One root of this equation becomes very large, if the line
y = mx as it turns about the origin approaches one of the
lines y ="m 1 x,...,y = m n x ; for then

(m — m x ) (m - m 2 ) . . . (m—m n ) — > 0.
We may sum up this result in the convenient, if rather
inaccurate, form

The terms of highest degree in the Cartesian equation of a
curve equated to zero give the lines joining the origin to the
points at infinity on the curve.
Let us now discuss a few cases in more detail.
First suppose m 2 not equal to any one of m 2 , m 3 , ... , m„.
The line y = m 1 x + c meets the curve where
{p + (m 1 — m 2 ) ... {m-L — m^c} a;™" 1 + (q + re + . . .) x n ~ 2 + ... = 0,
if we write

p = a + bm l + . . . + kni-^- 1 , q = A + Bm 1 + ... ,
r= b + ... + (n—l)Jemj n - 2.
Hence any line parallel to y = ra^x meets the curve in one
and only one infinite point.f with the exception of the line I
for which p + (% - m 2 ) (m 1 — m 3 ) . . . (uij — m„) c = 0.
It follows at once that any projection of the curve touches
the projection of I at the point where it crosses the vanishing
line, i. e. I is an asymptote (Ch. I, § 4).
