Sir Witt1aAm Rowan Hamirton’s Researches respecting Quaternions. 251 
On the Extension of the Theory of Multiplication of Quaternions to other 
numeral Sets. 
29. This seems to be a proper place for offering a few remarks on the treat- 
ment of the general equation (214), which may assist in the future extension of 
the present theory of multiplication of quaternions to other numeral sets; and 
may serve, in the meanwhile, to throw some fresh light on the process which has 
been employed in the twenty-fourth and twenty-fifth articles, for discovering a 
mode of satisfying that general equation, in the case when the exponent 7 of the 
order of the set is 4. 
Let 2, 7, ..%,_, be a system of m symbolical multipliers, which we shall 
assume to be unconnected with each other by any /inear relation; and let us 
establish the following formula, analogous to (225), 
CH iy Dios a Pee oP aL aay (269) 
Then, operating by the characteristic 2.7, on the equation (214), we shall trans- 
form that equation into the following : 
OS) (0 Ges — Gat ee) 5 (270) 
and may satisfy it by supposing 
Ga UG ates (Io ta Geeta 3 (271) 
for we shall then have 
3 DA Ff == 0b hy SS Oey es SS Ch the Senos (22), 
We are therefore to endeavour to satisfy the symbolical condition, 
Dial adie COUSt=t = a2 (273) 
this constant g, being independent of ¢ and wu, and the 7 symbols z,, 7,, &c., being 
still unconnected by any linear relation. When this shall have been accomplished, 
we may then employ the formula, 
Mr tga (274) 
which will give 
SS CRO SCF A Se Oe (275) 
and therefore will agree with the formula (153). And thus the equations of 
