316 THEORY OF COLLINEATIONS. 
formation is to revolve the whole plane through 180° about 
the line of invariant points as an axis. 
THEUREM 31. ‘The mixed group of Euclidian motions contains, 
besides the continuous group e@;( £1), o* collineations of type II 
and <* collineations of type 1V; these collineations interchange « 
and ~ and transform all plane areas into equal but non-congruent 
plane areas. 
Exercises on Chapter 4. 
1. Verify synthetically and analytically the structural for- 
mulas of all perspective groups given in Art. 315. 
2. Show that the group G,’’(AIN) contains H,/(Al) asa 
subgroup. 
3. Verify synthetically and analytically the structural for- 
mulas of all groups of type II, first class, as given in Art. 317. 
4. Verify by both methods the structural formulas of all 
groups of type II, second class, as given in Art. 318. 
5. Verify the structure of all groups of type I, first class, 
as given in Art. 319. 
6. Verify by both methods the structure of all groups of 
type I, second class, as given in Art. 320. 
7. Verify by both methods the structure of all groups of 
type I, third class, as given in Art. 321. 
8. Show that group G,’(A/), contains singular transforma- 
tions of both types III and V; and the group G,(AA’), con- 
tains singular transformations of type III. 
9. Show that groups G,(AA’),, G,(ll’),, G,(Al), (when 
r is rational), each contain singular transformations of 
type II. 
10. Show that the group G,(A),-_, contains © singular 
transformations of type II. 
11. Show that the singular transformations of type II in 
