122 Proceedings of the Royal Irish Academy. 



IN'ext, qziqi2 ( ) qi-T'^qzr^ changes i-^ to i^, i% to /g, and i^ to ?i, and 

 no other unit is affected. 



Repeating this process an odd number of times, it is evident that 



-^2m+l = 9'2m+l) 1 ?15 2m • • • 2'5l9'l49'3l9'l3 



— q-lm-^X') 1 q\i 1m ■L2m-\ 



— 9'2m+l) 1 -^ Imj 



Pzm+1 and -P'am being the functions defined in Art. 57, for the operators 

 here, and in that article have the same effect on any vector, and the 

 scalar parts of the functions Fzmny ^^^^ ^'im are here (as there) equal 

 to unity. 



The function F^m = i^h ■ • • 4^'2m = ioA . . . 4,(1 + iAm)P2,»-i can 

 be formed without much trouble from Pzhs-i- 



61. The functions Pjm and P^mH have been treated in considerable 

 detail, on account of their 'direct application to the Theory of 

 Substitutions. 



If Xi, X2, . . . Xn are n letters, their order may be considered as 

 defined by 



p = A'i4 + Xti^, + . . . + xJn, 



if the order of the coelficients of the units in this equation is made to 

 follow the order of the units in the series «i, i^, . . . 4- Operating on p 

 ^y 2i2 ( ) 2'i2~\ the result is ■ = 



- Xiii + Xih + x^is + . . . + xj,^ ; 

 that is, disregarding signs, the order of the letters is changed to 



/y /y /y /y • 



*t'2j «^lj ^Z • • • ^n » 



or the positions of x^ and x^, are interchanged. 



By combination of the q^t functions, any substitution whatever may 

 be effected by the operator Q ( ) Q"^, in which Q - '^q,f In this way, 

 all the operations in the Theory of Substitutions are made to depend 

 on the Associative Algebra defined by 



42 = -1, 44 + 44 = 0; 



and, by the device of putting part (Q) of the operator before the 

 operand and part (Q"^) after it, the comples laws of that theory are 

 reduced to the simple laws of combination of the units (^). 



The functions P of late articles give rise to cyclic substitutions. 

 A s before stated, any two functions 512 and 3-34 which have no common 

 suffix are commutative in order of multiplication, and give rise to 

 commutative symbols of operation which may be applied in any 

 order. 



