312 THEORY OF COLLINEATIONS. 
transformed into another line also parallel to /’, but at an 
equal distance from / on the other side. Angles are unaltered 
in magnitude but reversed in sense, and linear magnitudes 
are reversed in sense but unaltered in length. The effect of 
such a collineation is to revolve the whole plane through two 
right angles about the line /’ as an axis, and then to slide the 
whole plane along /’. With the origin on / the collineation 
T’ is given by v7, = —x, andy,=y-+t. 
Let t=0 in the above collineation 7’ of type II; 7’ then 
reduces to an involutoric collineation of type IV. The effect 
of such a collineation is to revolve the whole plane through 
two right angles about /’ as an axis. 
THEOREM 29. The mixed group emG,(¢e’) contains, besides 
the continuous group eG@,(¢/), o4 hyperbolic collineations of type 
I, o# collineations of type II, and o? involutoric¢ collineations of type 
IV. All collineations interchanging @ and ©’ transform plane fig- 
ures into similar but non-congruent plane figures. 
374. The Group of Euclidian Motions. The group of 
Euclidian motions in the plane is a subgroup of the group of 
similarity G,(ow’). Every collineation in G,(o’) transforms 
all plane figures into similar and congruent figures and alters 
all linear magnitudes in a constant ratio R; shape is an inva- 
riant of the group G,(ow’), but size is not invariant. 
All collineations in G,(o’) for which R is unity form a 
subgroup of the group of similarity; all such transformations 
are common to the two groups G,(/.),-_, and G,(ow') and 
form a subgroup of each. All collineations of this subgroup 
conserve size as well as shape. Every plane figure is trans- 
formed into an equal and congruent plane figure. But such 
a collineation is evidently brought about by a rigid motion of 
the whole plane into itself. Hence the group of collineations 
of the plane which conserves the size, shape and congruity 
of every plane figure is the group of all Euclidian motions in 
the plane. 
375. The Group of Motions is G,(ow’),-_;. It was shown 
in Art. 370 that the path-curves of the one-parameter group 
