GROUPS OF TyPEs II, III. 251 
Making B’=0 and B,)=0 in the conditional equations of 
G,(AlL) we get 
Gyn, Omebacee, oe 
kek = tAN ak i=2) 
Ee a) espe 
The same normal form and conditional equation may be ob- 
tained by making c = 0, c’ = 1, c, =0 and ¢,' = 1, in the nor- 
mal form of the group G,/( Al’) and its conditional equations. 
306. The Group G,/(AA‘l). If we make A,’ = A’ in the 
normal form and conditional equations of the group G,’ (Il’), 
we fix the point A’ on the w-axis and obtain the normal form 
and conditional equations of G,/(AA‘l’), the fundamental 
group of type II. The normal form remains the same as in 
G,'(ll) and the conditional equations become 
Cl) ee hia, Ce et (85 ) 
The same results may be obtained by making c=0, ¢’/=0, 
¢,=1 and ¢,/=1, in the normal form and conditional equa- 
tions of G,/( AA’). 
We have now verified the list of all varieties of groups of 
type II that can be compounded out of the ©’ two-parameter 
groups G,/(AA'l’). 
THEOREM 57. There are six varieties of groups of the first class 
of type II, viz.: G.’( AA), Gl(4A’), Gy (Ul), G,/ (Al), Gs (AV) 
and G,/ (Al). 
307. Groups of the Second Class. It was shown in Art. 
258, that the group G.’( AA’l’) breaks up into one-parameter 
groups G,’(AA’l’), when we put k = a‘ and keep a constant. 
In like manner all groups of the first class of type II, which 
have among their conditional equations these two, viz.: 
k, = kk, and t, =t-+t,, break up into subgroups characterized 
by a constant a. The groups of the first class which meet 
these conditions are G,’(AA'l’), G,/(AA)), G,/ (ll), G,{(A’l), 
G,/(A'l’). We therefore have five varieties of groups of the 
Becondiclass, viz.: G,/(AAl).. (G,’(AA’)... G(W),, G;' (Al), 
