Sir Witi1am Rowan Haminron’s Researches respecting Quaternions. 211 
vice versd: a change which may be expressed, according to known analogies of 
notation, by prefixmg the sign — to the symbol of the simple or complex relation 
which has thus been altered: for example, the equations (48), (49) give, by this 
change of signs, 
A, — Ag = — a, 4, — A) = —a, &e.3 (62) 
and 
q— ev =— gq, &e. (63) 
Hence we may write, as a consequence of the formula (55), the following : 
—q=(—a, —a, — a, — a,); (64) 
that is, for any ordinal quaternion, we have 
—l= (= Roy = Rp — Roy — Ry)» (65) 
with similar results for other ordinal sets. The notation may be abridged if we 
agree to write, for the present, such formule as the following : 
= = Sa So } (66) 
(REE sick. haan FPN el 
for then we can not only express the symbolical equation (65) under the shorter 
form, 
ay 1 SSP cope ep (67) 
but can compose, generally, characteristics of ordinal derivation, which shall 
express the joint or combined performance of several simultaneous or successive 
acts of separation, inversion, transposition, and recombination of the constituent 
relations of any ordinal set. Thus if we operate twice successively on an ordinal 
couple (a, 2,), by the characteristic of derivation R_, ,, we obtain thereby the 
two new or derived couples : 
B_ 1,0 (ap a) = (— ay a) 
R21,0 (a, a,) = R_1,0(— a ay) (68) 
= (— oy —a,) ==> (a » ay) 3 
of which the last is merely the original couple (a,, a,) with its sign changed; so 
that we have the symbolic equation, 
hein — le (69) 
This symbolic result, presented under a slightly different form, was made the 
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