GROUP STRUCTURES. 269 
of A’and A,’ all elations in the group H,’(l’) may be pro- 
duced. 
Analytic Method. Let k=a‘, t=0 and A’=0 but lim 
= = in equations (4) G,/( Al’) Article 317; these reduce to 
y+ Bax (5) 
and Y= 1+t a ~ 
Are x 
Ti cite 
These equations represent the group of elations //,’(/). 
In like manner the structural formule for the remaining 
groups of this class may be verified. 
319. Structure of Groups of Type I. First Class. The 
nine groups of type I, first class, show the following struc- 
tural formule : 
G,(AA') = G:(AA’A”) +H, (Al) + (AV) +H (A), 
G3 (AA!) = 01G (A) + G.! (Al) + Hp (ll) + He(A) +i () + Hy (AD, 
Gi(A) =0%G,(A) +00! G! (AAV) +H; (L) + He! (1) +282 (A), 
Gi (ll) =02G,(A)+200'G, (Al) + H(A) + He! (A) +2H2 (Wl’), 
Gy (A, Ul’) = 0? G2 (A) +00'G,! ( Al’) + 001 Hp (A) + Mi (A, 0”) 
+001 Hy!(t), 
G;(Al) =03G,(A) +02Gs! (Al!) + 02G2! (AV) + Gs! (AL) + Ha (1) 
+ H3(A) + He! (1) + Hy! (A), 
Go(l) . =04G,(A) +02G, (AU) +003 Gy! (Al) + 01G," (Al) 
+H; (1) +00? Ha (A) + Hy! (1) +00" Hy! (A), 
Go(A) =04G,(A) +03G,/ (Al) +03G)' (Al) + 01G; (Al) 
+ H,(A) +002 Hp (ll) + H./(A) +00! Hy’ (1), 
Gs = 0 §G, (A) +005 G,! (Al) +002G,"( Al) + 04H; (A, 1”) 
+03 Hy (Al). 
The verification of these structural formule presents no 
special difficulties. 
320. Second Class. The structural formule of the four 
following groups are exhibited thus: 
G,(AA'l’)r = 01G,(AA'A”’)r+ H,(Al)+S. T., 
G(AA')r = 0*G,(AA’A")r+ H(t) +S.T., 
G(r = 0*G (AAA) r+ H(A) +S. T., 
G(Al)r =@'G,(AA’A”)r+H(A) +H, (1) 
+ G,/"(Al) +S. T. 
Synthetic Method. From the group G,(AA’), take two 
collineations T(A A’A’’), and T,(AA’A,'’),, where A’ and 
