Sir Witt1Am Rowan Hamitton’s Researches respecting Quaternions. 207 
contrary, that the former ordinal relation, as compared with the latter, is one of 
comparative earliness : and because, in the first case the moment p is too Jate, 
while in the second case this moment is too early, to satisfy the analogy (34), we 
may still call the first formula (37) a momental non-analogy of subsequence, and 
may call the second formula (37) a non-analogy of precedence. By compound- 
ing several such momental non-analogies, or even one such, with any number of 
momental analogies, into one system, we shall compose a complex non-analogy 
between two pairs of momental sets, which may easily be denoted on the plan of 
recent notations ; thus, if we make, for abridgment, 
ut 
LL au ut uw 
Q = (55 Win SoZ )b 71 
i wt “i aw wa 
Q ='(A5 9 A, » Ao s Ag ); i 
retaining for Q and q’ the same meanings as in the equations (1), (2), and then 
(38) 
write the formula 
Q 10% (=) <<, =) S— Q, (39) 
we are to be considered as expressing concisely hereby a complex non-analogy be- 
tween two pairs of momental quaternions, @, 9’, and q”, Q’”, which may be resolved 
into the following system of mixed analogies and non-analogies between four 
pairs of moments : 
/ "I « . 
M,o/” — M,Q” > M,Q’ — M,Q@3 OF, Ag — Ap > Ap — Ay | 
mo” — uo’ =u’ — MQ; Ay’ — Ay = Ai — A L (40) 
M,Q’” — M,0" < M,Q’ — M,Q; IE mS CR INA oe 
Mtr i — / 5 wat Wa ee 
MQ’ — MQ” =™,o' — MQ; Ie ee IN SINE IN j 
A little consideration suffices to show, by the aid of the fundamental notion 
of TIME, which enters essentially into this whole theory (as least as the subject is 
here viewed), that every simple or complex analogy or non-analogy of the kind 
considered in the present article admits of alternation ; that is to say, if we call 
the moments B and ¢, or the sets q’ and q”, the means, and call the moments a 
and p, or the sets q and q’”, the extremes, of the analogy or non-analogy, it is 
allowed to interchange the means or to interchange the extremes among them- 
selves, without destroying the truth or changing the character of the formula. 
For example, under the conditions (40), we may write, instead of (39), either 
of the two following forms : 
VOL. XXI. 2F 
