206 Sir Wrmuram Rowan Hamitron’s Researches respecting Quaternons. 
or sets belonging to each separate pair. Representing (as in the former Essay ) 
by the notation 
D—c=B—4A, (34) 
the existence of an analogy of this sort between the two pairs of moments, a, B, 
and c, D, or the supposition of an exact similarity between the two ordinal rela- 
tions of p to c, and of B to A; we may, in like manner, denote by the formula, 
qe’ — a’ =e’ — Q, (35) 
the complex analogy which may be conceived to exist between the two pairs of 
quaternions, or other momental sets, Q, Q’, and Q”, Q’”, belonging all to any one 
determined order 7, that is, containing each » moments. ‘This analogy (35) 
requires, for its existence, in the view here taken, that the m constituent ordinal 
relations between moments which compose, by their mental and symbolic combi- 
nation into one system, the complex ordinal relation of the set Q’”’ to the set Q”, 
should, separately and respectively, be exactly similar to those 7 other constituent 
ordinal relations between moments, which collectively compose the other complex 
ordinal relation of the set @’ to the set a; for then, but not otherwise, do we 
regard the one complex ordinal relation as being in all respects similar to the 
other. In symbolical language, the complex set-analogy (or analogy between 
pairs of sets) of the n order (35) may be resolved into n momental analogies 
(or analogies between pairs of moments), namely, the following: 
MQ” o M,Q” = M,Q’ ‘Ay M,Q ; | 
ee Ny (36) 
of which each separately is to be interpreted on the same plan as the analogy (34). 
The two formule of momental non-analogies, or of dissimilar ordinal relations 
between pairs of moments, 
= CS: ih Sy ] a 
NE O< Pes i (37) 
may still be interpreted as in the former Essay; the first formula (37) denoting 
that the relation of the moment p to c is, as compared with the relation of B to a, 
a relation of comparative lateness ; and the second formula (37) denoting, on the 
