FUNDAMENTAL GROUPS. 207 
Hence S, belongs to the same set as S and S, and the set has 
the first group property. 
The inverse of S is found by solving the equations of S for 
«andy. Thus 
63a =hkl x, 
Se sy=ky, 
SZ — <1 - 
Hence the inverse of S is also in the set and the set possesses 
the second group property. The set therefore forms a group 
designated by H,(A,1); it is a one-parameter group, the 
cross-ratio k being the parameter of the group. 
THEOREM 30. The fundamental group of type [IV consists of all 
collineations of type 1V having the same vertex and axis: it isa 
one-parameter group whose parameter is the cross-ratio 4. 
245. Properties of the Group H,(A,l). This group 
H,(A,1) has essentially the same properties as the one-para- 
meter group G,(A’A) of our one-dimensional transformations 
of points on a line (Chap. I, Art. 27). It is needless to repeat 
the statement of those properties. In both groups the law of 
combination of parameters is k,= kk, and the parameters 
vary in precisely the same way. The transformations of the 
two groups G,(A’A) and H,(A,/) have a one-to-one corre- 
spondence and because of this property are said to be holoe- 
drically isomorphic. 
246. Fundamental Group of Type V. A perspective col- 
lineation S’ of type V leaves invariant a point A (the vertex), 
a line / (the axis) through A, every line of the plane through 
A and every point on the line /, Fig. 14, V. Along every in- 
variant line through A there is a one-dimensional parabolic 
transformation having only one invariant point, viz., A. 
The canonical form of S’ may be written (Art. 152): 
Roo ag ee ee es ' 
ee (51) 
where the vertex A is the origin and the axis / is the #-axis. 
Since t may have any value whatever, we see that there is a 
set of ~’ elations all having the same invariant figure. Let 
