[383 
613 
'poser’s 
NOTES AND REFERENCES. 
302, 305. The theory of curves in space was proposed as the subject of the 
Prize-question of the Steiner Foundation by the Academy of Sciences of Berlin in the 
year 1881 “ irgend eine auf die Theorie der höheren algebraischen Raum cur ven sich 
beziehende Frage von wesentlicher Bedeutung vollständig zu erledigen,” and the prize 
was divided between the two memoirs 
Halphen, “Mémoire sur la classification des courbes gauches algébriques.” Jour. 
École Polyt. Cah. lu. (1882), pp. 1—200, and 
Nother, “Zur Grundlegung der Théorie der algebraischen Raumcurven.” Abh. der 
Akad. zu Berlin vom Jahre 1882, pp. 1 to 120; both treating of the classification of 
curves in space. 
We have also the valuable memoir 
Valentiner, “Zur Theorie der Raumcurven,” Acta Mathematica, t. II., 1883, pp. 
136—230, which relates less directly to the question of classification. 
The three authors all refer to these papers in the Comptes Rendus, and make 
considerable use of my conception of the monoid surface. It would be out of place 
to attempt to give any account here of these memoirs : I only refer to such remarks 
or theorems contained in them as stand in immediate connection with the remarks 
which follow. 
The question of classification is much simplified by excluding from consideration 
the curves with singular points (that is actual double points and stationary points), 
and this is in fact done both by Halphen and Nother and in the present Note. 
The curves considered are thus curves with only apparent double points (adps.) viz. 
for a curve of the order d (I use Halphen’s letters) with h apparent double points, 
taking an arbitrary point as vertex, the cone through the curve is a cone of the 
order d, with h nodal lines, each of these meeting the curve in two (real or 
imaginary) non-coincident points. Such a curve is the partial intersection of the 
cone in question say U, = (%, y, z) d , = 0 with a monoid surface w, = ~ , 
where the inferior cone P, = (x, y, z) k , = 0, of the monoid surface, and the superior 
cone Q, = (x, y, z) k + l , = 0, of the monoid surface each of them pass through all the h