( 495 ) 



round R^, the latter counted n{)i — 1) times, tbrui the complete apparent 

 circuit, indeed a degenerated curve of class 2n {n — 1). 



For the vertical -projection the centre Y^ lies on j-^^; the apparent 

 circuit on the vertical plane consists therefore of n [n — 1) pencils 

 round points on the projection of i\, and of a pencil whose vertex 

 is the point at infinity on the ^i-axis and which pencil must be 

 counted n {n — 1) times. 



Mathematics. — "Surfaces, twisted curves and groups of points 

 as loci of vertices of certain systems of cones" by Prof. P. H. 

 ScHOüTE ^). First paper. 



1. We consider as given (/i-|-2), pairs of straight lines crossing 

 each other, {ai, a!i),{b,b'), where i assumes successively the values 

 1, 2, . . . , ^?i(7i-|-3). We represent by t^. a transversal of (a,, a',), by 



tb a transversal of {b, b'). Tlie points F emitting {n -\- 2), trans- 

 versals t(i., ti lying on a cone C" of orde\ n form a surface {P) of 



which the order is to be determined. 



However we remark first, that the ').{ji-\-2)^ given lines [ai,a'i)y 



{b, b') are lines of multiplicity 7i on (P). For, the cone C' with an 

 arbitrary point F of b as vertex and the trans\'ersals ta. emitted by 



this point as edges, cuts the line b' in n points and is therefore to 



be counted n times among the considered system of cones C'\ i.e. 

 once for each of these points of intersection. 



Moreover it is immediately evidejit, that each point of each of 

 the two common transversals ti^k and t\k of the pairs {a,,a'i) and 

 {(ih a' I.) is vertex of a cone of the system, as we find for this point 

 (n-[-2)j — I edges only. So these lines, 6(7z-|-3)^ in number, are single 

 lines of {P). 



2. In order to determine the order of (P) we try to find the 

 number of points P satisfying the conditions of the problem lying 

 on an arbitrary transversal ti , by means of a figure lying in an 

 arbitrarily chosen plane .t connected with our figure in space in the 

 following way. 



We consider the transversals ta. emitted by the points P of tt, 



and remark that they form a regulus {tb,ai,a'i) of which tb,ai,a'i 



1) Suggested by the last communication of Prof. Jan de Vbies (These Proceedings, 

 XIV, p. 259). 



