à 
4G 
[10 
10. 
ON THE THEORY OF ALGEBRAIC CURVES. 
[From the Cambridge Mathematical Journal, vol. iv. (1843), pp. 102—112.] 
Suppose a curve defined by the equation U = 0, U being a rational and integral 
function of the m th order of the coordinates x, y. It may always be assumed, without 
loss of generality, that the terms involving x m , y m , both of them appear in JJ ; and 
also that the coefficient of y m is equal to unity : for in any particular curve where 
this was not the case, by transforming the axes, and dividing the new equation by 
the coefficient of y m , the conditions in question would become satisfied. Let H m denote 
the terms of U, which are of the order m, and let y — ax, y — fix...y — Xx be the 
factors of H m . If the quantities a, /3 ...X are all of them different, the curve is said 
to have a number of asymptotic directions equal to the degree of its equation. Such 
curves only will be considered in the present paper, the consideration of the far more 
complicated theory of those curves, the number of whose asymptotic directions is less 
than the degree of their equation, being entirely rejected. Assuming, then, that the 
factors of H m are all of them different, we may deduce from the equation U = 0, by 
known methods, the series 
(1). 
and these being obtained, we have, identically, 
u= {y-ux-CL - ...) (y -fix-fi'- ...) ... (y -Xx- V - ...) (2),