REAL GROUPS. 259 
k! pe v6 
k per? 
9=0, k” =1 and the transformation along A’ A” is identical. 
Hence the subgroup of eG,(AA’A”’)-= ~ isa group H,(Al’) 
of real perspective collineations. 
THEOREM 62. The real elliptic group eG.( A A’ A”) contains 
one subgroup whose path-curves are conics and one whose path- 
curves are straight lines. 
is elliptic.. Since k= hk" k= =e 24") “When 
313. Real Groups of Type I. Having determined two va- 
rieties of fundamental groups of type I we go on to enumerate 
the groups of higher order that can be compounded out of 
these. 
The invariant figures (A), (1), (Al), (AA’), (l’), (A,U”), 
(AA‘l’) must be examined separately. The real groups 
G,(A), G,(1) and G,(A,/’’) contain both hyperbolic and ellip- 
tic collineations. The groups G,( Al) and G,(AA‘l’) contain 
hyperbolic, but no elliptic, collineations of type I. There are 
two varieties of groups leaving two points invariant, viz.: 
hG,(AA’) and eG,(A’A”). In the first case the two points 
A and A’ are real and the group h G,(AA’) contains only hy- 
perbolic collineations of type I (and lower types). In the 
second case the points A’ and A” are conjugate imaginary 
points and eG,(A’A’’) contains only elliptic collineations of 
type I. In like manner we have two varieties of groups leav- 
ing a pair of lines invariant, viz.: h G,(ll’) and eG, (Il’). 
Kvidently the groups h G,( AA’) and hG,(ll’) break up into 
subgroups of the second class for a constant 7, just as 
hG,(AA’A’’) does. Also the groups e G,(A’A”) and eG, (ll’) 
break up into subgroups of the second class for a constant c 
just as eG,(AA’A’’) does. 
The real groups of the third class G,(A),__,, G,(1),__, and 
G,(A,l’’),__,, contain both hyperbolic and elliptic collineations 
of type I. The real groups G,( AIK) and G,( AlS) contain 
only hyperbolic collineations of type I. There are two varie- 
ties of real three-parameter groups leaving a conic K inva- 
