314 THEORY OF COLLINEATIONS. 
centric circles about the point A. The collineations of this 
group are therefore rotations of the whole plane about the 
point A. It is evident that all rotations of the plane about a 
fixed point A form a one-parameter group. A rotation about 
A through an angle # combined with another about the same 
point through an angle 6, results in a rotation about the same 
point through an angle 6,=6-+06,. The characteristic cross- 
ratios of the two component collineations are respectively 
e*’ and e*”; the cross-ratio of the resultant is e’” =e: ei = 
e(@+h) , 
The path-curves of a one-parameter group of motions of 
type V are straight lines meeting at A, a point on the line at 
infinity; 7. e., they are parallel lines all in the direction of A. 
The collineations of this one-parameter group are therefore 
translations of the whole plane in the direction of A. It is 
evident that all translations of the plane in a given direction 
form a group; a translation in a given direction through a 
distance t combined with another translation in the same di- 
rection through a distance t,, results in a translation in the 
same direction through the distance t, = t+ t,. 
THEOREM 30. Thegroup of Euclidian motions Gs(¢e’),-_,; con- 
tains only collineations of types I and V; the former are Rotations 
of the whole plane about a point, the latter are Translations of the 
whole plane in a fixed direction. 
378. Subgroups of the Group of Motions. The group of 
motions contains ©’ distinct rotations and ? distinct trans- 
lations. All the rotations about a point form a group of one- 
parameter and there are * such points in the plane; hence 
all the rotations of the plane naturally fall into ©” one-pa- 
rameter subgroups of rotations, one subgroup for each finite 
point in the plane. These one-parameter groups of rotations 
do not combine to form two-parameter groups of rotations, 
for the resultant of two rotations is not always a rotation 
but is sometimes a translation. 
The * translations of the plane form a two-parameter 
subgroup of the group of motions. This is the group H,’(I..). 
