Sir Witi1am Rowan Haminton’s Researches respecting Quaternions. 205 
momental set (A,, 4,,-- An), regarded as an ordinator set, by comparing each 
moment of the one set with the homologous moment of the other set, primary 
with primary, secondary with secondary, and so forth, we shall obtain in general 
n different ordinal relations, which may, however, be combined, in thought and 
in expression, into one system, or ORDINAL SET; and this set, which may be said 
to be of the n order, will admit of 3” different forms, obtained by attributing 
separately to each of its m constituent ordinal relations each of the 3 forms 
> <=. For example, the complex ordinal relation which a sought momental 
quaternion Q’, regarded as an ordinand, bears to a given momental quaternion a, 
regarded as an ordinator, is composed of four ordinal relations between the homo- 
logous moments of these two momental sets, of which four relations each sepa- 
rately may be one of subsequence ( >), or of precedence (< ), or of simultaneity 
(=): and hence this complex ordinal relation of Q’ to @ may receive any one of 
3* = 81 different forms, of which one, namely, the case of quadruple momental 
coincidence, has been considered in the first article, and of which the others may 
be denoted on a similar plan. Thus to write the formula 
Or, =) <=) 0; (31) 
if @ and Q’ denote the quaternions (1) and (2), may be regarded as a mode of 
concisely expressing the following system of four separate ordinal relations be- 
tween moments, 
AG Ags Aq Anis Ag Ap as Aare (32) 
or, in the notation of the second article, 
M,Q@ >M,Q; M,Q’=M,Q; M,Q' <M,Q; M,Q' = MQ; (33) 
and similarly in other cases. 
4. Again, as we have compared two moments, or two sets of moments, or have 
conceived them to be compared with each other, with a view to discover the 
(simple or complex) ordinal relations existing between them, so we may now 
compare, or conceive to be compared, two pairs of moments, or of momental sets, 
with respect to their (simple or complex) analogy or non-analogy ; that is, with 
respect to the s¢milarity or dissimilarity of the two simple or complex ordinal 
relations, which are discovered by the two separate comparisons of the moments 
