54 
ON THE THEORY OF ALGEBRAIC CURVES. 
[10 
Addition. 
As an exemplification of the preceding formulae, and besides as a question interesting 
in itself, it may be proposed to determine the asymptotic curves of the ?* th order of 
a given curve having all its asymptotic directions distinct,—r being any number less 
than the degree of the equation of the given curve. 
Definition. A curve of the r th order, which intersects a given curve of the 
m th order in a number of points, = mr — (r + 3), is said to be an asymptotic curve 
of the r th order to the curve in question. Suppose, as before, £7=0 being the equation 
to the given curve, 
and let 6, cf>... o) denote any combination of r terms out of the series a.... and O', ^... &/, 
&c. ... the corresponding terms out of a'...A', &c. Then, writing 
ft 
V = E 
^ (m—2) \p (m—1) \pim) 
(y-fx ... - 
... [y — wx — Cl'... - 
/7>Wl 3 rrAYl 2 f-yAYL 1 
%Aj xAj tU , 
(where the quantities Cl'... n (m) are entirely determinate, since, by 
what has preceded, O', $'... Cl' satisfy a certain equation, 0", <fi", ... Cl" two equations 
0 im) , <l> (m) , ... Cl {m) (m — 1) equations), we have F=0 for the required equation of the 
asymptotic curve. It is obvious that the whole number of asymptotic curves of the 
order r, is n (n — 1) ... (w - r + 1), viz. 1 .2 ... r curves for each combination of 
n {n — 1) ... (n — r + 1) 
asymptotes. Some particular instances of asymptotic curves will 
1.2... r 
be found in a memoir by M. Plücker, Liouvilles Journal, vol. I. [1836, pp. 229—252], 
Enumeration des courbes du quatrième ordre, &c. The general theory does not seem to 
be one to which much attention has been paid.