10] 
ON THE THEORY OF ALGEBRAIC CURVES. 
53 
Also, if 6 be the number of points through which the curve W = 0 can be drawn, 
including the points of intersection of the curves U = 0, V = 0, then 
6 — cf) + (mn — t — t' — or 
0 = ^p(p+S)+^(p~m~ n + l)(p-m-n + 2) + ^- A-t-t'...~ t ( * -1) (13). 
Any particular cases may be deduced with the greatest facility from these general 
formulae. Thus, supposing the curves to intersect in the complete number of points mn, 
we have 
<f> = zP (p + 3) 4- \ (1 — 8) (p — m — n + 1) (p — m — n + 2) — mn, 
S being zero or unity according as p<m+n-1 or p>m + n-1. Reducing, we have, 
for p m + n — 3, 
0 ~ \p (p + 3) + 2 (p — m — n + 1) (p ~ m ~ n + 2) — mn, 
0 = 2 p(p + 3) + h(p~m — n+ 1) (p - m - n + 2) ; 
and for p > m + n — 3, 
<f> = ip (p + 3) - mn, 
0 = 2P (p + 3). 
Suppose, in the next place, the curves have t parallel pairs of asymptotes, none of 
these pairs being coincident. Then 
p m + n — t — 2, 
4> = hp (p + 3) + \ (p — m —n+ 1) (p — m — n+2) — mn, 
6 = \p (p + 3) + I (p — m — n+ 1) (p—m — n + 2) — t ; 
p> m + n —t — 2, p m + n — 2, 
$ = \P (.P + 3) + \ (p — m — n + 2) (p — m — n + 3) — mn + t, 
6 = ^p (p + 3) + ^ (p — m — n + 2) (p — m — n + 3), 
p >m + n — 2, 
cf) = ^p(p + 3) — mn +1, 
Ô = hp (p + 3). 
In these formulae, if t be equal to 2 or greater than 2, the limiting conditions are 
more conveniently written 
p^>m + n — t— 2; p m + n — t— 2 > m+ n — 4 ; p >m + n — 4. 
Similarly may the solution of the question be explicitly obtained when the curves 
have t asymptotes parallel, and t' out of these coincident, but the number of separate 
formulae will be greater. 
In conclusion, I may add the following references to two memoirs on the present 
subject : the conclusions in one point of view are considerably less general even than 
those of my former paper, though much more so in another. Jacobi, Theoremata de 
punctis intersectionis duarum curvarum algebraicarum ; Crellds Journal, vol. xv. [1836, 
pp. 285—308]; Plücker, Théorèmes généraux concernant les equations a plusieurs variables, 
d’un degré quelconque entre un nombre quelconque d’inconnues. D° vol. xvi. [1837, 
pp. 47—57].