402 



MR T. B. SPRAGUE ON A NEW ALGEBRA. 



P 



I F 



BHDH 

 MH 



i B ID3I : h» ah 



Ffe. 3. 



I 





*VP 



































The effect of r upon P's graph may be described as a rotation of it through two right 

 angles, about the line GH bisecting the square ABCD ; while the effect of i may be 

 described as a similar rotation round the line EF, which also bisects the square. When 

 the two operations are combined in either order, the effect is equivalent to a rotation 

 through two right angles, in either a positive or negative direction, about an axis per- 

 pendicular to the plane of the paper. 



The aggregate of the four permutations may be represented by 



(l+i)(l+r)V, or (l+r)(l+i)Y. 



The operation p, when performed on a permutation, has the effect of transforming it 

 (generally) into another, such that any constituent (a, b) in the first corresponds to a 

 constituent (b, a) in the second; so that p(a, b) = (5, a). For instance, p(13452) = 

 (15234). From this it appears that p~(ab) =pp(ab) =p(ba) = (ab) ; whence p 2 = 1. 



The two permutations, P and pP, are said to be conjugate to each other; and if it 

 happens that, as in the case of 15432, the operation leaves the permutation altered, it is 

 said to be self-conjugate. 



Now performing the operations, r, i, ir, on p¥, we have 



rpP = r(15234) = (51432) ; ipV = i(15234) = (43251) ; irpV = i(51432) = (23415) . 

 The graphs of these permutations are as follows : — 



rp? 

























ipP 





irpP 





■■■■ 





■KU 





nana 













Comparing the graph of pP with that of P, we see that the effect of p on the latter is a 

 rotation through two right angles about the line AC ; and similarly we see that the 

 effect of irp on the graph of P is a similar rotation about BD. 

 The aggregate of these four permutations may be denoted by 



(l-M)(l+r)j?P, or (l + r)(l + i)pV; 

 and the aggregate of all the eight permutations by 



(l + 0(l + >-)(l+P)P,or(l+r)(l + ;)(l+2>)P. 



