210 THEORY OF COLLINEATIONS. 
T is now characterized by two constants, k and r, and the 
group G,(AA’A’’) has the two parameters, k and r. 
Let r be fixed and let k alone vary. In this way we select 
from the group G,(AA’A”) aset of *collineations. We 
wish to show that this set forms a one-parameter group. 
To show this take two collineations 7 and 7’, from the set 
characterized respectively by the constants (k,7) and (k,, 7) 
and find their resultant. From equations (52’’) we have 
ke=tKkipand ky hk! 3 settine. b= andale— i ainetnese 
equations we get k,= kk, and k,!=k"k7 =(kk,)"=k,. Thus 
the resultant 7, is characterized by the constants (k.,7). 
Hence the resultant belongs also to the set and the first 
group property is established for the set. The inverse of 
T(k,v) is T(k',r); this also belongs to the set and the set 
is therefore made up of pairs of inverse collineations. Both 
group properties are therefore established and the set is a one- 
parameter group. This one-parameter group G,(AA’A”), is 
a subgroup of G,(AA’A”). The group G,(AA’A”) contains 
coo such subgroups, one for each value of 7. 
THEOREM 33. The fundamental group of type I, G,( 4 4/4”), 
contains o! one-parameter subgroups @,( A A’ A”),; each subgroup 
is characterized by a constant 7, and its variable parameter is &. 
250. Properties of G,(AA’A”’),. This one-parameter group 
has properties very similar to those of the group G,( AA’) of 
one-dimensional transformations discussed in Chap. I, Art. 
27. The identical collineation is in the group and is given by 
k= 1, the infinitesimal collineation of the group is given by 
k=1+6, Art. 27, where 6 is infinitesimally near to zero. 
Corresponding to k= 0 and k= there are two pseudo-col- 
lineations. The group is, properly speaking, discontinuous 
for these values of parameter k. It is continuous for all 
values of k except k=0 and k= —, 
251. Path-curves of G,(AA'A’’),. We wish to investigate 
the effect on a point P of the plane of all the collineations of 
