266 THEORY OF COLLINEATIONS. 
iS : 47 = ON 7] = ky ° 
FE ng a eh ine 1).), 0? Seat A oe Wich y pla ete a) armen 2g) 
eS Sn Tt ot Regs oe RyeN Sh 
lal) ket | (a= 
whence k, = kk, and B. = = ae aan Now let k, =}, 
; : jl=al il 1 
in these equations; we have k, = 1 and Die ae eee } 
j Sys Meat 1 
hence B,=0, Putting lim ae =t we find t=(1-’) (2-5). 
Hence S, reduces to 
beet 1% ere UE. 
Ue Th epi? way ieege 
and is therefore an elation. Since t may be made to assume 
all values by varying K or B and B,, all elations of the group 
H,'(A‘l) are contained in H,(ll'). The same method may be 
used to verify the structural formule of the other perspective 
groups. 
316. Structure of Groups of Type III. We found only 
three varieties of groups of type III, viz.: G,’’( AlS), G,/’(ALN), 
G,’’ (Al); these cases. are easily disposed of. The group 
G,’( ALS) contains only collineations of type II]. The group 
G,/’( ALN) contains the group H,’ (Al) as a subgroup (see ex- 
ercise 14 at the end of the preceding chapter). The group 
G,’( Al) contains, as we saw in Article 268, the groups 
H,/(A) and H,/(l) as subgroups. The structural formule 
of the groups of type III are shown as follows: 
G,! (ALN) = ~'G,(AlS) + H,'(Al). 
G,' (Al) eaG CALS) a. (Aaya ra uChye 
317. Structure of Groups of Type II. First Class. There 
are six varieties of groups of this class; we have found that 
the group G,’(AA'l’) contains the groups H, (A’, l’) and H,( Al) 
as subgroups; and hence we must expect that subgroups of 
types IV and V will appear in the groups of higher orders of 
this class. The structural formule of these groups are as 
follows :. 
