LINEAR RELATIONS. TYPE II. 191 
defined inany way. For if it should be found that D and two 
or more sets of linear relations or D and R, define a group, 
the number of parameters would have to be less than five. 
THEOREM 20. There are no five- six- or seven-parameter groups 
of type IT. 
225. Four-Parameter Groups of Type II. We shall now 
prove the existence of four-parameter groups of type II. We 
saw in case (1) above that conditions (i) and (11) were not 
sufficient to cause (iii) to vanish because of the presence of 
the terms a.3, and b,a, in (iii) not involved in (1) and (ii). 
b,=Oand 3,=0, or a,=0 and «.=0. Each one of these 
conditions is also sufficient, for on either supposition we have, 
Sincewails'a FOOL (a9). @,— 1 andia,— 1, or/0,— 1and @,—71; 
and either supposition causes (1111) to vanish. 
We found also in case (2) that a necessary condition for the 
vanishing of (jjjj) is either 6,=0 and 3,=0, or a,=0 and 
u,=0O. Each of these conditions is also sufficient, for on 
either supposition we have from (j) 6, =a, and 3,=u, and 
these cause (}}jj) to vanish. 
We found also in case (2) that a necessary condition for the 
vanishing of (jjjj) is either b,=0 and 3,=0, ora,=0 and 
u,.=0. Each of these conditions is also sufficient, for on 
either supposition we have from (j) 6,=a, and 3,=.u, and 
these cause (jjjj) to vanish. 
Thus cases (1) and (2) lead to the same result and we have 
a four-parameter group of type II, if b,=0 or a,=0. Tak- 
ing all combinations we have four cases to examine, viz.: 
b, = 0 with / a single or a double root of \(p) = 0, anda, = 0 
with / a single or a double root. The proof holds in all four 
cases. For these four cases the matrix M reduces respec- 
tively to 
|a, O Of a, O 0| \\a,; 6, O| a, 6, Ol 
(1) ja m 0], (2) /e 2 0|, (3)/0 a ol, (4) |o 1 0 
Nosibe 3 a3 63 7 ad; b; 1| Gis [a if | 
Let the invariant figure of type II, Fig. (14 II), be lettered 
