104 KAMSAS UNIVERSITY QUARTERLY. 



groups; the path curves of one of these subgroups are twisted cubics; 

 each subgroup leaves invariant a family of quadric cones, a family of 

 cubic cones, and a family of quartic surfaces. 



§3. Other Subgroups of GelApl). 



Five-parameter subgroups of G(,{ApV). — If m be kept fixed and 

 the other five parameters be allowed to vary, the resulting collinea- 

 tions form a five-parameter group GG(Apl)m. If we make mi = m a 

 constant in equations (7)-(12), we find also m2 = m,and equation (9) 

 is no longer independent. The remaining five equations show the five- 

 parameter group. There are co^ such subgroups in G6(Apl), one for 

 each real value of m. The group G5(Apl)m leaves invariant a net of 

 Go'-^ conies in the plane p. 



In exactly the same way it may be shown that when ni = n also 

 n9 = n, and we have another singly infinite system of five-parameter 

 subgroups G;-,(Apl)n. The group Gf,(Apl)n leaves invariant a net of 

 00- quadric cones with their vertices at A. 



Four-parameter subgroups of G&{Apl). — If n and h are both con- 

 stant while the other four parameters vary, the remaining cc^ collinea- 

 tions form a four- parameter subgroup of GeCApl). This is shown by 

 the vanishing of equations (8) and (10); the remaining four show a 

 four-parameter subgrouj^. There is a doubly infinite system of these 

 four-parameter groups, one for each value of n and h. One of these 

 groups, G4(Apl)nh, leaves invariant a singly infinite system of 

 quadric cones contained in I. 



If ra and k are both constant while the other four parameters 

 vary, the remaining oo* collineations form a four parameter group 

 G4(Apl)mk. This is shown by the vanishing of equations (9) and 

 (11); the remaining four equations show the four-parameter group'. 

 There is a doubly infinite system of these four-parameter subgroups, 

 one for each value of m and k. The group G4(Apl)mk leaves in- 

 variant a pencil of conies in the plane p. 



If m and n are both constant while h, k, g, t vary independently, we 

 have another four-parameter subgroup of G6(Apl). The six equa- 

 tions (7)-(12) reduce to 



t + ti = to, ht + hiti = hats, kt + kiti = kst-i, and 



(hm + kn)t'+ 2(h,in + k,n)tt, + (h,m + k,n)t= , 4. , + h.m + k^n. 2 . ^l 

 a H gt + glti = ^ U + g2to. 



The group G4(Apl)mn leaves invariant a net of oo- conies in p and 

 a net of oo^ cones through A. There is a doubly infinite system of 

 these four-parameter subgroujDS, one for each value of m and n. 



Three-parameter sxibgroups of Ge{Apl). — There are two triply in- 

 finite systems of three-parameter subgroups of G6(Apl); one of these 



