ASYMPTOTES 
43 
or 
Booh I 
.=-0. 
ar points, and if the 
of partial fractions 
?e* 
(10) 
it is better to go at 
md we substitute in 
n in x whose degree 
root. There will be 
ds to an equation of 
asymptote is 
nd drawn, it will be 
if the curve with the 
t at a non-singular 
r the curve goes to 
rections, or whether, 
3S of the asymptote 
We determine this 
,t one asymptote at 
us assume that the 
tien y — 0, x takes k 
mt. When y is ex- 
i, the preponderant 
md neither of these 
line but once at the 
en, x k ~ x and y both 
Chap III. 
change sign with x and y so that the values of y corresponding 
to numerically large x's with opposite signs are themselves 
opposite in sign, i.e. the curve goes to infinity on opposite sides 
of the asymptote. The reverse is true when k is odd. 
Theorem 4] If an asymptote tangent to a real curve at a non 
singular infinite point have even-point contact, the corresponding 
branch of the curve will go to infinity on opposite sides of the 
asymptote in the two directions. The reverse is true in the case 
when there is odd-point contact, the distant parts of the branch are 
all on one side of the asymptote. 
Suppose that our homogeneous polynomial f n {x, y) has a linear 
factor with the multiplicity k, which is not a factor of f n -x{x,y). 
This will not correspond to a singular point on the infinite line, 
but to ¿-point contact therewith. We may choose the axes so 
that this factor is y, and write 
y k <f>n-ki x >y)+fn-i{ x >y)+- = °* 
We now make a change of variable, writing 
y' k K-k( i > v') ■+ x 'fn-i( 1 . y') ■+ • ■• ■• = 0 
0)^0. 
Then by Theorem 4] of Ch. II we may write 
x'= a k y' k +.... 
Suppose a k >0. If k be even, x' is positive for all values of y' 
close to 0. Hence x is large and positive, and y changes sign 
with y’, i.e. all horizontal lines meet the curve far out to the 
right. When a k < 0 they will all meet it far out to the left. 
If k be odd x' and y' change sign together, hence y is essentially 
positive and each y will correspond to two numerically large 
values of x, one positive and one negative. 
Theorem 5] If a real curve have even-point contact with the 
line at infinity at an ordinary point, a finite line through that point 
will have one distant intersection with the curve, if the contact be 
odd-point, there will be two distant intersections or none. 
We can remember the rule by noting that all vertical lines 
have one real intersection with the parabola y — x 2 , but if we 
take the cubic y 2 — x 3 some vertical lines have two real inter 
sections, and some none.