204 Sir Witt1am Rowan Haminton’s Researches respecting Quaternions. 
Forms of ordinal Relations between Moments, or Sets of Moments ; and Com- 
parisons of Pairs of Moments, or Pairs of Sets, with respect to Analogy or 
Non-analogy. 
3. If the moment denoted by the symbol a’ be supposed to be essentially, as 
well as symbolically, distinct from the moment denoted by a, so that these two 
symbols denote two different moments in the progression of time, and that there- 
fore the momental equation a’ = a does nut hold good; then it is an immediate 
and necessary result of our notion or tuition of time, that the moment a’, since 
it is not coincident with a, must be either later or earlier than it. Using, there- 
fore, as in a former Essay,* the signs > <, which are commonly employed as 
marks of inequality of magnitude, to denote these two modes of ordinal diversity, 
and thus employing the formula 
NM Sea5 (28) 
to express, without any reference to magnitude, that the moment 4’ is /ater than 
A; but, on the contrary, using this other formula, in like manner without refe- 
rence to magnitude, 
alncinas (29) 
to express that a’ is earlier than a; so that the character > is here used as a 
sign of subsequence, whereas the mark < is, on the contrary, in this notation, a 
sign of precedence ; while the formula, or equation, 
ei (30) 
still expresses that the moment a’ is cotncident (or simultaneous) with a, so that 
the mark = is at once an expression of symbolic equivalence and also a sign of 
simultaneity ; we see that the comparison of any sought moment 4’, regarded as 
an ordinand, with any given moment a regarded as an ordinator, must conduct 
to one or other of these three forms of ordinal relation, (28), (29), (30) ; and that 
no such comparison of two moments can conduct to two of these three forms, or 
modes of relation, at once. In like manner, if we compare any se¢ of 2 moments 
(a', 44, ++ A/n_,), regarded as an ordinand set, with any other equinumerous 
* On Algebra as the Science of Pure Time.—Transactions of the Royal Irish Academy, vol. xvii. 
Dublin, 1835. 
