258 



KANSAS UNIVERSITY SCIENCE BULLETIN. 



XVII. Triple root, rank 3; double root, rank 3. 



1 + 2<» points|— 2 ranges, and 1 point of 1. 



2 + 2" + '»- lines — 2 lines, ranges of points; 2 «> lines in 2 



planes; axes of pencils of spaces; =»- lines joining two 



ranges of points. 

 24-2°° + ■»- planes — 2 planes; axes of pencils of spaces; 2°° 



planes through ranges of points; <^' planes through lines of 



1 pencil|of lines. 

 1 + 2°° spaces— 1 double space; 1 pencil of spaces. 

 XVIII. Quadruple root, rank 3; single root. 



1 + » points — 1 single point; 1 range. 



1 + t» + co2 lines— 1 range of points; " lines from point to 



range; ■»- lines containing <» pencils of planes. 

 l + oo + oo- planes — 1 plane from point to range; " planes 



through range; »- planes consisting of '» pencil of planes. 

 1 + ■» spaces — 1 single space, and a pencil of spaces through 



plane from point to range. 

 XIX. Quadruple root, rank 2; 1 single root. 



1 + "'- points— 1 single point; 1 plane of points. 



l-|-oo2-|-oo2 lines — 1 line, axis of <»- spaces; «>- lines from 



point to plane of points; "- lines in plane of points. 

 1_)_ oo2_f_ cc2 planes — 1 plane containing "^ points; ^' planes 



through line axis; o»- planes from point to lines in plane of 



points. 

 1+ <»- spaces — 1 single space; 1 pencil of »- spaces through 



line axis of plane. 

 XX. Quadruple root, rank 1; 1 single root. 



l-l_ co3 points — 1 single point; 1 space of points. 



c»3_|_ oo4 lines— "2 lines from point to space; "' lines in a 



space. 



00 3 _j. 00 4 planes — "* planes through a point and lines of a 



space; »■* planes in a space. 



1 + oos spaces — 1 single space; =»' spaces through a point. 

 XXI. Quintuple root, rank 3. 



2 + so points — range; and 2 points of range. 



3 + 2" lines — 2 lines containing pencils of planes; 1 line, 



range of points; 2 pencils of lines in 2 planes. 



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

 axis of pencil of spaces; 2 pencils of planes through 2 lines. 



2 + <" spaces — «= spaces through a plane; and 2 spaces con- 

 taining pencils of planes. 

 XXII. Quintuple root, rank 3. 



1 range of points, and 1 point of same. 



2 + «• lines — 1 line, axis of pencil of planes; 1 line, range of 



points; 1 pencil of lines. 



2 + <» planes — 1 plane, axis of pencil of spaces; 1 plane con- 

 taining a pencil of lines; 1 pencil of planes. 



1 pencil of spaces and 1 space of same, containing pencil of 

 planes. 



