37 
643] ON A QUARTIC CURVE WITH TWO ODD BRANCHES. 
or, what is the same thing, 
where 
or say 
a 2 + 
1 
a 2 
2 + to 2 , 
a 2 = I; {2 + to 2 + to V(4 4- ??i 2 )}, 
^ | {2 + to 2 — to V(4 + to 2 )}, 
so that to being positive a > 1, and the curve consists of two real branches included 
1 i 
between the lines x = a, x — -, and the lines x= — a, x = — ~ respectively; each of these 
lines touches the curve in a real point, viz. x having any one of the last-mentioned 
values, the value of y at the point of contact is y = -; and between each pair of 
lines we have the asymptote « = + 1 or x = — 1. Hence the curve has the form shown 
in the figure, and it is thereby evident, that each branch of the curve is met by 
any real right line whatever in one real point, or else in three real points. The 
numerical values in the figure are a = f, in = f, whence also x — cl or —y=l, and 
1 , 
X— — OL 01’ - , y = — 1. 
cc 
The curve has two nodes at infinity, viz. writing the equation in the form 
that is, 
(x 2 - z 2 ) (y 2 + z 2 ) — mxyz- = 0, 
x 2 y 2 + z 2 {x 2 -if- may) + z* = 0, 
it appears that the points (z = 0, x = 0), (z — 0, y — 0) are each oi them a node. Ihe 
first of these 0 = 0, x = 0) is the real intersection of the two odd branches: the other 
of them is a conjugate point.