NEWSON : REAL COLLINEATIONS. 97 



cc^ one-parameter subgroups. To show this, let k=a' and let a, n and 

 h be constants. There are oc^ coUineations in the group G4(ABlp) 

 which satisfy these conditions. In the plane p the plane coUinea- 

 tions form a one-pararaeter group of type III, and in the plane ABl 

 they form a one-parameter group of type II. t is the single inde- 

 pendent variable parameter. These cc^ coUineations in space evidently 

 form a one-parameter group. There are cc^ such groups in G4( ABlp), 

 one for each positive value of a and each real value of n and 

 h. Such a one-parameter group is designated by Gi(ABlp)anh. 

 The properties of the group Gi(ABlp)anh are those of a one-para- 

 meter parabolic group in one dimension. 



Theorem 17. The group Gi(ABlp) contains cx^ one-parameter 

 subgroups Gi(ABlp)anh; for each subgroup a, n and h are con- 

 stants and t is the variable parameter. 



Invariant curves and surfaces of Gi{ABlj>)anh.— The systems 

 of invariant surfaces whose intersections are the path curves of the 

 group Gi(ABlp)anh are determined as follows: Let (ABCD), where 

 C is on 1 and D in the plane p, be the tetrahedron of reference. 

 and let T be a coUineation of the group Gi(ABlp) which transforms 

 the point P= ( x, y, z, w) to Pi=( xi, yi, zi, wi). From the properties 

 of the invariant figure we easily obtain the following equations of 

 transformation : 



i7-7 = t. (2) 



ii = f+ntj + ft'+ht. (3) 



By eliminating t from these equations taken two and two we ob- 

 tain the following : 



x = Ca"^z, I 



X XX 



Z_n?^^ + ^!-^^— h|^^=C, II 



z log- a z ' 2 lug- a log a ' 



fy'-^-f hyz — wz = Czl III 



These equations represent "the invariant families of surfaces whose 

 intersections are the path curves of the group. Equation III is a 

 system of quadric cones with vertices at B and the line AB as a com- 

 mon element. 



Theorem 18. There are three distinct families of ruled surfaces in- 

 variant under all the coUineations of the group Gi(ABlp)anh. Two 

 of these families are transcendental surfaces and one is a family of 

 quadric cones. 



Two- and three-parameter suhgronps of Gi{ABlp). — If a is con- 



