Sir Witt1am Rowan Hamitton’s Researches respecting Quaternions. 209 
relations between moments, which may or may not be relations of identity, and 
which may otherwise be denoted by such symbols as B — a, 4’, — Ay &c., which 
have been already used as members of formule expressing analogies or non- 
analogies ; writing, for example, 
Ags Ay SSO) TAS A) ape - ] (48) 
wt wa 
ee f {ane / 
Ay — Ap =, A; — A; = a).. J 
and extending this notation so as to introduce the corresponding abridgments, 
/ 
a —a=q, a” —o" =’; (49) 
then we may not only transform the formula (31), or the system of the formulz 
(32), by writing 
O (S55 <5 =])Or (50) 
but also, on the same plan, may substitute for the expression of the complex non- 
analogy (39) this more concise expression, 
es S <=) q- (51) 
For in this notation (as in that of the former Essay), the first, second, and third 
of the three formule, 
atSOs “a'<OF fa 0; (52) 
express, respectively, that the ordinal relation between moments, denoted by the 
letter a, is one of lateness, or of earliness, or of simultaneity ; and in like manner, 
the three written assertions, 
Dea, Da, bs, (53) 
express, respectively, that the ordinal relation between the two moments of one 
pair, denoted by b, as compared with the relation between the two moments of 
another pair, denoted by a, is one of comparative lateness, comparative earliness, 
or comparative coincidence, that is, analogy. And to mark generally the unity 
of the conception of an ordinal set, or system of ordinal relations, such as was 
considered in the foregoing article, we may agree to denote such a system or 
set of relations by writing in parentheses, with commas interposed, the symbols of 
those separate relations; and thus may write the formula, 
Q —Q=(A— Ay Ar— Ay + An — Ani) (54) 
2F2 
