252 
The groups of 7 * 6 • 4 and 11 • 10 • 6, which I think ought 
to he called the groups of Galois, will always be remarkable 
as being among the earliest discovered and the most difficult 
to construct. They appear to be a complete family of them- 
selves. One is indeed strongly tempted to believe in the 
existence of other non-modular groups of ls{n-\-\)n(ii — 1) for 
higher prime values of n=Hp J r\, p being prime, since they 
exist for n— :5, n— 7, and w=ll. But the non-existence of a 
similar group for w=2 3 may be easily proved, in half an 
hour’s labour, by the method pointed out in the 94th article 
of my Memoir on Groups above quoted. It suffices to write 
four vertical rows of the powers to be examined. Instead of 
“ As there are N values of h” I should have written in that 
section, “ As there are N — 1 values 7 0 of h” And in the 
last line of page 383, art. 90, the word principal ought to 
be substituted for only. The latter is correct, if N=2p+1, 
where p is prime. This does not affect the reasoning. The 
groups E and E x should be represented thus — 
{234567890 a} u { 3 6 9 1 4 7 0 2 5 8 a}' 5 {3 8 1 9 5 0 7 2 4 6 a} 2 
X {6 084715293 a} 2 {6 518934720 a} 3 =E, 
{234567890 a} u {3 6 9 1 4 7 0 2 5 8 a} 5 {1843907256 a}, 
X {153 72846 09a}, {0 8 3 2 7 5 6 4 1 9 a} 3 =E l5 
of which the observations there following are true, except 
that “non-” should be written before modular in the next 
line, p. 390; as also in the third line of page 374. 
Triads can be formed with 23 elements to exhaust the 
duads of 23 six times, by perpetual additions of unity to all 
the elements following : — 
1-2-4, 1-2-6, 1-2-8, 1-2-9, 1-2*11, 1-2-12, 13-6, 1-3-7, 
1-310, 1-811, 1-3-12, 1-4-8, 1-4-10, 1*411, 1-4-12, 
1-5-10, l'5-ll, 1-5-13, 1-6-12, 1-6-13, 1614, 1-7-15. 
I have a simple tactical method of forming many such 
systems for all prime numbers, depending on the theory of 
difference circles, which requires the solution of no equation 
or congruence. 
