211. Number of Asymptotes to a Curve of the nth Degree. It is clear that since p(u)=0 is in general of the nth degree in μ, and ẞp'n(u)+Pn-1(u) =0 is of the first degree in B, that n values of μ, and no more, can be found from the first equation, while the n corresponding values of B can be found from the second. Hence n asymptotes, real or imaginary, can be found for a curve of the nth degree.

212. If the degree of an equation be odd it is proved in Theory of Equations that there must be one real root at least. Hence any curve of an odd degree must have at least one real asymptote, and therefore must extend to infinity. No curve therefore of an odd degree can be closed. Neither can a curve of odd degree have an even number of real asymptotes, or a curve of even degree an odd number.

213. If, however, the term yn be missing from the terms of the nth degree in the equation of the curve, the term μ" will also be missing from the equation $n(u)=0, and there will therefore be an apparent loss of degree in this equation. It is clear, however, that in this case, since the coefficient of μ" is zero, one root of the equation Pn(u)=0 is infinite, and therefore the corresponding asymptote is at right angles to the axis of x; i.e., parallel to that of y. This leads us to the special consideration of such asymptotes as may be parallel to either of the axes of co-ordinates.

214. Asymptotes Parallel to the Axes.

Let the curve arranged as in equation (A), Art. 209, be

If arranged in descending powers of x this is

а ̧x2+(a ̧y+b12) x2-1+... = 0).

Hence, if a vanish, and y be so chosen that

+bn-1yn - 1

(A')

(B)

the coefficients of the two highest powers of x in equation (B) vanish, and therefore two of its roots are infinite. Hence the straight line a1y+b=0 is an asymptote.

In the same way, if an=0, an-1x+bn-1=0 is an asymptote.

Again, if a=0, a=0, b, =0, and if y be so chosen that ay2+b2y+c2=0,

three roots of equation (B') become infinite, and the lines represented by

ɑy2+b2y+c2=0

represent a pair of asymptotes, real or imaginary, parallel to the axis of y.

Hence the rule to find those asymptotes which are parallel to the axes is, "equate to zero the coefficients of the highest powers of x and y.”

Ex. Find the asymptotes of the curve

x2y2 — x2y — xy2+x+y+1=0.

Here the coefficient of x2 is y2-y and the coefficient of y2 is x2 - x. Hence x=0, x=1, y=0, and y=1 are asymptotes. Also, since the curve is one of the fourth degree, we have thus obtained all the asymptotes.

2. The co-ordinate axes are the asymptotes of

xy3+x3y=aa.

3. The asymptotes of the curve x2y2=c2(x2+y2) are the sides of a

This gives a very easy way of obtaining the asymptotes.

* Suppose the single factor t-μ1 to occur in on(t). Let

But if be the partial fraction corresponding to the factor t – μ1,

A

Ex. Find the asymptotes of the curve,

Here

(x2- y2)(x+2y)+5(x2+y2)+x+y=0.

Pn−1(t)

+

+

Pn(t) (2t+1)(t − 1)(t+1) 2t+1 t-1 t+1

Hence the asymptotes are

216. Particular Cases of the General Theorem.

We return to a closer consideration of the equations

of Art. 209.

Pn(u)=0;.

ẞpn'(u)+pn-1(u)=0,

(E)

(F)

It is proved in Theory of Equations that if an equation such as fn(μ)=0 have a pair of roots equal, say then pn'(μ1)=0.

be

I. Let the roots of ñ(μ)=0 bе μ, μ2, ..., μn, supposed all different, so that pn'(u) does not vanish for any of these roots. Also, suppose pr(μ) and pn-1(μ) to contain a common factor μ-μ, say, then pn-1(μ1)=0, and therefore B1 =0.

Hence the corresponding asymptote is y=ux and passes through the origin.

II. Next, suppose two of the roots of the equation Pn(μ)=0 to be equal, e.g., μ2=μ1, then pn'(μ‚)=0. In this case, if on-1(u) do not contain μ-μ, as one of its factors, the value ẞ determined from equation (F) is infinite. The line y=μ1x+ẞ1 then does indeed cut the curve in two points at an infinite distance from the origin, but it makes an infinite intercept on the axis of y and there

fore this line lies wholly at infinity. Such a straight. line is not in general called an asymptote, but it will however count as one of the n theoretical asymptotes discussed in Art. 211.

III. But if μn(u)=0 have a pair of equal roots each =μ1, we have pn'(μ1)=0, and if μ, be also a root of Pn-1(u)=0 the value of B cannot be determined from equation (F). We may however choose B so that the coefficient of an-2 in equation (D) of Art. 209 vanishes, that is so that

2

Þn”(u)+ßp'n-1(u)+$n-2(u) = 0,

from which two values of B, real or imaginary, may be deduced. Let the roots of this equation be ẞ, and B'. We thus obtain the equations of two parallel straight lines y=μ1x+B12

which each cut the curve in three points at an infinite distance from the origin. In this case there is a double point on the curve at infinity (see Art. 249).

It is clear that in this case any straight line parallel to y=μ1 will cut the curve in two points at infinity. But of all this system of parallel straight lines the two whose equations we have just found are the only ones which cut the curve in three points at infinity, and therefore the name asymptote is confined to them. The one equation which includes both straight lines is obtained at once by substituting y-u, for B in the equation to obtain B and is

(y− μ‚x)2μ‚”(μ‚)+2(y—μ‚x)$'n-1(μ1)+2&n−2(μ,) = 0.