92 
And on the partition, 
10 = 3 . + 8' • 1' + 2 • 2, (M, = 1 = M,' = M 2 = F 2 ) 
we can construct, if we take S 3 = 6 S' 3 = 3 ; 
7T 1 0 
1 • 1 • 2 • 6 • G • 2 2 
woven groups each of 6 • 3 • 2 2 • 1 • 1 • 2 = 144 substitutions, 
all ditferent groups from those above enumerated. 
(10!') If we take for each of the r groups Gjj the R cyclical 
permutations of 123 • • II, and write 
Sjj = R, l r —{trr), 
we obtain the group of 
A" B 1 C c J J irn irb ire • • irj 
. substitutions of which Cauchy has demonstrated the existence 
in the IXth section of his “Memoire sur les arrangements,” &c., 
in the Exercises de Mathemalique et de Physique Analylique. 
It is evident that the largest woven group constructible on 
our partition of N is of 
(?rA) a (’’’By 1 (ttC)' • ' • ira irb ire • • * • = O 
substitutions. This group has no derived derangement, i. e. it 
is a maximum group, and has 
7tN 
"cT 
equivalent groups, itself 
included. 
This theorem M is, as I believe, in that generality, new. 
On the Construction of Functions. 
(104) Theo. N. Let G be any group (l Aj A 2 • • A^_i) of 
L substitutions, made with N elements (12* • N). 
Let 
■n a /3 7 6 
P = Xj zf xj ■ • • Xy 
be the product of N different powers (^0) of the N vari- 
ables Xi Xi • • • • x . 
Let 
4>=P 1 + P 2 + • • r x 
be the sum of the N terms, obtained by performing on the 
subindices of P the substitutions of G. 
