçon. Names may also be used for the different 
forms of infinite branches, but we have first to consider the distinction of hyperbolic 
and parabolic. The leg of an infinite branch may have at the extremity a tangent ; 
this is an asymptote of the curve, and the leg is then hyperbolic; or the leg may 
tend to a fixed direction, but so that the tangent goes further and further off to 
infinity, and the leg is then parabolic ; a branch may thus be hyperbolic or parabolic 
as to its two legs ; or it may be hyperbolic as to one leg, and parabolic as to the 
other. The epithets hyperbolic and parabolic are of course derived from the conics hyper 
bola and parabola respectively. The nature of the two kinds of branches is best under 
stood by considering them as projections, in the same way as we in effect consider the 
hyperbola and the parabola as projections of the ellipse. If a line il cut an arc aa', so 
that the two segments ab, ha' lie on opposite sides of the line, then projecting the 
figure so that the line il goes off to infinity, the tangent at b is projected into the 
asymptote, and the arc ab is projected into a hyperbolic leg touching the asymptote 
at one extremity; the arc ba' will at the same time be projected into a hyperbolic 
leg touching the same asymptote at the other extremity (and on the opposite side), 
but so that the two hyperbolic legs may or may not belong to one and the same 
branch. And we thus see that the two hyperbolic legs belong to a simple inter 
section of the curve by the line infinity. Next, if the line il touch at b the arc aa' 
so that the two portions ab', ba lie on the same side of the line il, then projecting 
the figure as before, the tangent at b, that is, the line il itself, is projected to infinity; 
the arc ab is projected into a parabolic leg, and at the same time the arc ba' is 
projected into a parabolic leg, having at infinity the same direction as the other leg, 
but so that the two legs may or may not belong to the same branch. And we thus 
see that the two parabolic legs represent a contact of the line infinity with the 
curve,—the point of contact being of course the point at infinity determined by the 
common direction of the two legs. It will readily be understood how the like con 
siderations apply to other cases,—for instance, if the line il is a tangent at an inflexion, 
passes through a crunode, or touches one of the branches of a crunode, &c. ; thus, if 
the line il passes through a crunode we have pairs of hyperbolic legs belonging to 
two parallel asymptotes. The foregoing considerations also show (what is very important) 
how different branches are connected together at infinity, and lead to the notion of 
a complete branch, or circuit. 
The two legs of a hyperbolic branch may belong to different asymptotes, and in 
this case we have the forms which Newton calls inscribed, circumscribed, ambigene, &c. ; 
or they may belong to the same asymptote, and in this case we have the serpentine 
form, where the branch cuts the asymptote, so as to touch it at its two extremities 
on opposite sides, or the conchoidal form, where it touches the asymptote on the same 
side. The two legs of a parabolic branch may converge to ultimate parallelism, as in 
the conic parabola, or diverge to ultimate parallelism, as in the semi-cubical parabola 
y- = and the branch is said to be convergent, or divergent, accordingly ; or they 
may tend to parallelism in opposite senses, as in the cubical parabola y = a?. As 
mentioned with regard to a branch generally, an infinite branch of any kind may have 
cusps, or, by cutting itself or another branch, may have or give rise to a crunode, &c.