228 Sir WimtrAm Rowan Hamirton’s Researches respecting Quaternions. 
Successive complex Derivation : Conception of a numeral Set. 
17. Suppose that, after deducing q’ from q, by the complex derivation or sym- 
bolical multiplication (136), we again derive another ordinal set q’’ from q’ by 
another multiplication of the same sort, with the same constant coefficients of 
derivation, c, but with a new system of variable numerical multipliers, m ; which 
supposition we shall, on the same plan as before, express as follows : 
(m, Xo +--+ me Xp+-- +m, Xi) 7 =q". (143) 
Making now, in imitation of the expression (137), 
Gi (Gos thay yte anes); (144) 
we shall have, as expressions analogous to (138) and (135), the following : 
Ber aN Beads (145) 
A y= Easy A} (146) 
and thus the result of this swccessive multiplication will be a determined and 
known set, q’’. In the next place, let this resulting set, or swecessive symbolical 
product, q/’, be divided by the original set q, which was at first proposed as a 
multiplicand; we shall then obtain, by the method described in the foregoing 
article, a symbolical quotient of the form, 
q’ = q = my Xo -- Men Ke tee + My Xa (147) 
in which, on the same plan as in the formula (142), and with the same system of 
eliminational coefficients of the form /, determined by (141), we have, 
Mi, = IE, hn, (ay = a). (148) 
Substituting for a/’ its value, given by (145), (146), and by (138) or (140), and 
eliminating the numerical denominator / by (141), we find that we may write : 
Myer = Vp ye Mp, My Ne, os, ott 3 (149) 
if we establish, for conciseness, the following formula, including 7’ separate ex- 
pressions for so many separate numbers : 
n,, Yr! (2, : Long Cy, sys a, «) = (2. . bs, 3 (150) 
in which it is to be observed that the sum which enters as a divisor is the same 
