208 THEORY OF COLLINEATIONS. 
another elation of the same set be S,’ having the constant t,. 
The equations of S,’, are 
(HO eae by) este (51’) 
1+he@,’ Ae he 
Eliminating x, and y, we get the equations of the resultant 
S,’ as follows: 
S! i %®=—— —— (51’) 
tite? 9s Tie? 
where t,=t+t,. Thus S,’ belongs to the same set as S’ and 
S,’.. The inverse of S’ is found by solving the equations of S’ 
for x and y. Thus 
Sates = v1 eee (57) 
L—ta,? 
The inverse of S’ is also in the set; both group properties 
are satisfied and hence the set is a group designated by 
H/(Al). It is a one-parameter group, ¢t being the para- 
meter. 
THEOREM 31. The fundamental group of elations in the plane 
is a one-parameter group, whose parameter is ¢. 
247. Properties of the Group H,'( Al). The group H,/(Al) 
of elations in the plane and the group G,’(A) of one-dimen- 
sional parabolic transformations, Art. 29, are holoedrically 
isomorphic. They each contain a parameter t, which com- 
bines according to the parabolic law, t+ t,=t,. Hence the 
properties of H,’'( Al) need not be discussed in detail but may 
be inferred at once from the properties of G,'(A), stated in 
Chap. I, Art. 29. 
B. FUNDAMENTAL GROUP OF TYPE I AND ITS SUBGROUPS. 
248. Fundamental Group of Type I. It was shown in 
equation (21) Chap. II, Art. 148, that, when the invariant 
triangle of a collineation of type I is taken as the triangle of 
reference, the equation of a collineation T reduces to the 
form: 
con—ieaes 
T: sm=ky, (52) 
S21 =2, 
