A NEW THEORY OF COLLINEATIONS IN SPACE, III. 

 Collineations of Type V in Space. 



BY H. B. NEWSON. 



A. — Synthetic Forecast. 



Invariant figure of T. — The fundamental invariant figure of a 

 collineation in space of type V consists* of a plane p, a line 1, and a 

 point A ; A and 1 are both in p and A is on 1. Let this invariant 

 figure be denoted by (Apl), and let T be a collineation of type V, 

 leaving (Apl) invariant. The collineation T and its invariant figure 

 (Apl) are both self-dualistic. Along the line 1 and in the pencil of 

 planes through 1, T produces a one-dimensional parabolic transforma- 

 tion ; in the plane p and in the bundle of rays through A, T produces 

 two-dimensional transformations of type III. 



The group GeiApl). — The two-dimensional transformations of 

 type III in p, leaving the lineal element Al invariant, are oc^ in num- 

 ber and form a three-parameter group with one- and two-parameter 

 subgroups. This three-parameter grouj) leaves invariant the system 

 of Qo'^ conies touching 1 at A; a two-parameter subgroup leaves in- 

 variant a net of oo^ of these conies having three points in common at 

 A ; a one-parameter subgroup leaves invariant each of a pencil of cc} 

 conies having four points in common at A. 



Dualistically the two-dimensional transformations of type III in 

 the bundle of rays through A are oc^ in number, and form a three- 

 parameter group with one- and two-parameter subgroups, leaving in- 

 variant the system of ccP quadric cones having their vertices at A and 

 touching p along the line 1; a two-parameter subgroup leaves in- 

 variant a net of oc^ of these cones having three elements in common 

 along 1 ; a one-parameter subgroup leaves invariant each of a pencil 

 of Qc^ cones having four elements in common along A. 



These two two-dimensional three-parameter groups are independent 

 of each other, and hence the three-dimensional transformations of type 

 V, leaving (Apl) invariant, depend upon six parameters; /. e., they 

 are crJ' in number, and form a six-parameter group G6(Apl). 



Theorem 1. There are cc*^ collineations of type V in space leaving 

 the figure (Apl) invariant ; these form a six-parameter group Gg(ApI). 



♦Kansas University Quarterly, Vol. IX, p. 70. 



[99]-K.D.Qr.-A x 3-July, '01. 



