Sir Wirtram Rowan Hamiton’s Fescarches respecting Quaternions. 219 
restored, when it undergoes two successive inversions : the opposite of the opposite 
of arelation being the same with that original relation itself. Thus, in particular, 
for the case n = 4, the characteristic of octadic transposition, w, of which the 
symbolic square was unity, is connected with the sign —, or — ], prefixed, as a 
characteristic of inversion, to the symbol of an ordinal quaternion. 
11. Again, with respect to the sign of seMI-INVERSION, Y(—1), we may 
observe that 7f the exponent n of the order of the ordinal set be an even number, 
= 2mm, then we shall have in general, as a symbolic fourth root of unity, the fol- 
lowing characteristic of momental transposition, which may be obtained by 
changing r to 1, s to 4, and 7 to m, in the formula (18): 
m ,=1; (104) 
M4m_y, 0, 1,-. 4dm— 
and which takes the particular form (15), when m is changed to 1. And be- 
cause the symbolic square of the first member of (104) acquires the form (103) 
by restoring 7 in the place of 2m, we see that an ordinal set, if it be of an even 
order, such as is an ordinal couple or quaternion, may always be semi-inverted, 
and therefore operated on by the sign /(—1), in, at least, one way, through 
the medium of that momental transposition, performed on a momental set of an 
evenly even order, which is indicated by this first member. For example, when 
we operate on a momental quaternion (A’,, A’ Ay A,) by the characteristic 
M3, o, 1,2 We obtain the new momental quaternion, 
(Ap Ads Aty Ao) = Ma,o,1,2 (As Ais Aw» 41) 5 (105) 
and it is evident that, as was remarked in the second article, and as is included 
in the more general assertion (104), four successive transpositions of this sort re- 
produce the momental quaternion which was originally proposed to be operated 
on. But we now see, further, that if, on the plan of the article immediately pre- 
ceding the present, we connect, in thought, this momental quaternion with the 
ordinal couple, 
(Ags At) — (Ag 4,) = (Ao — Aw 41 — 4,)> (106) 
we shall thereby connect the foregoing operation of momental transposition with 
an operation of ordinal derivation, which must admit of being symbolically repre- 
sented by the sign »/(—1), and which here consists in passing from the couple 
(106) to this other ordinal couple : 
(Ay Ag) — (At, Ay) = (Ay — Aly Ad — Ag)- (107) 
