APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 109 



of L^ and thus L^ itself may be determined. L3 may be found in a similai- 

 manner. 



Through the C^ a pencil of quadrics may be passed of which the 

 four cones are the singular surfaces. The traces of all these surfaces 

 form a pencil of conies, and as four of them are circles (L„ L^, L3, L^), 

 the whole pencil consists of a system of coaxial circles with the radi- 

 cal axis p. The tetrahedron MjMjMjM^ is self-polar in the pencil 

 of quadrics through C^. Any ray through the vertex, for instance 

 M,, containing two points A and B of the C^, cuts the opposite polar- 

 plane M.^MgM^ in a point P so that ( MjPAB ) = — 1. The tangents 

 to the C^ at A and B meet in a point of the polar-plane. 



4. Through MjM2 pass two planes H and H* forming a har- 

 monic pencil with the planes MjMjMj and MjMjM^. The traces /?, 

 A*, J93, ji?4 of these planes form a harmonic pencil. Each of these 

 auxiliary planes cuts the curve in four points A, B, C, D and A*, B*, 

 C*, D*, and as these are all in harmonic planes with respect to the 

 foregoing polar-planes it follows that, two by two, they are also situ- 

 ated on rays through M3 and M,. Thus the rays A A*, BB*, CC*, 

 DD* pass through M^ and AC*, BD*, CA*, DB* through M,. On 

 account of the harmonic division the double secants BD and B*D* 

 of the C^ meet in a point of MjMj; similarly AC and A*C* meet in 

 a point of MjM2. These two points are harmonic with Mj and M.,. 

 By analogy similar relations are found on the other edges of the self- 

 polar tetrahedron. As A may be any point it follows that every 

 secant of the G ^ 'intersecting two opposite edges of the tetrahedroii , 

 intersects the curve in another point. 



As there are three pairs of opposite edges, the systems of double 

 secants of the C^ intersecting these pairs form three ruled surfaces 

 and these are of the 4th order and 4th class. 



In conclusion the theorem may be stated: — 



The eight points of the C^ in two planes harmonic with respect 

 to any two planes of the tetrahedron lie four tiines in groups oj 

 four on rays through the vertices and six times in groups of eight 

 on two planes through the edges of the tetrahedron. Each tangent 

 at one of the points of the group is met hy four other tangents in 

 points which are the intersections of the given tangent with the 



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