224 THEORY OF COLLINEATIONS. 
In the second case let ht = t’; substituting this value of t¢ 
in the equations of 7” we have 
Hial x 
il: 
1 1 i> Be 
a 
= t/2 t!. 
Yi yh yh i 
Now let both a and h approach © and let the limit of 2 
be A’, while t’ remains finite. Equations (57) reduce to 
—=— 4 A’t! 
Yi y 
Se oy (63 ) 
SS Sa St i! 
Yr y 
The collineation represented by these equations leaves in- 
variant every point on the «/-axis and every line through the 
point (A’0); it is therefore a perspective collineation of type 
V. Equations (16) represent the two-parameter group of 
elations H,’(l). 
THEOREM 48. The group G,/ (AZ) contains two two-parameter 
subgroups of type V, viz.: H./( A) and H,/ (7); thus @,’ ( ALY) for 
a=0= H,(A) and G,” ( ALN) for a= oa = Hy (2). 
269. Properties of the Groups G,'’( ALN) and G,'’(AlS). 
There are * conics touching the line J at the point A; and 
there are ~-/ circles touching / at A. Each one of these cir- 
cles is the circle of curvature of a system of ©* conics touch- 
ing / at A. The group G,”(AIN), for which a is constant, 
transforms into itself and thus leaves invariant the net N of 
coo* conics whose common radius of curvature is 2a. It con- 
tains the group of elations H,’( Al). . 
The group G,’’( AlS) whose parameter is ¢ and for which 
the law of combination of parameters is t, =¢t-+t,, is isomor- 
phic with the parabolic group G,’(A) of one-dimensional 
transformations. Its properties are therefore known and 
need not be restated in detail. 
