504 
[613 
613. 
ON THE GROUP OF POINTS G, 1 ON A SEXTIC CURVE WITH 
FIVE DOUBLE POINTS. 
[From the Mathematische Annalen, vol. vin. (1875), pp. 359—362.] 
The present note relates to a special group of points considered incidentally by 
MM. Brill and Nother in their paper “Ueber die algebraischen Functionen und ihre 
Anwendung in der Geometrie,” Math. Annalen, t. vii. pp. 268—310 (1874). 
I recall some of the fundamental notions. We have a basis-curve which to fix 
the ideas may be taken to be of the order n, =p+l, with \p (p — 3) dps, and 
therefore of the “ Geschlecht ” or deficiency p; any curve of the order n — 3, =p — 2 
passing through the (p — 3) dps is said to be an adjoint curve. We may have, on 
the basis-curve, a special group Gq of Q points (Q %> 2p — 2); viz. this is the case 
when the Q points are such that every adjoint curve through Q — q of them—that 
is, every curve of the order p — 2 through \p {p — 3) dps and the Q — q points—passes 
through the remaining q points of the group: the number q may be termed the 
“ speciality ” of the group: if q= 0, the group is an ordinary one. 
It may be observed that a special group Gq is chiefly noteworthy in the case 
where Q — q is so small that the adjoint curve is not completely determined: thus 
if p = 5, viz. if the basis-curve be a sextic with 5 dps, then we may have a special 
group G 6 2 , but there is nothing remarkable in this; the 6 points are intersections 
with the sextic of an arbitrary cubic through the 5 dps—the cubic of course intersects 
the sextic in the 5 dps counting as 10 points, and in 8 other points—and such cubic 
is completely determined by means of the 5 dps and any 4 of the 6 points. But 
contrariwise, there is something remarkable in the group G± about to be considered: 
viz. we have here on the sextic 4 points, such that every cubic through the 5 dps 
and through 3 of the 4 points (through 8 points in all) passes through the remaining 
one of the 4 points. 
The whole number of intersections of the basis-curve with an adjoint, exclusive 
of the dps counting as p(p— 3) points, is of course = 2p — 2: hence an adjoint 
through the Q points of a group Gq meets the basis-curve besides in R, = 2p — 2 — Q >