k>s 



MR T. B. SPRAUUE ON A NEW ALGEBRA. 



And mt(ab)^m(a ) b — l)=(a,2b — 2) 



tm(a,b)=t(a,2b)=(a,2b - l)=t-\mt(ab) 

 so that tm —t_ x mt ; m< = £ -2 m . 



Since £> affects both the constituents and their places, the order in which it is combined 

 with another of the operations, is never immaterial. 



We have pl(ab)=p(2a,b)=(b,2a) 



lp(ab)=l(b,a)=(2b,a) 



pm(ab)=p(a,2b)=(2b,a) 



mp(ab)=m(b,a)=(b,2a) : 

 so that pi = mp ; pm = Ip . 



The relations we have establisht may be conveniently tabulated as follows : — 



We have 

 ll-\ab) = (ab); and if we suppose l-\ab) = (A,b), then ll-\ab)^=l(A,b)=(2A,b); or (a&)=(2A,&); 



so that 2A=a ; that is to say, 2 A = a or (a + n) ; and therefore A = a/2 or (a + n)/2 ; and 

 we have to take the former value if a is even, and the latter if a is odd ; n being, as 

 already stated, always odd when we deal with the I and m operations. We may say, 

 then, that Z -1 (ct&)=(a/2, b). Similarly m _1 (a&)=(a, b/2). 



The permutations that can be formed from P by means of I and m, may be symbolized 



by 



(1 + 1 + P+ . . +l"- ] )(l+m+m 2 + . . . +m"- l )P, or by LMP, 

 if we put l + l + l'+ . . . +l"~ l = L, and l+m+m 2 + . . . +m"- 1 = M. 



