FUNDAMENTAL GROUPS. 213 
Fic. 24. 
of the invariant triangle are common tangents and the third 
side is the chord of contact. 
THEOREM 57. The group G2( A 4’ A”) contains three special 
subgroups for which the path-curves are conies having double con- 
tact; these are the subgruups for which 7= — 7, 2, 7/2. 
255. Rational and Irrational Values of r. The subgroups 
of G,(AA’A”) for which r is rational have important prop- 
erties not possessed by the subgroups for which 7 is irrational. 
The equation of the family of path-curves is 
ie —3(O) (53 ) 
These curves are algebraic or transcendental according as r 
is a rational or irrational number. The vertices of the inva- 
riant triangle are singular points on all curves of the family 
when 7 is irrational; one vertex is a singular point when r 
