210 Sir Wittram Rowan Hamirton’s Researches respecting Quaternions. 
or, more concisely, by the abridgments (48) and (49), if we confine ourselves to 
the case of an ordinal quaternion, 
G) Sep eyen os) (55) 
. 
Operations on an Ordinal Set ; Coordinate Characteristics of Quaternion- 
Derivation. 
6. We may now treat this last expression for an ordinal quaternion in the 
same way as the expression for a momental quaternion was treated in the second 
article. Let r,, R, &c., be characteristics of ordinal separation, analogous to the 
characteristics of momental separation, M,, M, &c.; we may then, with their help, 
decompose the equation (55) into four others, as follows : 
Rei == ays Gi ==yans) Rage aes" R.qy=— ans (56) 
we may therefore write, for any four ordinal relations, a, b, c, d, between mo- 
ments, the identical equations, 
aR, (a5/D, €,0)5— b= Rk, (a,b,c; di) see. 5 (57) 
and, for any ordinal quaternion, we may write the corresponding identity, 
q = (Bod Biq> Req, Bq) 5 (58) 
or more concisely, by abridgments analogous to those marked (13), 
IU==S (Risen eG) Resp Re) aR eas (59) 
with formulz of the same kind for ordinal sets of higher orders. Characteristics 
of ordinal transposition are easily formed on the same plan; and we may write, 
for example, as the expression of one such transposition performed on the ordinal 
quaternion (55), 
Pay Chess (Ey er Oro a)e (60) 
and may hence deduce this symbolic equation, analogous to (15), 
Papa sik (61) 
If, instead of thus transposing the ordinal relations, we transpose, in the ex- 
pression of any one relation, the two related moments, or momental sets, we then 
obtain, in general, a new ordinal relation, which is the enverse or opposite of the 
old relation, or is that old one with its sign (or signs) changed, each constituent 
relation of earliness being altered to a relation of lateness (in the same degree ), and 
