44 
REAL CURVES 
Booh I 
Chap. Ill 
Suppose that a curve has an ordinary singular point of order 
r at infinity, with r distinct parallel asymptotes running there. 
These r asymptotes have nr intersections with the curve of 
which r(r+l) are their intersections with the line at infinity 
counted (r+1) times. Hence, by the total intersection theorem 
of the last chapter, we are able to prove* 
Theorem 6] If a curve have r 'parallel asymptotes running to 
a point of multiplicity r, their finite intersections with the curve 
lie on a curve of order n—r—1. 
§ 2. Real singular points 
In most of the work which we have done so far, in connexion 
with singular points, we have assumed that we had only to deal 
with ordinary singularities. We shall postpone to the next book 
a theoretical discussion of the properties of singular points that 
are not ordinary ones or cusps, but shall give at this point a 
discussion of the method of plotting a curve in the vicinity of 
a singularity of a complicated nature. Much of what we do here 
will prove of great value later. 
Suppose that we have a singular point at the origin. When 
x and y approach 0, the terms become infinitesimal of different 
orders. We wish to find a set of terms which are of the same 
order, lower than the infinitesimal orders of the others. These 
terms alone will give us a partial representation of the curve in 
that vicinity. Let the curve be 
J^A^yPi = 0. 
i 
It is conceivable, and in fact, highly plausible, that the curve 
can be developed in the vicinity of the origin in a set of series 
of fractional or integral powers of x, of the form 
y = y'xv-^-atfPt 1 -^... 
For points very near the origin, we may content ourselves with 
the first term of the series, we want then to find such a value 
¡jl that if we put y = xv, divide out a suitable power of x and 
then let x become 0, y' will approach a finite value. As an 
approximation to the curve, we content ourselves with those 
terms where x has the same power, after this substitution, which 
is lower than its power in the other terms. How do we find 
* Cf. Hyashi. 
them ? It is well, 
elaborate change 
We wish to rei 
value, and mom 
to r and s such 1 
rotp+6 
OCp < ... < 
whereas in all tt 
method was disc« 
Let us start ir 
plane, and mark 
take the expone: 
and mark the co 
from {a p) f} p ) to ( 
The points («¿,/3 
line, whose equa 
The terms we 
lie on the other f 
procedure is, th( 
The origin in i 
is surely some 
original equatioi 
marked point o 
Let the half-lin 
origin rotate pc 
more marked pc 
P v Let the hah 
about P 2 till it ] 
the most distan 
* Cf. i