Mathematics. — "A Representation of the Line Elements of a Plane 

 on the Tangents of a Hyperholoid." By Prof. Jan df, Vriks. 



(Communicated al the meeting of March 24, 1923). 



1. In order to arrive at a representation of the line elements 

 (/-*,/) of a plane «, we consider a hyperholoid H which touches « in 



A, and which cnts it along the straight lines n^ and n^. Let R be 

 the projection of P on H ont of the point (J of H, ^ tiie tangent 

 plane at R, r the intersection of q with the |)lane (.*/; we consider 

 r as the linage of the line element formed by P and /. 



If, inversely, ?• is a tangent of H, R the point of contact, P the 

 projection of R, / the projection of r, the line element (/*,/) has 

 the tangent /■ for image '). 



We shall call the straight lines of U which cnt each other in 0, 

 hj and h,; h, cuts <e in a point i?i of a,, b, passes through a point 



B, of (7,. 



2. If / passes through 5, and F coincides with B^, R is the 

 point of contact of the plane bj, and any tangent r lying in this 

 plane, may be considered as the image (5,,/). Hence (S,,/) is a 

 sinqular element and its image is the plane pencil (?•) round R. If 

 I revolves round i),, the plane pencil (r) describes the parabolic 

 bilinear congruence with the directrix /*,, formed by the tangents 

 which have their points of contact on i,. Analogously the line ele- 

 ments (-S,,/) are singular. 



If B is an arbitrary point of the straight line b^B^B.^, R lies 

 in 0. The line element {B,b) is therefore also .lingular and is repre- 

 sented by the plane pencil {o) of the straight lines that touch H in 

 and lie in the tangent plane m. 



Hence, inversely, any tangent o is singular, as it represents all 

 elements (B,b). But at the same time it is the image of all the 

 elements of which the point P lies in the intersection of o with o, 

 for r is projected out of by any plane which contains r. The 



1) A fine representation of the hne elements of x on the points of space may 

 be found in the thesis of Dr. G. Schaake. (Afbeeldingen van figuren op de 

 punten eener lineaire ruitnte, P. Noordhotl", ]9!22). 



