400 



MR T. B. SPRAGUE ON A NEW ALGEBRA. 



Any 



Fig. l 



column, 



Fie. 2. 



permutation of the n numbers may be represented geometrically by means of a 

 square, divided by parallel lines so as to contain n~ equal square cells ; 

 and this representation I call the " graph " of the permutation. The 

 graph of 13452 is given in fig. 1. 



We now see that s and t are operations of precisely the same 

 kind; s operating on the rows of cells in the graph, and t on the 

 columns. 



If now we form a square containing 81 cells, by placing together 

 four squares similar to fig. 1, and then removing the outermost row and 

 we see that any square containing 25 of the cells is the graph of one of the 



permutations in the set, and that there are 25 

 such squares, corresponding to the 25 permutations 

 mentioned above. 



This representation of the permutation renders 

 evident certain properties of the graphs, and suggests 

 corresponding properties of the permutations. For 

 instance, in the present case, any graph which has 

 a diagonal in either of the lines ab or cd or ef, &c, 

 is symmetrical with regard to that diagonal : the 

 corresponding permutations are 



•sP = (52341), sH? = (12354), sH*P, &c. 

 sP = (52341), sH~ 1 P = (54123), s 3 £~ 2 P,&c. 



Also the graph which has the line ae for its side, is symmetrical with regard to both its 

 diagonals : the permutation in this case is sP = (52341), where it is to be noticed that 

 the sums of the constituents equidistant from the central one, 3, are each double of that 

 central one. 



Permutations can, of course, not be added, the one to the other ; but we may interpret 

 such an expression as (s + £)P or sP + tP, as denoting the aggregate of the two permu- 

 tations sP and tP. Consistently with this, we may represent the aggregate of the ri 2 

 permutations, which belong to the same set as P, by the symbol (l+s-M 2 + . . . . 

 + s"- l )(l+t + t 2 + .... +r- 1 )P, or by STP, if we put S=l+s + s 2 + .... +s n -\ 

 T = l+t + ?+ . . . + t"~ 1 ; also by (l+t + t 2 + . . . + i"- 1 )(l +s-M 2 + . . . + «*- 1 )P| 

 or TSP. 



It is often convenient to represent the permutation P by (c^o^ • • • ct n ) \ then 



sP^a x — l,a 2 — 1, . . . a n — 1), 



where Gauss's symbol, =, is used to indicate that n is to be written instead of where 

 it would occur. More generally, it indicates in this paper that n, or any multiple of it, 



