n 
46 
[494 
494. 
EXAMPLE OF A SPECIAL DISCRIMINANT. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. XI. (1871), 
pp. 211—213.] 
If we have a function (a, ...Qx, y, z) n , where the coefficients {a, ...) are such that 
the curve {a, ...\x, y, z) n = 0 has a node, and a fortiori if this curve has any number 
of nodes or cusps, the discriminant of the function (that is, the discriminant of the 
general function {*§x, y, z) n , substituting in such discriminant for the coefficients their 
values for the particular function in question) vanishes identically. But the particular 
function has nevertheless a special discriminant, viz. this is a function of the coefficients 
which, equated to zero, gives the condition that the curve may have (besides the 
nodes or cusps which it originally possesses) one more node; and the determination of 
this special discriminant (which, observe, is not deducible from the expression of the 
discriminant of the general function (*][#, y, z) n ) is an interesting problem. I have, 
elsewhere, shown that if the curve in question (a, ..fix, y, z) n = 0 has 8 nodes and 
k cusps, then the degree of the special discriminant in regard to the coefficients 
a, &c., of the function is = 3 (n — l) 2 — 78 —11«: and I propose to verify this in the 
case of a quartic curve with two cusps. 
Consider the curve 
Qnx 2 y 2 + 12rz 2 xy + (4gx + 4iy + cz) z 3 = 0, 
where x = 0 is the tangent at a cusp; y = 0 the tangent at a cusp; and z = 0 the 
line joining the two cusps. 
For the special discriminant we have 
Snxy 2 + 3 ryz 2 + gz 3 = 0, 
3 nx 2 y + '3rxz 2 + iz s = 0, 
z {Qrxy + (3gx + 3iy + 4>cz) z] = 0 ; 
the last of which may be replaced by the equation of the curve.