Sir Witt1aM Rowan Haminton’s Researches respecting Quaternions. 217 
that is to say, if we operate successively on any ordinal quaternion q by the 
three modes of coordinate derivation, 7,7, /, in their order (first by 2, then by J, 
and finally by £), the result will be the original quaternion itself. And if we 
make, for abridgment, in the notation of the sixth article, 
i= Ry, -0, 3 23 
nt > 4 
: Ry, -3, -0, 19 (97) 
rf . 
k = Rs, 2-1 —0? 
| Il 
so that the results of the operation of these three new characteristics, 7’, 7’, k’, on 
the quaternion (73), are, respectively, 
i’q = (b, —a,d, —c); 
Jq=(e, —d, —a, b); (98) 
k’q = (d, ce, — b, —a); 
we shall then have not only the relations, 
{= —4 9 50k = — hy (99) 
but also these others, 
= a WO | 
Jj =I =1; (100) 
i Si | 
on which account we may call these three new signs, 7’, 7’, k’, as compared with 
the signs 2, y, k, coordinate characteristics of contra-derivation, performed on an 
ordinal quaternion. 
Connexions between the coordinate Characteristics of Quaternion- Derivation 
and those of Octadic Transposition, introduced in the foregoing Articles. 
10. It may serve to throw some additional light on the foregoing relations 
between the coordinate characteristics, 7, 7, /, of quaternion-derivation, if we 
point out a connexion which exists between (1st) the system of these three signs 
and the sign —, which enters with them into the formula (4), on the one hand, 
and (2nd) the system of the four characteristics of octadic transposition, ,, w,, w., 
and w, which were considered in the second article, on the other hand. In 
th 
general, an ordinal set of the n" order, since it involves ” constituent ordinal 
262 
