86 Proceedings of the Royal Irish Academy. 



16. Por the sake of facilitating various operations on the general 

 functions of n units, it seems to be useful to introduce certain new 

 symbols or characteristics of operation. 



The first of these is ^, the analogue of the symbol of conjugation 

 in Quaternions. For present purposes, the effect of the symbol jfmay 

 be defined as a change of sign of all the units in q, and an inversion 

 of the order in which they occur. From this definition, it is manifest 

 that ^^q =K.Kq^q. 



If q^ is homogeneous and of order m, 



^U = (-) ?m- 



To prove this, if i-S-i . . . «'„ is a product of 7?i distinct units, its conjugate 

 is {-y"hJm-i • • . «2«'i, or it is (-)"' (-)'"~H;„.i«„,_2 . . . ^2^li„^, by the law of 

 interchanges, (zi«2 = - i^ii) ; or finally, the conjugate is 



(_)-(_)-! (_)-2 . . . {-)\-yi,i, . . . 4. 



Hence, if m = 0, or 3 (mod. 4), Kq,^ = + q,n ; 



and if w = 1, or 2 (mod. 4), X'q^ = -$'«»; 



and in general, if q = q^o) + ?(i) + Sm + S{-o)> 



^i = 9(0) - 9(1) - ?(3) + ^(3), 



provided ^'(o) is the sum of products in q whose orders = (mod. 4). 



When using this symbol, it must be remembered that w, the pro- 

 duct of all the n units, obeys the relation 



iCo) = (-) ^ w, 



or that Kfji = co, n = 0, or 3 ; and Kin = - w, w = 1, or 2 (mod. 4). 

 In particular for Quaternions, 



Kdi = CO, or ijlc = - I = - Tiji. 



Again, take the conjugate of iiq,Jii where q^ is a homogeneous 

 function of the units which does not contain «i, 



K. i,qj^ = HE:q,'Jx = {-y"''Jrq,n, 

 by the rule of interchanges. But 



and the conjugate of the condensed product is equal to the conjugate- 

 of the uncondensed product, or in symbols 



