256 KANSAS UNIVERSITY SCIENCE BULLETIN. 



type. In the case where one element of a range or 

 pencil is specially singled out by some property, as for 

 instance, when one point of a range is the vertex of a 

 pencil of lines, it will be counted separate from the in- 

 finity of elements to which it belongs. 



The first seven types occur when the matrix is of 

 rank 4. 



I. 5 points — all distinct. 

 10 planes. 

 10 lines. 

 5 spaces. 

 II. 4 points — 1 double, 3 single. 

 7 lines. 

 7 planes. 



4 spaces. 



III. 3 points — 2 double, 1 single. 



5 lines. 



5 planes. 

 3 spaces. 

 IV. 2 points — 1 double, 1 triple. 

 3 lines. 



3 planes. 



2 spaces. 



v. 3 points — 1 triple, 2 single. 



4 lines. 



4 planes. 



3 spaces. 



VI. 2 points — 1 quadruple, 1 single. 

 2 lines. 

 2 planes. 



2 spaces. 



VII. 1 point— quintuple. 

 1 line. 

 1 plane. 

 1 space. 

 VIII. 3 single roots — 1 double, matrix of rank 3. 



3 + oo points — 3 single points, 1 range of points. 



4 + 3 00 lines — 1 line containing range of points; 3 lines, axe 



of pencils of planes; 3 pencils of lines. 



4 + 3 oo planes — I plane, axis of space pencil; 3 planes con- 

 taining pencils of lines; 3 pencils of planes. 



3 + oo spaces — 3 single spaces; 1 pencil of spaces. 

 IX. 2 double roots, rank 3; 1 single root. 



1 + 2 M points -1 single point; 2 ranges of points. 



2 + 2 =0 + «=- lines — 2 lines containing ranges of points; 2 pen- 



cils -I- oo pencils of lines. 

 2 + 2 " + »= planes — 2 planes, axes of pencils of spaces; 



2 + ■» pencils of planes. 

 1 + 2 0° spaces — 1 single space; 2 pencils of spaces, through 

 two planes. 



