NEWSON : REAL COLLINEATIONS. 91 



one-dimensional transformations along AB and AC, respectively, are 

 identical, and there results in each case a two-parameter subgroup of 

 typeXIinhG3(ABCl). 



In terms of the parameters a, r and t the subgroup of type VIII 

 results when a=Gc; the subgroup of type IX results when r=0; the 

 two subgrovips of type IX result when r=l and cc respectively. 



Theorem 6. The group hG3(ABCl) contains one two-parameter 

 subgroup of type VIII, one of type IX, and two of type XI. 



Suhjroups of types VI, VII and X in hGz{ABCl). — If t = and 

 k = l, the transformation in the plane ABl is identical and leaves in- 

 variant all points in the plane. The corresponding collineations in 

 space are of type VI, C being the vertex and ABl the axial plane. 

 The remaining parameter k' gives us a one-parameter subgroup of 

 type VI in hG3(ABCl). In like manner, if t^=0 and k'=l, we have 

 a one-parameter subgroup of type VI whose vertex is B and whose 

 axial plane is ACl. 



If k=l and k'=l, the coUineation in the plane ABC is identical, 

 the parameter t gives us a one-parameter subgroup of type VII in 

 hG:;(ABCl); A being the vertex and ABC the axial plane. 



If t=^0 and k'= k, the one-dimensional transformations along AC 

 and BC are both identical ; there results a one-parameter subgroup of 

 type X in hG3(ABCl). 



In terms of a, r and t the subgroup of type VII is given by a=l ; 

 the two subgroups of type VI are given by t=0 and r = l, t=0 and 

 r= GO, respectively ; the subgroup of type X is given by t=0 and r=0. 



Theorem 7. The group hG3(ABCl) contains one one-parameter 

 subgroup of type VII, two of type VI, and one of type X. 



Other special subgroups of liGi{ABCl). — There are only three 

 other special subgroups of hG3(ABCl) to be noticed; these are 

 when the path curves in the plane ABC are conies. These path 

 curves are conies for three values of r, viz., r= — 1, 2, 1/2. When 

 r=2 the conies have double contact at B and C; when r= — 1 or 1/2 

 the conies have double contact at A and C, A and B, respectively. 

 These are two-parameter subgroups of hG3(ABCl). 



§5. The Elliptic Case eG3(ABCl). 



Parameters of eGi{ABCl). — In the elliptic subtype of type II, 

 where the i^oints B and C are conjugate imaginary, the theory is 

 somewhat ditferent from that of the hyperbolic subtype. In the 

 plane ABC the two-dimensional collineations of the elliptic sub- 

 type and the parameters are given in the form k^ exp.(c + i)^; thus 

 c and d are the parameters. The three parameters of eGs are there- 



