108 UNIVERSITY OF COLORADO STUDIES 



case of a quadrilateral is obtained in a very simple manner by this 

 descriptive-projective m ethod. 



But this method also yields the theorem of Clebsch proved in §4. 



3. The polar-lines of the pencil of planes through M,M, with 

 respect to the two given cones lie respectively in the polar-planes Pj, 

 Pj (traces p^ and p^ of MjMj with respect to the same cones. Let 

 P, and Pj intersect each other in the line g. The pencil of planes 

 through MjMj and its corresponding pencil of polar-lines in P, cut g 

 in an involution of points Ij. The same pencil of planes and its cor- 

 responding pencil of polar-lines in Pj likewise cut g in an involution 

 I,. If we now construct the double-elements M3 and M^ of I, and I,, 

 then Mj and M^ have each the same polar-plane with respect to both 

 of the given cones. Hence, if P is any other point of intersection 

 of the given cones, i. e., a point of the C^, and CP produced cuts the 

 polar-plane of C in T and the given cones in F and G, we have 

 again (CPTF) = -1, (CPTG)== — 1, i. e., F and G must coincide. 

 In other words, any ray through M^ and a point of the C ^ intersects 

 the same curve in another point. The same is true of M^. 



Both J/j and M\ are therefore vertices of cones of the second 

 order containing the 6'^ completely, and what has been said with 

 regard to the first two cones also holds for these, and finally for any 

 two among the four. 



In the projection, M3 and M^ lie on the traces L3 and L^ of the 

 last two cones, and the C3 is also tangent to L^ and L^ at M, and M^. 

 The tangents at M, and M^ also pass through S^*. In conclusion the 

 theorem holds: The points J!/,, J/^, Jfg, M^ being the projections of 

 the vertices of the four cones passing through the C^ form a 

 quadruple. 



To construct the traces of the other two cones, for instance of 

 L,, jBnd first the trace S^^ of M^ Mj, which is obtained as the inter- 

 eection-point of the traces p^ and p^ of the polar planes of M2 and M^ 

 with respect to the corresponding cones. S^^ is therefore the point 

 of intersection of p,^ and MjM^. Through Sj^ draw a trace joining S^^ 

 with the point of intersection of M^Mj and L,; this trace intersects 

 M^M, produced in a point of L,. In a similar manner another point 



