218 Sir Witr1am Rowan Hamitton’s Researches respecting Quaternions. 
relations, which are each between two moments, or because it is a complex ordinal 
relation between two momental sets, which are each of the n order, may be re- 
garded as containing, in its first conception, a reference to 2” moments; and 
these moments may always be supposed to be collected, in thought and in expres- 
sion, into a new momental set, of twice as high an order as the ordinal set which 
was proposed. In symbols, the ordinal set (54), which may be thus denoted : 
o’—Q— (45.45. - An) = (Ags Ay - + An_1)s (101) 
may naturally suggest the consideration of the following momental set, with 
which it is connected : 
/ 
(Ags Asie ae AeA ACs Ue Abert) 5 (102) 
and if the latter set be given, the former can be deduced from it. Hence every 
operation of transposition performed on the 2” moments of the set (102), is con- 
nected with, and determines, a certain corresponding change of the » ordinal 
relations of the set (101). Yor example, if in the formula of momental transpo- 
sition (18) we make s = 2, r= 1, then, with reference to a certain operation on 
the momental set (102), which consists here in exchanging the places of each mo- 
ment A with the corresponding moment a’, we obtain the symbolic equation, 
Maney ote (103) 
which implies that a repetition of this process of transposition would restore the 
set (102) to its original state. But the same operation on this momental set cor- 
responds to, and determines, a certain other operation, performed on the ordinal 
set (101), which consists in changing the sign of each constituent ordinal rela- 
tion, and in therefore changing, by the sixth article, the sign of the ordinal set 
itself, or in operating on that ordinal set by the characteristic —, or — 1; we 
might therefore, in this way, be conducted to the known result, or principle, that 
the sign —, or the coefficient — 1, is a symbolic square root of unity. And we 
might be led to express in words the corresponding conception, by saying that as 
two successive interchanges of the places of two moments, or of two momental 
sets, regarded respectively as ordinand and as ordinator, do not finally affect their 
ordinal relation to each other; the second transposition of these two moments or 
sets having destroyed the effect of the first: so too, and for a similar reason, the 
character (as well as the degree) of an ordinal re/ation is not changed, or is 
