NEWSON: ON THE CONSTRUCTION OF COLLINEATIONS. 71 



cables leaves invariant all points on the line Ih and the triangle 

 ABC. Such a collineation is of type YIII. If B and 6' coincide 

 the collineation is of type IX. If ^ and ^ or ^ and 6' coincide the 

 collineation is of type XI. If A with Ih through it coincides with .Vi 

 tlie collineation is of type X. If A, B, and 6'all coincide the colline- 

 ation is of type XIII. If A and B both coincide with ^ the colline- 

 ation is of type XII. In each of these types the invariant figure has 

 at least one line of invariant points. 



Theorem 8. The method of constructing a collineation in space hy 

 means of two twisted cubics lying on the same cone gives rise to all 

 of the thirteen knovm tyjyes of such collineations . 



13. Qo ■' Different Constructions of the Same Collineation. 

 The construction of a given collineation Thy means of the two cubics 

 6^ and Ci is evidently not unique. The vertex ,S'i of the cone on which 

 the cubics lie may be any point in space. For each point in space 

 there is a different pair of cubics by means of which a transformation 

 Tcan be constructed ; in other words there are in the general case 

 3c -^ different constructions of the same transformation T 



14. Collineations with the Same Invariant Tetrahedron. 

 Passing through the five points AB CBS there are oo"-' cubics; for a 

 cubic is determined by six points and each point counts as a double 

 condition. There are oo i cones of second order with vertex at Si and 

 passing through A BCD. On each cone there are co i of the cubics 

 through the five points ; hence on each cone there are x - pairs of cu- 

 bics, each pair giving rise to a different collineation with the same 

 invariant tetrahedron ABCU. Since there are oo i such cones, there 

 are y.'^ collineations which have the same invariant tetrahedron. 



