287 



straight line p, envelop a curve (p) of class (« + ^)j of which p is 

 a ^-fold tangent. 



Through a point S pass {a -\- ^) tangents of {p)\ from this it is 

 evident that the null-points on tlie rajs of the pencil S form a 

 curve {SY'^^. Now, aS is always one of the null-points, so that an 

 arbitrary raj of the pencil bears onlj (^i — 1) points X outside S. 

 Consequentlj [Sy--^^ has an (« -|" l)-fold point in S. 



Analogouslj we find that (.s%-|-,9 has the straight line s as (ji-|-l)- 

 fold tangent, while a straight line .v^ is a (/?-|-2)-fold tangent of 

 the curve (.9^),^._^^?. 



3. The curve (P/+/5 is of class (« + /?) (« -f /? — 1) — «(«—1). 

 Through I* pass therefore {2n -{- ^) [ii — 1) more tangents, which 

 touch it elsewhere. To them belong evidentlj the straight lines /^aS^, 

 as S^ re|)ieseiits two coinciding null-points. Consequevtly the null- 

 rays bearing a double null-point envelop a carve of class (2« -j- ^) 



0?'— 1) — 4 



The complete enveloping tigiire contains moreover the g^ class- 

 points S^. 



It is of course possible that the enveloped curve breaks up. This 

 e. g. happens with the null-system that arises if each tangent of a 

 pencil (c") is associated to the {n — 2) points, in which it moreover 

 intersects the c" (satellite points of the point of contact). 



We have to distinguish then between the envelope of the 

 stationarj tangents, which each bear one double null-point, and the 

 envelope of the bitangents, which each contain tioo double null- 

 points. The curve (P) is now the so-called satellite-curve ^). 



In a similar waj we find: lite locus of the points N, for which 

 tnw of the null-rays n have coincided is a curve of order {a -j- 2^) 

 (« — 1)— 4 



4. The curves (/>)a-f,3 and (7)«-(-^ have the <( null-rajs of the 

 point p(/ in common. To the remaining common tangents the singular 

 rajs s ii\n\ s^ evidentlj belong '■). There are therefore (n-|-^)* — « — 'j — o^ 

 rajs n, a null-point ^V of which lies on />, another null-point iV' on 7. 



This number has another meaning jet. If jX describes the straight 



1) Cf. my paper "On linear systems of algebraic plane curves'' (These Pro- 

 ceedings vol. VII, page 712) or "Fuisceaiix de courbes planes'' (Archives Teyler, 

 série II, t. XI, p. 101). 



2) If ji = 1. (})) and (q) have, besides the a null-rays of: pq, only singular rays 



in common; consequently we have o -{- <y* = (-('^ -^ (t + ^- The tangents and points 

 of contact of a tangenlial pencil provide an example of this. 



19* 



