60 
A positive group G is a maximum positive group, when no 
derangements of G or of any factor of G can be added to it so 
as to complete a positive group of an order inferior to \ IIN. 
To every mixed group J, i.e. a group containing positive 
and negative substitutions, derangements JP, JQ, . . . can be 
added so as to complete the entire group of the order IIN. 
A mixed group J is a maximum mixed group when ho 
derangements of J or of any factor of J can be added to it to 
complete a group of an order inferior to IIN. 
To every group H, whether mixed or positive, derange- 
ments can be added so as to complete the entire group of 
E(N, which is always a mixed group if N >2. 
The suspicion expressed in the final paragraph of my 
memoir On the Theory of Groups (Memoirs of the Literary 
and Philosophical Society of Manchester, 1861), that the 
effect of variation of exponents in the auxiliary group g (Art. 
34) on the number of equivalent groups constructible, would 
have to be reconsidered, turns out to be correct. The groups 
enumerated in theorem H (Art. 37) always exist, but they 
are not always the whole of the equivalents. If the partition 
of N is 
N = A -a = A A a', 
A being a prime number, my estimate of the effects of variation 
of exponents is correct for N^A'A^A 2 , whatever prime A 
may be, and also for N=A‘A 2 =:3-2 2 =:8 ; but it is defective 
for higher values of N. I took it for granted that if the 
auxiliary be 
1 2 3 4 5 6 7 8 
2 1 4 3 6 5 8 7 
3 4 1 2 7 8 5 6 
1 3 2 1 8 7 6 5 (g) 
5 6 7 8 1 2 3 4 
6 5 8 7 2 1 4 3 
7 8 5 6 3 4 1 2 
8765432 1, 
no variation of exponents would give different equivalent 
