MB T. B. SPEAGUE ON A NEW ALGEBRA. 409 



The number of such permutations is u 2 ; and, combining this with our former results, we 

 see that the total number of permutations that can be derived from a single one, by 

 means of the operations r,i,p,s,t,l,m, is 8n 2 u 2 ; and they may be symbolized by 



(l+i)(l+r)(l+_p)STLMP. 



Of course in many cases these will not all be different permutations. 



It is clear that all our symbols are subject to the associative law ; for instance, 

 (ir)p — i(rp) ; for each of these simply denotes that the three operations, p, r, i, are to be 

 successively performed. 



Let us next consider how we are to interpret such inverse operations as (ir)~ l . 



Let (ir) _1 P = Q ; then P = irQ, and operating with ir on both sides, irP = ( ir) 2 Q = Q, whence 

 (ir) _1 P = irF, or (ir)~ 1 = ir. 



Next let (ip)~ 1 V = Q .'. P = ipQ, whence %P = pQ, and piP = Q; or (ip)- 1 P=piP, and, (ip)~ 1 =pi. 



Again, let (ips)~ a P = Q .\ F = ipsQ,. Then iP=psQ,; piP — sQ; s- l pil? = Q; so that (ips)- 1 ~P = 

 s' l p%?, or (ips)- 1 = s~ l pi . 



These examples show clearly the process by which the interpretation of similar, but more 

 complicated inverse operations, is to be arrived at ; for instance, we see that 



(Istirp) ~ 1 =prit~ 1 8~ 1 l~ 1 . 



I will conclude by giving a few examples of the uses that may be made of these 

 symbols. 



Any permutation in the same set as P, may be denoted by s h t k P, where h and k may 

 each have any value from to (n— 1). Is it possible for sY'P to be the same as P? or. 

 in other words, is it possible for the same permutation to occur twice in a set ? If so, 

 we have P = sYP = sY(sYP) ■■= s' 2 Y 2 *P ; and similarly 



P = .s'fP = s 2, 'PP = s 3 'f i T= . . . =(s h t k ) n - l ~P . 



Siuce s"= 1, t n = 1, we have to reject any multiples of n that occur in the indices, or, in 

 other words, to retain only the residues of the indices to modulus n. Confining our 

 attention on the present occasion to the case where n is prime, we know from the theory 

 of numbers that the residues of h,2h, 3h, . . . (n— l)h consist of the numbers 1, 2, 3 

 . . . (n— 1), arranged in a different order ; and the same is true of the residues of 7c, 2k, 

 3k, . . . (n — 1)&. Hence the series of operators contains two which may be denoted by 

 st? and sH ; and it follows that P = .s^P, 



so that ( ai a. 2 . . . a n )=(a, 2 -g,a 3 -g . . . cf-g^-g); 



whence a x =a,-g, a 2 =a 3 -g, . . . a^c^-g ; 



