406 



MR T. B. SPRAGUE ON A NEW ALGEBRA. 



for (l +i)(l + >')■ This follows so easily from the principles we have establisht, that it 

 seems unnecessary to give the demonstration. 



When we say that 8n 2 permutations can be derived from P, it is not to be understood 

 that these are all necessarily different from each other ; in fact, for values of n less than 

 6, the number of possible permutations, n ! , is less than 8n 2 . For larger values of n, all 

 the 8n 2 permutations may be different, or some of them may be identical. 



Permutations may be transformed in various other ways, of which I will only 

 mention two. If n is odd, and each constituent in the permutation is multiplied by 2, 

 and the resulting number (diminished by n if necessary) substituted as a new constituent, 

 we shall get a new permutation. This operation I denote by I, so that l(ab)=(2a, b). 



For instance £(13452)= (21354). 



Again, if the constituent, a, remain unchanged, but be transferred from the place b, 

 to 2b, we get a new permutation which I denote by mP ; and we have m(a, &)=(a, 2b). 



Thus m(13452) = (41532). 



These operations, like all the others we have considered, are periodic ; and it is obvious 

 that they have the same period. If this is u, then l a = 1, m u = 1. 



The following table shows, for odd values of n not greater than 25, the period, 

 and the sequence in which one constituent is substituted for another in a permuta- 

 tion which is operated on by I. 



Sequences. 



, 2, 1 . . . . 



,2,4,3,1 .... 



, 2, 4, 1 ... . 



, 6, 5, 3 ... . 



, 2, 4, 8, 7, 5, 1 ... . 



, 6, 3 . . . . 



, 2, 4, 8, 5, 10, 9, 7, 3, 6, 1 ... . 



,2,4,8,3,6,12,11,9,5, 10, 7, 1 ... . 



, 2, 4, 8, 1 



, 6, 12, 9, 3 ... . 



, 10, 5 ... . 



, 14, 13, 11, 7 ... . 



, 2, 4, 8, 16, 15, 13, 9, 1 ... . 



,0,12,7, 14,11,5,10,3 .... 



, 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1 ... . 



, 2, 4, 8, 16, 11, 1 ... . 



, 6, 12, 3 ... . 



,10,20,19, 17, 13,5 .... 



, 14, 7 ... . 



, 18, 15, 9 . . . . 



, 2, 4, 8, 16, 9, 18, 13, 3, 6, 12, 1 ... . 



, 10, 20, 17, 11, 22, 21, 19, 15, 7, 14, 5 ... . 



, 2, 4, 8, 16, 7, 14, 3, 6, 12, 24, 23, 21, 17, 9, 18, 11, 22, 19, 13, 1 ... . 



, 10, 20, 15, 5 ... . 



11 



Period. 



S 



3 



2 



1, 



5 



4 



1, 



7 



3 



1, 



j) 



>> 



3, 



9 



6 



1, 



» 



>> 



3, 



11 



10 



1, 



13 



12 



1, 



15 



4 



1, 



?> 



>> 



3, 



» 



» 



5, 



» 



» 



7, 



17 



8 



1, 



» 



>> 



3, 



19 



18 



1, 



21 



6 



1, 



}> 



}t 



3, 



» 



>> 



5, 



7> 



» 



7, 



J) 



>J 



9, 



23 



11 



1, 



J> 



>> 



5, 



25 



20 



1, 



» 



)> 



5, 



