404 MR T. B. SPEAGUE ON A NEW ALGEBRA. 



(H- i ,)(l+iXl+r)=(l+i)(l+r)(l+2?). 



Bui (l+iXl+p)(l+r) = (l+i)(l+r+p+pr) 



= I+r+2y+pr+i+ir+ip) + ij)r 

 = l + r+i + 2p + ir+2ip; 

 and (l + r)(l+p)(l+i) = l+r+i + 2p+ir+2rj) , 



(1 + i)(l + p)(l + i) = 2 + 2i +p + ip + rp + irp , 

 (l + r)(l+p)(l +r) = 2 + 2r+p + ip + rp + irp . 



Looking now at the graphs in figures 3 and 4, we see that the effect of the operation 

 ip on a graph, is a rotation through a right angle in a positive direction ; and the effect 

 of rp, is a similar rotation in the negative direction ; so that the effect of the repetition of 

 ip, is precisely the same as that of a repetition of rp ; and the performance of either 

 operation four times in succession, gives us the original graph, and the original permuta- 

 tion ; this being exactly what is indicated by the various equations we have obtained. 

 It is useful for some purposes to have a single symbol to denote the operation ip, and I 

 therefore put ip =j ; so that j 2 = ir, j 3 = rp, j* = 1 ; also j(ab)=^(b, 1 — a). 



Now it will be noticed that each of the operations r, i, p, has the effect of rotating 

 the graph to which it is applied, through two right angles about an axis in the plane of 

 the paper, so as to place the back of the graph upwards : hence any two of these opera- 

 tions will place the front upwards again ; and a combination of all three, will place the 

 back upwards again. We may therefore group the 8 permutations according to the side 

 of the graph which is upwards : thus 



P, irP, rpV, ip~P, show the front of the graph ; 

 rP, iP, pF, irpF, „ back „ 



We may convenient^ speak of the operations and graphs relating to the upper four 

 permutations as being " obverse " ; and those relating to the lower four may, for want of 

 a better name, be called " converse". We cannot, as in the case of a coin, call them 

 " reverse", as we have given a different meaning to that word. Using the symbol j, and 

 bearing in mind that ir =j 2 , ip =j, rp =j 3 , the permutations P, j~P, j 2 P, j 3 !*, are 

 obverse, or show the front of the graph ; and the aggregate of these four permutations 

 may be denoted by ( 1 +j +j 2 + j 2 )F ; or by JP, if we put J = 1 + j +j 2 +j 3 . But, from 

 the nature of the operations, the same aggregate may be denoted by 



JirF, or JrpF, or JipP . 



Henc< • J = Jlr = Jrp = Sip ; and similarly J = irJ = rpj — ipj . 



According to the ordinary rules of algebra we should be entitled to conclude from these 

 equations that ir= 1, rp= \,ip = \; but such a conclusion is not legitimate in the pre- 

 sent case. Putting for ir its equivalent, j 2 , we have 



3w = (i + j +f +jw = / +f +j* +f =j* +r+i +j = J 

 since j* = 1 ; and similarly for the others. 



