038 
Mathematics. — “On Steineriin points in connexion with systems 
of nine @-fold points of plane curves of order 30.” By Dr. W. 
VAN DER Woupe. (Communicated by Prof. P. H. Scouts). 
(Communicated in the meeting of December 28 1912). 
§ 1. In a former commnnication') has been indicated what is 
the locus of the point ferming with eight given points a system of 
nine nodes of a non degenerated plane sextic curve ; here will be 
treated a more general problem including the preceding one as a 
particular case. 
To that end we remark that -by nine arbitrarily chosen points 
D,, D,,..., D, a curve of order 39 passing o times through these 
points is determined; in general however this Cs, is a cubic curve 
counted 9 times. So the problem we propose now is: “Eight points 
D,, D,,..., D, being given, to determine the locus of the point D, 
under the condition that the nine points D; can be g-fold points 
of a curve C3, not degenerating in the manner mentioned. 
§ 2. As we shall find by and by this problem is very closely 
related to the following one: “Let B, B... B, be the base points 
of a pencil (3’) of cubic curves, and uw, any curve of this pencil. 
On u, lie (0?—1) points S each of which forms with 4, a Steinerian 
pair“) of order e@. To determine the locus of these points S, if wu, 
describes the pencil (3’)”. 
§ 3. We start by treating the first of the two problems. 
So the eight points D,, D,..., D, are given and we have to 
determine the locus of the ninth point D, satisfying the condition 
stated. In the quoted memoir the case 9 = 2 has been treated; for 
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convenience sake we repeat here the principal results. 
Then we oceeupy ourselves with the case @ = 3 before passing to 
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1) W. v. p. Wovpe, “Double points of a cg of genus O or 1 (Proceedings of 
Amsterdam, vol. XIII, p. 629). 
Compare also Dr. V. Snuper, “Tre involutorial birational transformation of 
the plane of order 17” (American Journal of Mathematics, vol XXXIII, p. 328). 
2) Two points P and Q of uw; form a Steinerian pair of order , if it be possible 
to inscribe in «z one and therefore an infinity of closed polygons with 27 vertices, 
the sides of which pass alternately through P and Q. Literature: Steiner (Jour- 
nal of Crelle, vol. XXXII, p. 182); Kürrer (Math. Ann, vol XXIV, p. 1); ScHRÖTER 
(Theorie der ebenen Kurven drifter Ordnung, § 31). For the treatment by means 
of elliptic functions see Cregson: Vorlesungen über Geometrie. 
