FUNDAMENTAL GROUPS. 223 
This is independent of C and hence the polar of A’ with re- 
spect to each conic of the pencil is the same straight line 
through the origin. 
If A’ is the point at infinity on y= 0, its polar is the com- 
mon diameter of the pencil of conics. The equation of the 
common diameter, or line of centers, is found by making 
x’= a. We thus get for the line of centers, 
tate — 0). 
Hence h is the negative cotangent of the angle which the line 
of centers makes with the invariant line y = 0. 
THEOREM 47. The constant 4 in the group G,”( A/S) is the 
negative reciprocal of the slope of the line of centers of the pencil of 
invariant conies of G,”( AlS) . 
268. Special Subgroups of G,/’( Al). The fundamental 
group of type III G,”’( Al) contains two two-parameter sub- 
groups which require special attention, viz.: when a= 0 and 
when a@ = o, 
First let a = 0 in equation (57); we then have 
a 1 1 x 
Te = 7aand Sigieia aes (62) 
1 £ 
Yi 
y 
1+ta+hty ° 
Ores and. 4, 
x 
1+ tx, +hty 
These equations show that all lines through the origin are 
invariant lines and all points on the line «+hy=0 are inva- 
riant points. The collineation 7’ reduces in this case to S’, 
a perspective collineation of type II. The system of equa- 
tions (58) reduce to 
je Pe 
het; = hoe h ft. 2) 
The group G,’’( ALIN) for a = 0 is therefore the two-parame- 
ter group of elations H,’(A). 
