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Mathematics. — "An article on the knowledge of the tetrahedral 

 complex." By Dr. Z. P. Bouman. (Communicated by Prof. Jan 

 DE Vries). 



§ 1. When for an arbitrary ray out of a tetrahedral complex P,- 

 represents the point of intersection with the face Ak Ai Am of the 

 tetrahedron, then 



where R represents tlie given anharmonic ratio of the complex and 

 2)^ (^i = 1 . . Q) are the Plücker coordinates of lines. 



By using the condition necessary for each ray of the complex, 

 namely 



Pi Pa + P^ P, + Pz Pe = Ö 

 the equation of the complex becomes 



^ Pi Pi + ^ P^ Pi -^ C p^ p, = 0, 

 wiiere the anharmonic ratio is given by 



_B-A 



A given tetrahedral complex can always transform itself projectively 

 into another one with the same anharmonic ratio in regard to the 

 faces of tiie rectangular system of coordinates and the plane at 

 infinity. 



§ 2. After having executed this transformation we can examine 

 whether a surface with two independent parameters can be found in 

 such a manner that the normals to be erected in an arbitrary point 

 on the OD^ number of sui-faces passing through that point, are rays 

 of the given tetrahedral complex. 



To this end we make the two determining points to lie infinitely 

 close to each other on each ray of the complex, so that each ray 

 is determined by one point {a;, y, z) and the direction {dx, dy, dz) in 

 that point. The coordinates of lines now take the form : 



p^ =: w dy — y d.t, p^ ■= y dz — z dy, p^^=. z dx — x dz, 

 p, = — dz, Pi — — dx, Pe = — '^y- 



So the equation for the complex becomes : 



A {x dy — y dx) dz -\- B {y dz — z dy) dx -\- C {z dx — x dz) dy = 0. 



If now every ray of the complex is to be at right angles to a 

 surface z=zf{x,y), then we have for each ray in each point of the 

 surface : 



