251 



The substitution ISayr,' GlOu is 



1537284609«-a943867520 l=a 073684259 1=:6>, 

 whose circular factors are 209568, 437, a \. 



It can be demonstrated, either a priori, or by simple in- 

 spection of the above table of the 55 triads, that none of the 

 twelve just written transposes a consecutive pair of the circle 

 2 9 5 6 8, i.e., none has a transposition which fractures 

 that circle into one of five and another of one. And as each 

 of the twelve has one element of each of the three circles 

 undisturbed, it must have for one of its four transpositions a 

 non-consecutive pair of the circle 2 9 5 6 8, while the other 

 three transpositions will of necessity \iQ junctures of the four 

 resulting circles into one circle of eleven. 



Theor. A transposition of tioo letters of any circle always 

 fractures that circle into ttco : a transposition of elements of 

 txoo circles always unites those circles into one. 



From this follow many important theorems on substitu- 

 tions. Hence we deduce easily, as above for substitutions of 

 the seventh order, that there are 12*1 Or substitutions of the 

 eleventh order, represented by 12 * 2 ' 55 • 6 triplets A(B0)6, 

 each substitution by at least six different triplets having the 

 same A. Hence r-=\ or r=ll, of which the former only is 

 the true value ; and there are 120 substitutions of the 

 eleventh order. We prove readily that no quadruplet A B C D 

 is irreducible ; hence no quintuplet, &c. And the 55 substi- 

 tutions of the second order, 15 7, 5 T 1, &c., form with their 

 products a group of 



6 6 • 45-1-55 • 26 -f 55 • 23 -V 55 • I2 -f- 12 • lOn -j- Ij = 11 • 10 • 6 

 substitutions. This group is maximum, and has 9*8"T*5'4*3'2 

 equivalents. 



We thus find that, in the discussion and construction of 

 these hitherto difficult groups, we can dispense with con- 

 gruences, and with the still more formidable apparatus of 

 imaginary subindices, which MM. Betti and Mathieu have 

 so skilfully handled. 



