ON A QUARTIC CURVE WITH TWO ODD BRANCHES. 
[From the Messenger of Mathematics, vol. vi. (1877), pp. 107, 108.] 
It is a known theorem that the branches of a plane curve are even or odd; viz. 
two even branches, or an even and an odd branch (whether of the same curve or of 
different curves) intersect in an even number (it may be 0, and this is to be under 
stood throughout) of real points; but two odd branches (of the same curve or of 
different curves) intersect in an odd number of real points *. 
In particular, a right line is an odd branch, and hence it meets any even branch 
of a curve in an even number of real points, and an odd branch in an odd number 
of real points; or (what is the same thing) an even branch is one which is met by 
any right line whatever in an even number of real points; and an odd branch is one 
that is met by any right line whatever in an odd number of real points. 
It is to be observed, that the simple term branch is used to denote what has 
been called a complete branch, viz. the partial branches which touch an asymptote at 
its opposite extremities are considered as parts of one and the same branch, and so 
in other cases. Thus a quadric curve, whether ellipse, parabola, or hyperbola, is one 
even branch; a cubic curve is either one odd branch, or else it is an odd branch 
and an even branch; and generally a curve of an odd order has always an odd number 
of odd branches, and a curve of an even order has always an even number of odd 
branches. 
A curve without nodes has at most one odd branch; for if there were two, these 
would intersect in a real point, which would be a real node on the curve. In parti 
cular, a quartic curve having two odd branches must have a real node; this however 
may be, as in the instance about to be given, a node at infinity. 
A simple instance of a quartic curve with two odd branches is that represented 
by the equation 
(¿c 2 — 1) (y 2 + 1) — 2 mxy = 0, 
* The two branches must be distinct branches; a branch whether odd or even does not of necessity 
intersect itself (have upon it any real node), but it may intersect itself in an odd, or an even, number of real 
points.