( 7t'0 ) 



tangential planes, the four planes i(,, it.^. <r,,, «, uf" the [tencils of 

 rays and tlie planes «5, «„, a., «, parallel to the former and sym- 

 metric to these with respect to ; so those qnadratic surfaces are 

 a part of the tangential net J^, determined by seven of those eight 

 base-planes «,, enclosing together a regular octahedron. So our series 

 of surfaces being formed by the surfaces common to J^^j and ^V(, can 

 be regarded as the intersection of those nets. 



The tetrahedron of which the origin (J and the points AV. , y^.,Z,, 

 at infinity of the a.xes of coordinates are the vertices is common 

 polar tetrahedron of all surfaces of the two nets JSfp and Kt. In 

 connection with this ^\'^, has instead of a single infinite numbei- of 

 cones six pairs of planes, a pair thruugli each of the edges of the 

 tetrahedron, and Nt contains instead of a single infinite number of 

 surfaces reduced to conies six i)airs of points, a pair on each of the 

 edges of the tetrahedron. So we find the most general projective 

 transformation of the series common to N^ and JSft^ by starting from, 

 an arliitrary tetrahedron, an arbiti'ary point and an arbitrary plane 

 through this point and by then I'epresenting to ourselves the surfaces 

 having the given tetrahedron as polar tetrahedron, passing through 

 the given point and touching the given plane. 



We prepare the deduction of the three characteristic numbers of 

 our series of surfaces by determining the locus of the points of contact 

 with one of the eight base-planes, say «j. By considering the indicated 

 relation 



between the two lines l^ and l\ (fig. 3), in which a^ is cut by the 



quadratic surface belonging to l^, we find immediately that 5,, (',, Z), 



on A,A„ A,Ai, A^A, describe projective series of points when l^ 



rotates round 0^ and that /', envelops a conic described in triangle 



Fig. 3 



