ON THE EXPRESSION OF THE COORDINATES OF A POINT OF 
A QUARTIC CURVE AS FUNCTIONS OF A PARAMETER. 
[From the Proceedings of the London Mathematical Society, vol. vi. (1874—1875), 
pp. 81—83. Read February 11, 1875.] 
The present short Note is merely the development of a process of Prof. Sylvester’s. 
It will be recollected that the general quartic curve has the deficiency 3 (or it is 
4-cursal); the question is therefore that of the determination of the subrational* 
functions of a parameter tvhich have to be considered in the theory of curves of the 
deficiency 3. 
Taking the origin at a point of the curve, the equation is 
(x, yY + (x, y) 3 + (x, y) 2 +(x, y) = 0; 
and writing herein y — \x, the equation, after throwing out the factor x, becomes 
(1, \) 4 a; 3 + (l, \) 3 a; 2 +(l, \fx + (l, X) = 0; 
or, say 
where we write for shortness 
ax 3 + 3 bx 1 + 3 cx + d = 0, 
a, 6, c, d = ( 1, X) 4 , ¿(1, X) 3 , i(l> x ) 2 > (!» Mi 
viz. a, b, c, d stand for functions of X of the degrees 4, 3, 2, and 1 respectively. 
The equation may be written 
(ax + b) 3 — 3 (b 3 — ac) (ax + b) + a-d — 3abc + 2b 3 = 0; 
* The expression “ subrational ” includes irrational, but it is more extensive; if Y, X are rational 
functions, the same or different, of y, x respectively and Y is determined as a function of x by an equation 
of the form Y—X, then y is a subrational function of x. The notion is due to Prof. Sylvester. 
40—2