THE GENERAL LINEAR GROUP. S15 
It is evident geometrically that the resultant of any two trans- 
lations is a translation and hence that all translations of the 
plane form a group. A translation of the whole plane from 
O to P followed by a translation from P to P, is equivalent to 
a translation of the whole plane from O to P;. 
The two-parameter group of translations H.’(/..) breaks up 
into oo’ one-parameter subgroups of translations, one for 
each point on the line at infinity, 7. e., one for each direction 
in the plane. 
379. The Mixed Group mG,(EM). We found in Art. 373 
that the transformations in mG,(o.’) which interchange w and 
wo’ are of types I, I], and IV. All of these transformations 
which change areas into equal areas belong to the mixed 
group of Euclidian motions. These we now proceed to ex- 
amine. 
The two characteristic cross-ratios k and k’ of a hyperbolic 
transformation T of type I which interchange w and «’ satisfy 
the condition k-+k’=0; Art. 373. If T leaves all areas in- 
variant in magnitude but reversed in sense, / and k’ must 
satisfy the relation kk’ = —1; Art. 361. From these two con- 
ae k=1 [: Steet Bae ac 
ditions we have };,__, Or ,_, 3 in either case we see that 
the collineation is no longer of type I but of type IV, 7. e., it 
is a perspective collineation. Thus the mixed group mG,( HM) 
contains no hyperbolic collineations of type I. 
All collineations of type II in emG,(o0’) interchange w and 
w’ and transform all plane areas into equal but non-congruent 
plane areas. Consequently the ~* collineations of type II in- 
terchanging » and w’ belong to the mixed group mG,( HM). 
The effect of such a collineation, Art. 373, is to revolve the 
whole plane through an angle of 180° about some line of the 
plane as an axis and then to slide the whole plane along the 
axis. 
All transformations of type IV in mG, (ww’) evidently trans- 
form areas into equal but non-congruent areas and hence 
belong to the group mG,(HM). The effect of such a trans- 
