pond: collineations in four dimensions. 



259 



XXIII. Qiiinluple root, rank 3. 



1 range of points. 



1 + "o + "0- lines — 1 line, range of points; each point of this 

 range is the vertex of a plane pencil of lines. Each line of 

 1 pencil is the axis of a pencil of planes. 



! + '«' + "■- planes — I plane, axis of pencil of spaces; 1 pencil 

 of planes through line, range of points; "^ planes through 

 the lines of one pencil of lines; 1 pencil of spaces. 



XXIV. Quintuple root, rank 2. 



1 + »- points — ■»- points in plane and 1 point of same. 



1 -I- co2 -)- oo3 lines — I line, axis of pencil of planes; "- lines in 



plane; <x'-' lines from point where line cuts plane. 

 1 + °°'- + =^-' planes — 1 plane of points; »=2 planes through 



line; »°^ planes through lines in plane and lines radiating 



from point. 

 1 + >»- spaces. 

 Quintuple root, rank 2 — same as XXIV, except that line of axis 



of ™- planes lies in the plane of points. 

 Quintuple root, rank 1. 



1 -|- co3 points — 1 space of points and 1 point of same. 



oo4 -)- to3 lines— =0' lines in the space, »»3 lines radiating from 



point in the space. 

 00* -(- 003 planes— «>* planes determined by lines radiating from 



the space and lines in the space; » planes in the space. 

 1 + "3 spaces. 

 XXVII. Matrix, rank o. 



Llentical collineation leaving all points of the R^ invariant. 



XXV. 



XXVI. 



■1— Univ. Sci. Bull.. Vol. VII. No. H. 



