Mathematics. — ''Axes of Rotation and Planes of Synunetvy of 

 Quadratic Surfaces of Revolution through 5, 6 and 7 Given 

 Points." By H. J. van Veen. (Communicated by Prof'. 

 Jan de Vries). 



(Communicated at the meeting of April 29, 1922). 



§ 1. Let there be given five points A^ A, Bj {j= 1, 2, 3). I consider 

 the complexes P^ and r^ belonging to the points /i, Bj m\(\ A^ Bj '). 

 Generally a common ray / of F, and r, is the axis of an OMhrongh 

 the 5 points; for / is the axis of an 0' throngh A^ Bj and of an 

 0' throngh A,Bj; these two 0"s have in common the 3 parallel 

 circles on which the Bj lie; hence they coincide. An exception 

 exists for the straight lines in the perpendicular bisector plane of 

 the join of 2 of the points Bj, and also for the straight lines of 

 V . The field degree of the congruence of axes is therefore 



3.3 _3_2. 2=r 2. 



At the same time there must be split off: the sheaf of the rays 

 which are perpendicular to the plane through the points Bj. Let D 

 be the centre at infinity of this sheaf; both the complex cones of 

 a point P touch the plane throngh P M, M^ and D along P D, hence : 



the axes of the 0^'s thjvugh 5 given points form a congruence of 

 the sheaf degree 7 and the field degree 2, C^-^. 



^ 2. To C^-2 belong the complex rays of F, lying in the perpen- 

 dicular bisector plane of a straight line A^ Bj, hence: 



the 10 perpendicular bisector planes of the joins of the 5 giveii 

 points are singular pkuies of the order 3. 



§ 3. In the two null systems belonging to F^ and F, the curves 

 I-/ and k^* (6>"s through 4 points ^ 16) are associated to a |)lane 

 pencil round a point of V^ ; these curves pass through 0, 

 touch D at Z) and have accordingly six more points in common. 

 Consequently through there pass six straight lines on which the 

 two pairs of points which through the two null systems are asso- 

 ciated to them, have one point in common. 



1) Gf. my paper "Axes of Rotation of Quadratic Surfaces through 4 Given 

 Points." 



