208 Sir Wix1am Rowan Hamitton’s Researches respecting Quaternions. 
Qi ales =, Gis =) Q” 22 gs } (41) 
(1h (> a ae =) Q’ —Q”. 
We may also employ inversion, that is, we may substitute extremes for means, 
and means for extremes, provided that we, at the same time, change each of the 
two signs of ordinal diversity between moments, and every complex sign of 
ordinal non-analogy between momental pairs, to the contrary or opposite sign, 
by changing > to <, and < to >; thus we may write the complex non-analogy 
(39) under this other or znverse form : 
oe = a4(<, =; SS, =)e—q. (42) 
And with the same conceptions, and the same plan of notation, we are led to 
regard the following formula of guadruple momental analogy, 
tr 
oq Se(= =) ==) eG (43) 
as being only a fuller expression of that complex analogy between the two pairs of 
quaternions Q, 9’, and Q”, Q’”, which is more briefly denoted by the formula 
(35). 
5. Consistently with the same modes of interpreting formule for the expres- 
sion of any simple or complex analogy or non-analogy between pairs of moments 
or of sets, or of any similarity or dissimilarity between simple or complex ordinal 
relations, if we agree that the symbol 0, when it occurs as one member of any 
such formula, shall be regarded as a symbol of the relation of ordinal identity, 
writing thus for any two identical moments, or identical sets, 
a—a=0, ga—Q=0; (44) 
we may then not only write 
AI —O0h SQ: 10: 10) (45) 
as transformations of the equations (30) and (3) ; but also 
Ac WAS) OseemAt — Al-<aan()! (46) 
as transformations respectively of the two formule of ordinal diversity, (28) and 
(29); and may write 
= i(3) <=) 05 (47) 
instead of the formula (31). And if we employ small Roman letters, with 
or without accents or indices, such as a, a, &c., to denote generally any ordinal 
