216 Sir Witt1Am Rowan Haminton’s Researches respecting Quaternions. 
giving 
ki =93 th ==. (92) 
The importance and singularity of these results (86) (88) (90) (92) induce us 
to collect them here into one view, as follows : 
3 Sie | 
i=t; = —v (93) = (8) 
ht o5) he | 
9. It ought, however, to be observed, that when once the fundamental for- 
mula, or continued equation (A), has been established, no new operations of 
actual derivation of quaternions, by inversions and transpositions of ordinal rela- 
tions between moments, such as have been performed in the foregoing article, 
are necessary, for the deduction of these equations (B). Thus if we knew, by 
any process independent of the actual derivations (84), that ? = ijk = — 1, or 
that ?q = ijkq = — q, whatever ordinal quaternion q may be, we could infer 
immediately that 
jkq = —Pjkq = —i.tjkq = —1(—q) = 1g, (94) 
and thus could return to the symbolic equation (86), or to the essential part of 
the relation (84), from the equations (A). Again, from those equations (A) we 
can infer that 
kg zijkg=—qrkq=k.kq, (95) 
and, therefore, suppressing the symbol kq of the common operand, which may 
represent any ordinal quaternion, we obtain the first equation (90), namely, 
ij =k. Operating on this by 7, and changing 7” to — 1, we find the second 
equation (92), = — Jj. Operating with this on —hq, we obtain again 7 = jk. 
Operating on this by 7, we get ji = — h; that is, we are conducted to the second 
equation (90). Operating with this on — 7q, we find the first equation (92), 
namely, ki = j. And, finally, operating on this equation by /, we are brought 
to the equation (88), namely, kj = — 2, which completes the symbolic deduction 
of (8) from (4). 
Either by a deduction of this sort, or by actually performing the operations 
indicated, we find also that 
ka =; (96) 
