NEWSON: KEAL COLLINEATIONS. 39 



a point in space. It is evident, since these parameters are independ- 

 ent, that there is a point in space corresponding to every transforma- 

 tion of the group hGs (ABCD). All transformations whose 

 parameters satisfy the relations 



k'=k^-'- and k"=ki-'"+''^ 

 form a one-parameter subgroup of hGs (ABCD). Hence the curve 

 of intersection of the two cylinders whose equations are y=x^"^ and 

 z=:x^''+'^'* represents a one-parameter group hGi ( ABCD ) is and the in- 

 dividual points on the curve represent the individual transformations 

 of the group. If we give to r and s all real values we have a system 

 of Gc^ curves which represents the system of cc- one-parameter sub- 

 groups of liGs (ABCD). 



An examination of the equations y=x^''" and z=x^''+''^ shows that 

 one branch of the curve lies in the first octant for all values of r and 

 s; and if r is an irrational number, the curve lies wholly in the first 

 octant. If r and s are both rational, r with odd numerator and odd 

 denominator, s with even numerator and odd denominator, the curve 

 lies in the first and second octants. If r and s are both rational, r with 

 even numerator and odd denominator, s with odd numerator and even 

 denominator, the curve lies in the first and third octants. If r is ra- 

 tional with odd numerator and even denominator while s is irrational 

 or rational with odd numerator and odd denominator, the curve lies in 

 the first and fourth octants. If r is rational with odd numerator and 

 s rational with odd numerator and even denominator, the curve lies in 

 the first and fifth octants. If r and s are both rational, each with odd 

 numerator and odd denominator, the curve lies in the first and sixth 

 octants. If r and s are both rational, r with even numerator and odd 

 denominator, s with odd denominator, the curve lies in the first and 

 seventh octants. If r and s are both rational, r with odd numerator 

 and even denominator, s with even numerator and odd denominator, 

 the curve lies in the first and eighth octants. 



The curves of the family, y=x^"'" and z=x^''+'% contain every point 

 in the first octant, but not every point in the other seven octants. 

 Consequently the group hGs ( ABCD ) contains transformations which 

 are not included in any of its one-parameter subgroups. Such a 

 transformation has one or more of its cross-ratio parameters negative 

 and such that their values do not satisfy algebraic equations of the 

 form k'"=k""', k""=k', where 1, m and n are integers. 



Transformations commo7i to two or more one-parameter suhgroups 

 of hGi {ABCD). — In order to find all points common to any two 

 curves of the family representing the system of one-parameter sub- 

 groups of hG;j (ABCD), we solve the simultaneous system of equations 



V = X^'^ Z = X^"'' + '"; Vr=:x'"'', Z = X^"''+''''. 



