NEWSON : REAL COLLINEATIONS. 89 



C. In like manner eliminating t from (2) and (4), (1) and (2), and 

 (3) and (4) we have, respectively, 



(l-r)"- 



za ^==Cy, II 



x-iy=-Cz^ III 



rw 



ya^=Ox. IV 



Equations I, II, III give families of invariant cones whose vertices 

 are respectively at C, B, and A. Equation IV represents an invariant 

 family of ruled surfaces not conical. 



We have thus found four families of ruled surfaces which are in- 

 variant under all the collineations of the group hGi(ABCl). The 

 path curves of the group are the cc- common intersections of these 

 families of surfaces. 



Theorem 3. There are four distinct families of ruled surfaces 

 invariant under all the collineations of the group hGi(ABCl); three 

 of these are families of cones. The cc* curves of intersection of these 

 invariant surfaces are the path curves of the group. 



§2. Two-parameter Subgroups of hGslABCl). 



Tivo- parameter groups leaving invariant a family of surfaces. — 

 If r remains constant while a assumes in turn all real values between 

 and 00, and we have oc- one-parameter groups, all of whose trans- 

 formations leave invariant the family of cones given by equation III, 

 for the equation of this family of cones is independent of a. The path 

 curves of the a-J- one-parameter groups all lie on these cones. This 

 system of cc- collineations forms a two-parameter group hG2(ABCl)r; 

 the parameters of this group are a and t. The group hG3(ABCl) 

 contains oc^ two-parameter subgroups, one for each real value of r. 



In like manner if a is constant and rvaries, we get a two-parameter 

 group, leaving invariant the family of cones given by equation I. 

 Again, if a and r vary in such a manner that a^^"^'^ is a constant, we 

 get a two-parameter group leaving invariant the family of cones given 

 by equation II. Finally, if a and r vary so that a'' is constant, we 

 have a two-parameter group whose invariant family of surfaces is 

 given by equation IV. 



Thus we see that the group hG3(ABCl) contains four singly 

 infinite systems of two-parameter subgroups ; three of these systems 

 leave invariant families of cones, and one system leaves invariant a 

 family of ruled surfaces. 



Theorem 4. The three-parameter group hG3(ABCl) contains four 

 singly infinite systems of two-parameter subgroups ; these are given 

 by r = const.; a = const.; a^'^"=con8t., and a''=const. 



