MR T. B. SPRAGUE ON A NEW ALGEBRA. 407 



When a is prime, the period is n—1, or a factor of it. When n is composite, 

 the period is the " totient " of n, or a factor of it. (As the word totient has not yet come 

 into general use, it may be useful to explain that the totient of a number is the very con- 

 venient name given by Sylvester to the number of numbers less than it and prime to it.) 



The I operation leaves the constituent n in a permutation, unaltered ; or l(n, b) = (n, b). 

 The m operation always leaves the last constituent in the permutation unaltered ; or 

 m(a, n) = (a, n). 



When n = 5, the period is 4, and we have 



/ (13452) = (21354) ; m(13452) = (41532) ; 



£ 2 (13452) = £(21354) = (42153) 

 £ 3 (13452) = £(42153) = (34251) 

 £ 4 (13452) = £(34251) = (13452) 



m 2 (13452) = m(41532) = (5431 2) ; 

 m 3 (13452) = m(54312) = (35142) ; 

 m 4 (13452) = m(35142) = (13452) . 



We have now to investigate the laws according to which the symbols I and m com- 

 bine with the others. We have dealt with 2 pairs of operations ; r , i ; and s, t ; and in 

 each pair one changes the constituent and leaves its place unaltered, while the other 

 leaves the constituent unaltered, but changes its place : thus 



r(a&)=(l — a, b) ; i(ab)=(a, 1 — b) ; 

 s(a&)=(a— 1, b); t(ab)=(a,b — l). 



And we have seen that ri = ir, rt = tr ; si = is, st = ts . 



If now we consider any two operations, one of which affects the constituent only and the 

 other its place only, we see that it is immaterial in which order the operations are 

 performed. 



But s, r, I, affect the constituent, a , 



t, i, m, „ its place, b . 



Hence, if any one of the three, 5, r, I, is combined with any one of the three, t, i, m, it is 

 immaterial in which order they come. But when two operations both affect the 

 constituent, or both affect its place, their combined effect is different when their 

 order is changed. 



Thus lr(ab)=l(l - a,b)=(2 - 2a,b) 



r£(a6)=r(2a,6)=(l - 2a,b)=slr(ab) 

 so that rl = sir ; Ir = s~hi . 

 Again ls(ab)=l(a-l,b)=(2a-2,b) 



sl(ab)~s(2a,b)={2a - l,b)=s-Hs(ab) . 

 so that si =s~ Hs ; Is = s 2 l . 

 Also oni(ab)=m(a,l - 6)=(a,2 - 26) 



im(a&)=i(a,2c>)=(a,l — 2b)=tmi 

 so that im = tmi ; mi = t~ Hm . 



VOL. XXX VIT. PART II. (NO. 19). 3 



