28 TWo- AND THREE-PARAMETER GROUPS. 
33. The Three-Parameter G,. It was shown (theorem 6) 
that the resultant of T and T,, any two projective transforma- 
tions of the points on a line, is again a projective trans- 
formation; also, (art. 4.) that the inverse of every such 
transformation is a projective transformation. From this we 
infer that all projective transformations of the points on a 
line form a group. This is called the general projective group 
G,. It isa group of three parameters ; for the equation of T 
contains three independent parameters, viz.,a@:b:c:d. If 
these coefficients, a, b, c, d, be made to vary continuously, all 
the resulting transformations belong to the group G,; and 
conversely all transformations belonging to the above group 
are obtained by continuously varying the coefficients in T. 
Such a group is evidently continuous. If the equation of T 
be put into the normal form, 
a4 = ho (14) 
the three natural parameters, A, A’, k, may be made to vary 
continuously, thus generating the group G,. The group G, 
contains ©! two-parameter groups G,( A), one for each point 
on the line. It contains, as we have already shown, ~’ groups 
G,(AA’) and «/ groups G,/(A). 
34. The Mixed Group mG,(AA’). The one-parameter 
continuous group, G,(AA’), is made up of transformations, 
each of which leaves the points A and A’ separately invari- 
ant. The points A and A’ may be interchanged by certain 
transformations of the points on the line. The aggregate of 
all transformations, which leave the pair of points AA’ inva- 
riant, either separately or by interchanging them, is called 
the mixed group, mG,(AA’). 
The only transformation of the points on a line interchang- 
ing a pair of points is an involutoric transformation. Let the 
four points A’, A, P, Q form a harmonic range and let T be 
the involutoric transformation of the group G,(PQ). T will 
interchange 7A’ and A. Since there are ~’ pairs of points 
