88 Proceedings of the Royal Irish Academy. 



If, as in Art. 15, 



Ig, = 9(0) + qm - q{2) - Sizn 



^q = qw - qn) - q{2) + s-o), 



Klq = q^o) - qn) + qio) - ^(s) = I^q- 



18. Chiefly on account of symmetry, a third new symbol tT'may 

 be introduced which merely changes the signs of the units without 

 changing the order. It is evident that IK= KI= J, and that the 

 symbols are associative and commutative in operation. The laws of 

 their combination are contained in the symbolic equations 



P = J^ = K'~ = IJK=KJI=+l, 



or, more fully, by 



I=JK=KJ, J=KI=IK, K=IJ^JI, P^J^= K^=l. 



Por a product, J^{pq) = I{Kq^p) = JpJq. 



Taking any function q = ^(o) + ^(i) + q^z) + qiz,, 



previous results afford the relations 



^(0)? = §'(0) = i(i + /+./+ir)^, 



Vn)q = qiz) = i{l-I-J'+^)q- 



19. By the aid of the symbol ^it is easy to deduce some usuful 

 formulae, as follows : — 



The product of any two functions p and q may be written in the 

 forms (see Art. 15) — 



Pq = {P(o) +P{i) +Pi2) +i^(3)) too) + ?{i) + qw + 2'(3)) 



= ( ^(01 + ^(1) + F"(2) + V^3))2Jq. 



Taking conjugates, the relation 



^{Pl) = (^(0) - ^(1) - q{2) + q{3)) {PiO) -P{1) -P(2) +P(3)) 



= ( ^(0) - ?^(1) - ^(21 + ^i3))pq 

 is found, and this, when combined with the former, affords, on addition 

 and subtraction, expressions for (^(0)+ V(fi))pq, and for (^(1)4- Vi^yjpq- 

 Separating the parts of these which are even and odd in the units, the 

 values of F(o)^?, '^wPq^ '^{■i)Pq, and V^^i^pq are found. As the formulae 



