70 KANSAS UNIVERSITY QUARTERLY. 



tion of the quartics ; thus we have left four i^oints of intersection of 

 the two cubics. 



Since any element of the cone Aleuts C and C\ in a pair of corre- 

 sponding points, it follows that A, B, C, D are self-corresponding or 

 invariant points of the coUineation. In the general case these four 

 points form a tetrahedron. It is at once evident that the four faces 

 of the tetrahedron are invariant planes and the six edges are invariant 

 lines of the collinealion. 



Theorem 7. ^i collineatton T constructed l)y means of two cubics 

 C and C\ leaves invariant in the general case the vertices, faces and 

 edges of a tetrahedron. 



10. The Five Primary Types of Collineations. There are 

 thirteen types of collineations in space, each characterized by its in- 

 variant figure. When the two cubics C and Cj intersect in four dis- 

 tinct points, the coUineation T constructed as above is of type I. 

 If two points of intersection of the cubics C and C^ coincide but not 

 at the vertex of the cone, the coUineation is of type II. If the four 

 points of intersection A, B, C, D coincide two and tw^o, the coUinea- 

 tion is of type III. If three of the four points of intersection coin- 

 cide, the coUineation is of type IV. If all four points of intersection 

 coincide, the transformation is of type V. These constitute what are 

 called the five primary types of collineations in space. 



11. If one of the four points of intersection of C and C^^ coincides 

 with S^ , the vertex of the cone, the common tangent to C and C^ at 

 8^ is an invariant line of the coUineation. The lines joining S^ to 

 the three remaining points of intersection B, C, D are likewise inva- 

 riant lines; thus there are four invariant lines through ^S^, and 

 therefore all lines through S^ are invariant lines of the transforma- 

 tion. All points of the plane B, C, D are also invariant points of the 

 transformation; hence our transformation Tin this case is of type 

 VI. In case two of the four points A. B, C, I) coincide at S^ so that 

 the plane of invariant points passes through S, we have a coUineation 

 of type VII. These two cases are both perspective collineations. 



12. The Six Remaining Secondary Types. The cone ICon which 

 the cubics C and C^ lie may degenerate into two intersecting planes. 

 The two cubics in this ease each degenerate into a conic and a line I, 

 respectively ^i, not in the plane of the conic, meeting the conic in a 

 single point. The two conies lie in the same plane and generally in- 

 tersect in four jDoints SABC. The two lines I and U coincide 

 throughout and pass through one of the i^oints of intersection, say A, 

 of the two conies. 



The coUineation constructed by means of these two degenerate 



