211 
[588 
589] 211 
F, G, H 
T) subtend 
quadrangle 
îerefore not 
G, F, G, H 
this is the 
i, B, C, D 
). There is 
589. 
as a given 
mt to five 
m the five- 
etween the 
ON RESIDUATION IN REGARD TO A CUBIC CURVE. 
[From the Messenger of Mathematics, vol. III. (1874), pp. 62—65.] 
H as given 
determining 
a x, y, z. 
The following investigation of Prof. Sylvester’s theory of Residuation may be 
compared with that given in Salmon’s Higher Plane Curves, 2nd Edition (1873), pp. 
133—137 : 
If the intersections of a cubic curve U 3 with any other curve V n are divided in 
any manner into two systems of points, then each of these systems is said to be the 
residue of the other; and, in like manner, if starting with a given system of points 
on a cubic curve we draw through them a curve of any order V n , then the remaining 
quadrangle, 
assumed to 
required to 
0. 
intersections of this curve with the cubic constitute a residue of the original system of 
points. 
If the number of points in the original system is = Bp, then the number of 
points in the residual system is = 3q; and if we again take the residue, and so on 
îe two-fold 
es, in order 
n ; that is, 
bedron. It 
se lines are 
ce in these 
positions of 
Id relation, 
e, must in 
indefinitely, the number of points in each residue will be =0 (Mod. 3); viz. we can 
never in this way arrive at a single point. But if the number of points in the original 
system be dp ± 1, then that in the residual system will be Sq + 1; and we may in 
an infinity of different ways arrive at a residue consisting of a single point; or say 
at a “residual point,” viz. after an odd number of steps if the original number of 
points is =Sp — 1, but after an even number of steps if the original number of points 
is =3p+l. But starting from a given system of points on a given cubic curve, the 
residual point, however it is arrived at, will be one and the same point; this is 
Prof. Sylvester’s theorem of the residuation of a cubic curve. For instance, starting 
with two given points on the cubic curve, the line joining these meets the curve in 
a third point, which is the residual point; any other process leading to a residual 
point must lead to the same point. Thus if through the 2 points we draw a conic, 
meeting the cubic besides in 4 points; through these a conic meeting the cubic besides 
27—2