548 
Proceedings of the Royal Society of Edinburgh. [Sess. 
With regard to unireplacements a preliminary restriction is, for 
simplicity, required ; that is, to ensure (5) below universally. When a uni- 
replacement does not treat all the fictits similarly ; when, for instance, it 
changes the sign of some and not that of others ; then no unireplacement 
is to be made except on that one system (and its replacements). The uni- 
replacements are Q or K; certain others to be defined ; and the last in 
combination with Q or K. Any such other is defined as a replacement in 
which any assigned fictits of a given set are negatived, that is, have their 
signs changed. Thus P is such a unireplacement, namely, the one in 
which all the fictits are negatived. QK is another, since acc. as £ 2 = ±1„ 
QK = 1, or QK = P. 
For some purposes it is desirable to extend the meaning of unireplace- 
ment to allowing all or some of the fictits to be multiplied by x /( — 1). 
Thus for some purposes it would be convenient to consider the replacement 
in which every fictit is multiplied by the scalar J(i 2 ) ( — ) r ^r 2} ; since, if we 
were dealing with a multiplex of order n, this would make U5 precisely 
homologous with a scalar or a fictit according as n is odd or even. But I 
have thought it best to eschew imaginaries, especially since, for practical 
purposes, we can choose i 2 to be either +1 or — 1. 
With this meaning accepted above, 
*C 2 t = q (4) 
= R U -Rq (5) 
= Sjfej? . (6) 
(6) is proved by putting p = x 1 v 1 + x 2 e 2 + . . . -,q = yiv l + . . . S/pU u q = 
2® 1 2/ 1 S- y 1 ®’« u r 
A restriction that is not imposed refers to the independence of the 2” 
multits (other than the fictits which are covered by the fundamental laws). 
When n is odd the replaced meanings of S c and S n _ c are sometimes identical, 
namely, when the replacement is a complementary replacement. 
The special simplicities of unireplacements are then (1) R£ = l; (2) 
RR m = R (t R ; (3) R m S c = S C R M ; (4) a product of any number of unireplacements, 
is a uniproplacement or a uniretroplacement according as the number of 
constituent retroplacements is even or odd; (5) SpH u q = Sqh& u p. 
Any replacement R is converted from retro- to pro- or else from pro- to- 
retro- by a uniretroplacement such as K ; and its species is left unchanged 
by a uniproplacement such as P. Or generally 
Rvffiyj = Rp, RpR-Mr = RV ( /y\ 
RrRjR'„E3«r-.R; J * ' * ; 
In a certain obvious sense indeed uniretroplacements behave like minus- 
signs and uniproplacements like plus signs. 
