MR T. B. SPRAGUE ON A NEW ALGEBRA. 401 



may be added or subtracted, so as to make each constituent or its place, or both, 



positive, and not greater than n. 



Also, i? = (a 2 a 3 . . . a n a^) ; 



and s h t k V=(a k+l — h, a t+2 — h, . . . a n —h, a x — k . . . a k — h). 



In many investigations it is desirable to confine our attention to a single one of the 

 constituents of the permutation ; which we will take to be a standing in the 6 th place ; 

 and this constituent I denote by (a, b). Then it is easy to see that 



s(a,b)=(a — 1,6) ; t(a,b)=(a,b — 1) ; 

 and sH k (a,b) ={a — h,b — k) . 



I have now to describe three other kinds of transformation, which I denote by i, r, p, 

 and speak of as inversion, reversion, and perversion. 



The first, i, inverts the order of the constituents ; thus, 



i(13452) = (25431); 

 Ma^ . . . a n ) = (a„ . . . a 2 a{) . 



The second, r, gives a new permutation, in which each constituent is got by sub- 

 tracting the corresponding constituent of the original permutation from (n + 1) ; thus, 



r(13452) = (53214); 

 r^ctg . . . a n )={n + l—a v n + l — a 2 , . . n + l — a n ) 

 =(l—a v l-a. 2> . . . 1 — (6,0- 



We see that r, like s, affects the constituents ; and i, like t, affects their places. 

 Combining these operations, r and i, we have 



ir(13452) = i(53214) = (41235) ; 



H(13452) = r(25431) = (41235) = w(13452) . 



If we confine our attention to a single constituent (a, b), we have 



r(a6)=(l — a,b); i(ab)=(a, 1 — b) ; 

 ir(a6)=-i(l - a, 6)=(1 — a, 1 — b) ; 

 ri(ab)=r(a, 1 - 6)^(1 — a, 1 — b) ; 

 so that, generally, ir(ab) = ri(ab), or ir = ri . 

 Also r 2 (ab)=r(l — a, b)=(ab) ; 



i 2 (ab)=i(a, 1 — b)=(ab) ; 



whence r 2 =l, i 2 =l. These equations express the obvious property of the operations, 

 that the repetition of each leaves the original permutation unaltered. 



If, now, P being (13452), we represent P, rP, t'P, irP, by their graphs, we see that r 

 and i are operations of precisely similar character, one of which affects the rows of the 

 graph, and the other affects the columns. 



