POND: COLLINEATIONS IN FOUR DIMENSIONS. 257 



X. 1 double root, rank 3; 1 of rank 4; 1 single root. 



2+ oo points— 1 single; 1 double; 1 range. 



3 + 2 oo lines — 2 lines, axes of pencils of planes; 1 line, range 

 of points; 2 pencils of lines. 



3 + 2 oo planes — 2 planes containing pencils of lines; 1 plane, 

 axis of space pencil; 2 pencils of planes. 



2+ CD spaces — 2 single spaces; 1 pencil of spaces. 

 XI. 1 double root, rank 3; 1 triple root, rank 4. 



1+00 points — 1 triple point; 1 range. 



2+ CO lines — 1 line, range of points; 1 line, axis of pencil of 

 planes; I pencil of lines. 



2+ oo planes — I plane, axis of pencil of spaces; 1 plane con- 

 tains pencil of lines; 1 pencil of planes. 



1 + "0 spaces — 1 triple space; 1 pencil of spaces. 

 XII. 1 double root, rank 4; 1 triple root, rank 3. 



1+ <» points — 1 double point; 1 range. 



2+2" lines— 1 line, axis of pencil of planes; 1 line, axis of 



pencil of planes, and containing range of points; 2 pencils 



of lines. 



2 + 2 « planes — 1 plane, axis of pencil of spaces, and contain- 



ing pencil of lines; 1 plane containing pencil of lines; 2 

 pencils of planes. 

 1 + <» spaces — 1 space; 1 pencil of spaces. 



XIII. 1 double root, rank 4; 1 triple root, rank 2. 



1 _(_ oo2 points — 1 double point; 1 plane of points. 



1 + 2 ="■- planes— 1 line, axis of pencil of planes; »- lines 



from point to plane, each containing pencil of planes; <*>^ 



lines in a plane. 

 1 + 2 «o2 planes — 1 plane of =<j2 points; «■= planes through a 



line and points in plane; «>'- planes through lines from 



point to plane, each containing a pencil of lines. 

 1 + i»2 spaces — 1 double space; »- spaces through line. 



XIV. 1 double root, rank 3; 1 triple root, rank 2. 



CO + cc2 points — 1 plane of points; 1 range. 



003 ^ 0,2 lines — =<=3 lines from range to plane; «•- lines in plane 

 of points. 



°°^ + "3 planes — '"^ planes from range to points in plane; «•' 

 planes through lines from range to plane. 



CO + oo2 spaces — «> spaces through plane of points; «=- spaces 

 through range of points. 

 XV. 1 triple root, rank 3; 2 single roots. 



3 + <» points — 2 single points, 1 point of range, and range. 



4 + 3 " lines — 3 lipes, axes of pencils of planes; 1 line, range 



of points; 3 pencils of lines. 



4 + 3 »= planes — 3 planes containing pencils of lines; 1 plane, 

 axis of pencil of spaces; 3 pencils of planes; 3 + «> spaces. 

 XVI. 1 triple root, rank 2; 3 single roots. 



2 4-002 points — 2 single points; 1 plane of points. 



l + °°^ + 2«>- lines — 1 line, axis of pencil of planes and of pen- 

 cil of spaces; »- lines in plane; 2"- lines from 2 points to 

 plane of points. 



l + "- + 2a"- planes — 1 plane of w^ points; =«- planes from line 

 to points in plane of points; 2'»- planes through lines from 

 2 points to plane of points. 



2+ "2 spaces — 2 single spaces; »'- spaces through a line join- 

 ing 2 single points. 



