506 
ON THE GROUP OF POINTS 6r/ ON A 
[613 
The curves ft, v intersect in nv points, among which are included the seven 
points counting 2aa times: the number of the remaining intersections is therefore 
I recur to the case p= 5 and the group Gj, which is the subject of the present 
note: viz. we have here a sextic curve with 5 dps, and on it a group of 4 points 
Gi, such that every cubic through the 5 dps and through 3 points of the group, 
8 points in all, passes through the remaining 1 point. 
MM. Brill and Nother show (by consideration of a rational transformation of the 
whole figure) that, given 2 points of the group, it is possible, and possible in 5 
different ways, to determine the remaining 2 points of the group. 
I remark that the 5 dps and the 4 points of the group form “ an ennead ” or 
system of the nine intersections of two cubic curves: and that the question is, given 
the 5 dps and 2 points on the sextic, to show how to determine on the sextic a 
pair of points forming with the 7 points an ennead: and to show that the number 
of solutions is = 5. 
We have the following “Geiser-Cotterill” theorem: 
If seven of the points of an ennead are fixed, and the eighth point describes a 
curve of the order n passing a 1 , aa-, times through the seven points respectively, 
then will the ninth point describe a curve of the order v passing a,, a 2 ,.., a 7 times 
through the seven points respectively: where 
v — 8w — 32a, 
at] = 3ft — aj — 2a, 
a- = 3n — a 7 — 2a, 
and conversely 
ft = 8ft — 32a, 
a, = 3ft — aj — 2a, 
a 7 = 3ft — a 7 — 2a. 
(Geiser, Grelle-Borchardt, t. lxvii. (1867), pp. 78—90; the complete form, as just 
stated, and which was obtained by Mr Cotterill, has not I believe been published): 
and also Geiser’s theorem “ the locus of the coincident eighth and ninth points is a 
sextic passing twice through each of the seven points.” 
The sextic and the curve n intersect in 6ft points, among which are included the 
seven points counting as 22a points: the number of the remaining points is 
= 6ft — 22a. Similarly, the sextic and the curve v intersect in 6v points, among which 
are included the seven points counting as 22a points: the number of the remaining 
points is 6ft — 22a (= 6ft — 22a). The points in question are, it is clear, common 
intersections of the sextic, and the curves n, v: viz. of the intersections of the 
curves ft, ft, a number 6ft — 22a, = 6ft - 22a, = 3w + 3ft — 2a - 2a lie on the sextic.