86 
Let N=A o-}-B&-}-0c-4- . . 
A 7 B 7 C 7 .. 7 J; K being the least common multiple 
of A B C . . J, and a be • .j being any numbers. 
Let E < K be any one of the numbers A B C . . J ; 
and let E — 1 be a primitive root of the congruence 
x y 1 (mod. M). 
M being any one of the uumbers A B C..J; and such a 
root that it is also a root, primitive, or not, of the congruence 
#’■==1 (mod. X), 
X being one each in turn of the numbers A B C . . . J. 
We can form by the aid of this root E — 1, on the given 
partition of N, R* being as in theorem J, 
E c a-N y 
It* ira • irb • * • 7 rj A" B 6 * * ' J J > 
equivalent groups of K r substitutions, e being the multiplier 
of E in the partition (A a + B& -f . .) of N. These groups 
are all of the form 
G + G + ©2 G + .. G, 
when G is any one of the W groups of theorem Ch 
For example, let 
N=26=10-l + 8-2+4-2 (K=40). 
We have the primitive root 4 — l of the congruence 
x 4 — 1 (mod. 10) (Ee=42) 
which is also a root of the congruences 
a; 4 EEl (mod. 8) and 
#t==rl (mod. 4). 
One of theW= 
tt.(26) 
K 40 (7t.2) -2 10'8' 2 4 2 
groups G of theo. C is 
1234567890 a b c d ef g h ijkl m n p q 
2345678901 bcdefghajkli n p q ?n 
3456789012 edefg hah klijpqmn 
4567890123 defghabc l ij k q mn p 
=G 
of 40 substitutions (1 A A 2 . . . A 30 .) 
