Mathematics. — ''Numbers oj Circles Toachlng Plane Carves 

 Defined by Representation on Point Space." By L. J. Smid Jr. 

 (Communicated bj Prof. Hendrik de Vries), 



(Communicated at the meeting of June 24, J 922). 



The circles of a plane (degenerations included) may be represented 

 without exception through a one-one representation on the |)oints 

 of a projective space. (R. Mehmkk, Zeitschrift fur Mathematik nnd 

 Physik 24 (1879)). We can arrive at it among others in the 

 following way: 



Let W be an umbilical point of a quadric 0^' and let lu be the 

 tangent plane at that point, B a plane pai-allel to ?<;. A plane section 

 of 0^ with its pole relative to 0^ is projected out of W on J3 as 

 a circle with its centre, and inversely. We consider this pole as 

 the image of the circle. 



As a special case we may take for C^ a quadric of .revolution of 

 which W is a vertex. If moreover 0^ is a sphere, we get the repre- 

 sentation of Prof. Jan de Vries (Verhandelingen 29); if W moves 

 to infinity it becomes the representation of Dr. K. W. Walstra 

 (Verhandelingen 25). 



Prof. Hk. de Vries has studied cyclographically the circles touching 

 a curve C in B of the order n, the class v, passing e times through 

 both the circle points (with s different tangents in finite space which 

 cut Cat those points in e -f- 1 points), touching the line ^^^^ singly in ö 

 different points and having further no other singularities than d 

 nodes, x cusps, r bi-tangents and t inflexional tangents (Verhande- 

 lingen 8). 



We arrive at the same results through the above mentioned 

 representation. We shall only consider the principal ones. 



The curve C is projected out of IF on 0^ as a curve consisting 

 of the two generatrices through W, counted e times, and a curve k 

 of the order n = 2^1 — 2s passing (ju — 2f)-times through W. a pairs 

 of tangents at W coincide, because the parabolic branches of C 

 give rise to cusps of k in W . Further k has d nodes, x cusps and 

 (jLt — f) (ft — c — 1) apparent nodes and no stationary tangents. By 

 means of Plückrr's formulas we find other numbers characteristic of ^•. 



From the nature of the representation there follows that the points 

 of the surface L of the tangents of k represent the circles cutting 

 C at right angles. The tangent planes to 0" at the points of k 



