FUNDAMENTAL GROUPS. AD 
the group G,(A’AA”),. The infinitesimal collineation of the 
group 
Px=x+dou, 
PYi=Ytrdy, 
P2a=Z, 
moves P to P,; applied again it moves P, to P,; applied an 
infinite number of times it moves P along a certain curve in 
the plane called a path-curve. There are ‘of these path- 
curves so situated that every point in the plane lies on one 
and only one of them. The equation of this family of path- 
curves is found by eliminating k from the equations of the 
group, viz.: 
he hee Set (52) 
21 z 21 
Eliminating k we get 
ae me — ame ee — Ol ae ar tO (53) 
Thus the path-curves of the group G,(AA’A”)r are curves of 
order r and form a family whose parameter is C. Fig. 24. 
THEOREM 34. When the invariant triangle is taken as the tri- 
angle of reference, the family of path-curves of a one-parameter 
group G,; (AA’A”’), is given by x” 2~-" =Cy, where r is a constant. 
252. Geometric Meaning of r. It is not difficult to deter- 
mine, from the equation of the family of path-curves, the 
geometric meaning of 7. Take any point P=(w,, y,,z,) and 
draw a tangent at P to the path-curve through Pand join P 
to the vertices of the triangle (AA’A”). The equation of the 
tangent to the path-curve at P is 
EA te ( Cr) ye, — Us,e,1-2 (1 Or) — 0. 
The lines PA, PA’, and PA” are given by the equations 
PAS Sys — Cy 0. 
Aye 0 
PAL Ye. — 26, — O- 
For C= 1, the cross-ratio of the pencil P(TAA’A”) is found 
to be r. It is evident that this cross-ratio r is a constant for 
every point on every path-curve of the group G,(AA’A”),. 
