• MR T. B. SPRAGUE ON A NEW ALGEBRA. 403 



We have next to investigate the laws according to which r and i combine with p. 

 Confining our attention to a single constituent, we have 



rp(ab) = r(6a)=(l - b, a) ; 



ip(ab) — i(ba)=(b, 1 — a) ; 

 irp(ab)=i(l — b, a)=(l — b, 1 - a) ; 



pr(ab)=p(\ — a, b)=(b, 1 — a) ; 



pi(ab)=p(a, 1 — 6)=(1 — b, a) ; 

 pir(ab)=p(l — a,l — &)=(1 — b, 1 — a) . 



Comparing these, we see that ip =pr ; rp =pi ; irp —pir . 



The last of these follows from the two first : thus irp = i.rp = i. pi = ip.i =pri —pir . 



It is seen in the same way that rpr = ipi = irp = rip— pir =pri . 



Also prp=p.pi=pH = i; pip=p.pr=p 2 r = r. 



Again (ir) 2 = ir.ir = i.ir.r = i 2 r 2 = l ; 



(irp) 2 = irp . irp = ir. irp . p = (irfp 2 = 1 . 



These last relations also follow at once from the equations 



ir(ab)=(\ — a,l — b); irp(ab)=(l — b,l — a). 



Although r a =l, i 2 =l,^ 2 =l, (rp) 2 and (ip) 2 are not = l ; but 

 (rp) 2 = rp.rp = rp.pi = rp 2 i = ri = ir; 

 (ip) 2 = ip,ip = ip ,pr = ip 2 r = ir , 



Hence (rp) l = {(rpf} 2 =(rif = 1 ; (ip) i ={(ip) 2 } 2 = (ir) 2 = \ . 



We have seen that, by combinations of the three operations r, i, p, we are able to 

 obtain from any permutation, 7 others, making 8 in all ; and the relations we have just 

 established show that we cannot get more than these 8 permutations, in whatever way 

 we combine the operations. Thus, taking them two at a time, we have 



ir = ri, rp=pi, ip=pr; 



so that the 6 pairs of operations give us only 3 different permutations. 

 Again, taking them 3 at a time, 



irp = rip = rpr = ipi =pir =pri ; 

 rpi = ipr = p ; prp = i ; pip = r ; 



so that the compound operations in the upper line give us only one new permutation 

 and those in the lower line give us only permutations which we already have. We 

 saw that the 8 permutations are given by the development of (1 +*)(1 +»*)(!■ +£>)P ; and 

 it will be found that they are also given by the development of (1 +£>)(1 +*)(l +?')P ; in 

 other words, 



