MR T. B. SPRAGUE ON A NEW ALGEBRA. 405 



Since J = 1 +j +j 2 +f = (1 +j)(l + j 2 ) , 



and J = i + j5 + j2+j S=( i + j2 )( i + j S)} 



we have J = (1 + *p)(l + ir) = (1 + ir)(l + rp) . 



These relations are easily verified by actual multiplication ; each product being equal to 

 1 + ir + rp + ip ', and it will be found that the order of the two factors on either side may 

 be reverst. 



Similarly it may be shown that the aggregate of the four converse permutations may 

 be represented by JrP, or JiP, or JpP, or JirpF ; and also by rJP, or {JP, or pJP, or 

 irpJV, and therefore the aggregate of all 8 permutations by (l+r)JP, or (1+*)JP, or 

 (1 +p)JP, or (1+ irp) JP. 



I now pass on to investigate the laws according to which the symbols r, i,p, combine 

 with s and t. 



We saw that s(ab)=(a —l,b). t(ab)=(a, 6 — 1) . 



Hence rs(ab)=r(a — 1, 6)=(2 — a, b) ; 



6t(<z&)= s(1 — a, fr)=( — a, b) ; 

 rs _1 (a6)=r(a + l, &)=( - a, b) ; 



so that sr= rs~ l ; and it may similarly be proved that s -1 r= rs. 



Also rt{ab)=r(a, b — 1)=(1 — a, b — 1) ; 



tr(ab)=t(l-a, b)=(l -a, b-1); 



so that rt = tr. 



Again, is(ab)=i(a — 1, 6)=(c6 — 1, 1 — b) ; 



•si(a6)EEHs(a, 1 — 6)=(a — 1,1 — 6); 



so that si = is . 



And it(ab)=i(a, b—l)=(a, 2 — b); 



ti(ab)=t(a, 1 — 6)==(a, — 6) ; 

 ■££ - \ab)^i(a, 6 + 1) (a, - b) .; 

 so that ti = it~ 1 ; and similarly t~H = it . 



Lastly, ps(ab)=p(a—l, &)=(&, a — 1) ; 



sp(ab)= s(b, a) =(& — 1, a) ; 

 pt(ab)=p(a, b — l)=(fr— 1, a) ; 

 tp(ab)= t(b, a) =(&, a — 1) . 



Hence, ps = tp; pt = sp . 



If, instead of P, we take the permutation s h t k P, we can obtain from it a group of 8 per- 

 mutations by means of the operations r, i, p ; and we thus have altogether 8n 2 permuta- 

 tions, including P itself, derived from P. The aggregate of these 8n 2 permutations may 

 be represented by (1 +i)(l +r)(l +_p)STP ; where the factors (l+i)(l+r),l+p, ST 

 maybe permuted in anyway. Also TS maybe substituted for ST. and (l+r)(l+i) 



