228 THEORY OF COLLINEATIONS. 
c,=0 and c,' =1 in equation as chapter II, the normal form 
of type V. We thus get 
: t+ Ati y 
Ss’: «4= and. y, = —— (65) 
1+ty 1+ty 
S,' is of the same form with different A and ¢. 
a + Avtiy: 
SHe. 4. = a anda (65’) 
1+ty 1+tyi 
Eliminating «#, and y, we get 
w+ Astey 
So! %= eT NiChO qt ee (65’’) 
1+ toy 1+ toy 
where t,=t+t, and A,t,= At+A,t,. The first group 
property is thus established analytically, the second is shown 
to exist by solving (65) for « and y; and again we see the 
existence of the group H,’(l). | 
273. The Two-parameter Group H,'(A). Let us consider 
two elations, S’ and S,’, having the same vertex A and dif- 
ferent axes through A. Fig. 28(b). Every line through A is 
invariant under both S’ and S,’ and hence is invariant under 
their resultant. Along each invariant line through A both 
S’ and S,’ set up one-dimensional parabolic transformations 
with a common invariant point A. The resultant along each 
invariant line is therefore parabolic and hence the resultant 
of S’ and S,’ is again an elation with vertex at A. Thus the 
first group property is proved. Since the inverse of each 
elation is in the same one-parameter group, the second group 
property follows and we have established the existence of the 
two-parameter group of elations having a common vertex. 
The convenient symbol for the group is H,'(A). 
274. Resultant of Any Two Elations. Let S’(Al) and 
S,'(A,l,) be any two elations whose invariant figures are in 
the most general position with respect to each other. Let / 
and J, intersect in the point O and let g be the line joining A 
and A’, Fig. 27(c). Both S’ and S,’ leave invariant O and g, 
hence their resultant leaves both O and g invariant. Along 
the line g we have two one-dimensional parabolic transforma- 
