Sir Witi1am Rowan Hamitton’s Researches respecting Quaternions. 221 
tq = (A, — Aly Ag — Ag As — Azy Ap — Ad) 
JQ = (Ag — Ay Ag — Ay Ao — Ay A — Ai)5 (111) 
kg = (Ag — Agy Ag — Ady Al — Ay Ap — Ap) 5 
the characteristics of derivation 77k being easily seen to have the same effect and 
significance here as in the recent articles. Thus the three coordinate charac- 
teristics of quaternion-derivation, 7, /, k, correspond respectively to the three co- 
ordinate characteristics of octadic transposition, , w,, w,; and since the sign — 
has been seen to correspond in like manner, as a sign of ordinal inversion per- 
formed on the quaternion q, to the other octadic characteristic w, we see that a 
correspondence is at once established between the symbolic equations (22), re- 
specting transpositions of the moments of an octad, and the formule (72) or (a), 
respecting derivations of an ordinal quaternion. ‘The equations (25) corres- 
pond in like manner to the formule (93) or (8); the octadie characteristics, 
ww, WW, ww,, correspond to the characteristics of contraderivation of a quater- 
nion, 7’, 7’, k’; the equation (27) might remind us that 7, 7, k, 7,7’, k’ are, all of 
them, symbolic fourth roots of unity; and, finally, the equations (26) show, by 
the same kind of correspondence of relations, that we may write the following 
formule, which include the results (71, 4) and (96) : 
ijk = jki = bij = — 13 | (112) 
ye = ty. = fae = 1 J 
Addition and Subtraction, or Composition and Decomposition of Ordinal 
Relations between any Sets of Moments. 
13. The usual correlation between the signs + and — may be extended by 
definition to expressions involving those signs in conjunction with symbols for 
momental and ordinal sets; and thus, by the use already mentioned of zero, the 
following equations, 
(eY —a) +Qa=a, 
(e”— Q’) + (e— a) =0"—-Q, |; (113) 
0+2=a, 
together with those others which are formed from them by changing each @ to q, 
may here, as elsewhere, be regarded as identically true. At the same time, the 
