206 THEORY OF COLLINEATIONS. 
type III is G,’ (Al), Art. 227; that of type IV is H,(A,l), 
Art. 228; that of type V is H,/( Al), Art. 229. 
The fundamental groups of types I, II, and III each have 
more than one parameter and break up into one-parameter 
subgroups. The fundamental group of types IV and V are 
one-parameter groups. Each one-parameter group of 
collineations leaves invariant a family of curves called the 
Path-Curves of the group. The property of these path-curves 
will also be investigated in the present section. 
The efficient instrument for the investigation of these fun- 
damental groups, and in fact of any subgroup of G,, is the 
normal form of the collineation JT. We shall, therefore, in 
this and in $$ 8 and 9, make constant use of the normal forms 
of the various types of collineations. 
A. FUNDAMENTAL GROUPS OF TYPES IV AND V. 
244. Fundamental Group of Type IV. A perspective col- 
lineation S of type IV leaves invariant a point A (the vertex), 
and a line / (the axis) not through A, all lines of the plane 
through A and all points on the line /, Fig. 14, IV. It is fur- 
ther characterized by a constant, k, the cross-ratio of the one- 
dimensional transformations along each of the invariant lines 
through A. The equations of S may be reduced to the ca- 
nonical form, Art. 151, 
Pa, = kx 
St py = ky (50) 
Pz) = 2. 
Since / may have any value whatever it follows that there 
isa set of / perspective collineations leaving the same figure 
invariant. Let S, be a second collineation of the same set. 
The equations of S, may be written 
1% = ky a 
Sea ea (50’) 
Eliminating (7, y,z,) from the equations of S and S, we get 
the equations of their resultant S, in the form 
Po %. = Ko xy 
S22 psy: = key, Where k, = kk,. (50) 
