102 Proceedings of the Royal Irish Academy. 



Naturally, if both sets of units belong to the same space, the 

 operator may be much simplified. For examj)le, the equation 



jlj\ ' ■ 'J«, =hh ■ . . in 



must then be true, so that if ?'i is converted intoyi, ii ^^'^ojo, &c., and 

 «,^i into/^i, it will necessarily follow that i„, will be converted into/^. 

 Por this case, then, Q„,_i ( ) Q„^r^ will effect the required transforma- 

 tion. Here, also, it is not hard to see that 



Q,^, = 1 - IJJ, + ^jsjtitis - &c., 



in which the summation still extends from 5 = 1 to s = m, &c. ; but the 

 last sum consists, when m is odd, of products of ^ (?« - 1) of the unitsy 

 with the corresponding units i, and when m is even, it is halftYie sum 

 of products of ^m units/ with the corresponding units i. 

 For example, for four units, if 



because jdd-jz • Wih = iiHhiz • Hhiiii = 1- 



The Q functions considered in this article are all even in the units. 

 When both sets of units belong to the same space of m dimensions, 

 Q,„-i is of the order (w - 1) ; in the units i, when m is odd (when the 

 units y are supposed to be given in terms of ?'), and when m is even, 

 Q^i is of the order m.. 



An obvious remark may be useful. If the operators q{) q'^, and 

 P ( )P~^ ^^^ equivalent in the results they produce on all vectors, or if 

 qpq~^ = p>pp'^, then 5- = |-? to a numerical factor. For iJ~^2'p = pp'^q^ or 

 p''^q is commutative with all vectors, and is therefore a scalar. 



It is also useful to remark, when the units involved are contained 

 in a space of odd dimensions, if we multiply Q,n-i by the product of 

 the m units {p = iiii . . . ■?",„)> that the product pQm-i is odd in the 

 m units ; and that 



pQm-l{ ) Qm-{'^P'\ and Q„^_l{ ) Q,n_i 



have the same effect on all units contained in the wi-dimensional space, 

 and opposite effects on vectors perpendicular to this space — the first 

 operator reversing, the second retaining their directions. 



38. The operators q{ )q~'^ which change line vectors into vectors 

 are, of course, a particular class of linear vector functions. If 



<^p = qpq~^, Scrcfyp = Scrqpq~^ = Sq'^aqp = ;S<^~Vp = S(f)'crp. 



Thus the conjugate (^') of one of these functions (^) is its inverse (^"^). 



