FUNDAMENTAL GROUPS. PAT 
all pass through the points A and A’ and have these points as 
singular points. 
THEOREM 41. The family of path-curves of the group @//(44/Z)q 
A=z 
are transcendental curvesand are given by the equationz=Cya 44. 
260. Two Special Subgroups of G,/(AA‘l). Among the 
co subgroups of G,’/(AA’l), two are of special importance 
and require attention. When a=1, we have k=1t=1. 
This signifies that the one-dimensional transformations along 
AA’ and through A are both identical transformations, and 
hence all points on / and all lines through A are invariant 
under all collineations of the group. The one-dimensional 
transformation in the pencil through A’ is parabolic and hence 
along all the invariant lines through A the one-dimensional 
transformations are also parabolic. This particular subgroup 
of G,’(AA'L) is therefore a group of elations H,/( Al’) (l’ 
being the line AA’). When t= 0and k alone varies (this is 
equivalent to a= ©), the one-dimensional transformations 
along / and through A’ are both identical. In this case all 
points on / and all lines through A’ are invariant under all 
collineations of the group. The group G,/(AA’l),_. is, 
therefore, the group of perspective collineations of type IV 
ELE (GACT): 
A-«% 
The equations of the family of path-curves x=Cya 44 
reduces to x = Cy for a= 1 and to y= C/(A — 2) fora= 0. 
THEOREM 42. The two-parameter group G./( 44/1) contains 
one subgroup of collineations of type V, viz.: Hy ( Al’) for a = 1; 
and one subgroup of collineations of type LV, viz.: H,( A’l) when 
Ci— Con. 
261. Properties of the Group G,'(AA'l),. The parameter 
of the group G,’(AA’/), is t and the law of combination of 
parameters is t, =¢-+t,. Consequently the properties of the 
group are quite similar to those of the parabolic group of one- 
dimensional transformations G,’( A) (Chap. 1, Art. 29). 
