252 



The groups of 7 ' 6 * 4 and 11 '10 '6, which I think ought 

 to be called the groups of Galois, will always be remarkable 

 as being among the earliest discovered and the most diificult 

 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 ^{n-\-l)n(ji — 1) for 

 higher prime values of n=2p-\rl, p being prime, since they 

 exist for 7i=z5, nzrzl^ and 7i=:ll, But the non-existence of a 

 similar group for n=:2S 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 /*," I should have written in that 

 section, ^^ As there are N — 1 values 7 of A.'* And in the 

 last line of page 383, art. 90, the word principal ought to 

 be substituted for onfy. The latter is correct, if N=:S^-fl, 

 where p is prime. This does not affect the reasoning. The 

 groups E and E^ should be represented thus — 

 {234567890a}ii { 3 6 9 1 4 70 2 5 S^jg {3 8 1 9 5 7246a}3 



X{6084715293«}2 {6 518934720 43 =E, 

 {2345 6 7 89 0a}n {3 6 9 1 4 7 2 5 8a}g {l 8 43 9 72 5 G^ja 



x{l 537284609 a}o {O 8 3 2 7 5 6 4 1 9 rtjg rrEi, 

 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, 1-211, 1-2'12, 1-3-6, l-3'7, 

 1-3-10, l'3-ll, 1-3-12, 1-4-8, 1-4-10, 1-4-11, 1-4-12, 

 1-5-10, 1-5-11, 1-5-13, 1-6-12, 1-6-13, 1*6-14, 1-7-15. 



I have a simple tactical method of forming many such 

 systems for all imme numbers, depending on the theory of 

 difference circles, which requires the solution of no equation 

 or congruence. 



