222 Sir Wret1am Rowan Hamitton’s Researches respecting Quaternions. 
Y 
two symbols 0—q and—q will thus be equisignificant, each denoting the inverse 
or opposite of that complex ordinal relation between two sets of moments, which 
is denoted by the symbol q ; because the symbol — q has been already defined 
to denote that inverse relation, and therefore we have now the two equations, 
(—q) +q=0, (0 — q)-+q=0; and the other isolated, but affected symbol, 
+q, may in like manner be interpreted as bemg equivalent in signification to 
0+ q, and therefore to q. With the conceptions of addition and subtraction, 
or of composition and decomposition of ordinal relations, which correspond to 
these notations, we may write: 
(aiyb'.3. care aah. era (a crab by 5 5 (114) 
By oe eee ce BaS } (115) 
R, (q 1d) =R8q + 8,q;... 
or, using © and A as characteristics of swm and difference, we may establish the 
important identities : 
ep eSPI ACLS FAG ss “NAGE (116) 
Addition of ordinal sets is a commutative and also an associative operation ; that 
is, we have the formule, 
i ee a (117) 
(ost Mt er dears) > (118) 
the former of these two properties of addition being connected with the principle 
of alternation of an analogy, which was mentioned in the fourth article. An 
ordinal set, of any order 7, may always be regarded as the swm of other sets of 
the same order, in each of which only one constituent ordinal relation (at most) 
shall be a relation of diversity ; for we may write, generally, 
q = (8 q, 0,..) + (0, R, gq, ..) + &e. (119) 
Thus, for example, the ordinal quaternion (73) may be expressed as the sem of 
four others, which may be called respectively a pure primary (ordinal quater- 
nion), apure secondary, pure tertiary, and pure quaternary, as follows: 
(a, b, c, d) = (a, 0, 0, 0) + (0, b, 0, 0) + (0, 0, c, 0) + (0,0, 0, d). (120) 
