220 THEORY OF COLLINEATIONS. 
meter group, G,'/’(AlS). The group G,’( Al) contains ~, 
such subgroups, one for each pair of values of aandh; the 
group G,’’( AlN) contains ~’ such subgroups, one for each 
value of h. 
THEOREM 44. Each two-parameter group G@,” ( Al ) contains 
oo?! one-parameter subgroups @,” ( A/S) , one for each value of A. 
265. Path-curves of G,'/(AlS). The path-curves of the 
one-parameter group G,’’( AlS) are found by eliminating the 
parameter ¢t from the equations of the group: 
x1 
= at 
yi y 
/} ° 
ince Sn) Ve Pane (57) 
— =—4+t— +a@+ht. 
1 y y 
Eliminating t we get 
1 22 ha, ml a hz = 
yi fay? ay, y fay? 2ay, — 
or w+ 2hey+4aCy?=t4ay. (60) 
The path-curves of the group G,’’( AIS) are therefore a family 
of conic sections. 
From the equations of the family of conics we see that they 
all pass through the origin A and have the line / or y = 0, for 
a tangent. Two conics of the family have no point of inter- 
section except the origin; in fact the conics all have contact 
of the third order at the origin. They therefore form a pen- 
cil S of conics through four coincident points. Fig. 25. 
THEOREM 45. The path-eurves of the group @,” ( A/S) are the 
conics of a pencil S having contact of the third order at the invari- 
ant point A. 
266. Geometric Meaning of the Constant a. It was shown 
in Chapter II, Art. 139, that a collineation of type III is the 
limiting form of a collineation of type I, when the invariant 
triangle (AA’A”’) shrinks to a point. In this case in the 
equation k’ =k” we have r= 2 and the path-curves of the 
group G,(AA’A”’), are conics having double contact at A and 
