Sir WinirAmM Rowan Haminton’s Researches respecting Quaternions. 255 
i} g 
the mark of multiplication being allowed to be omitted, because its place is un- 
important to the result, in the successive multiplication of any three or more 
numeral sets. But we are not at liberty to write, generally, for any two such 
sets, as factors, the commutative formula, 
79 = 97 
since, although, by the equation (182), this last formula of commutation of 
factors holds good, not only for single nwmbers, but also when the factors are 
numeral couples, of the kind considered in the nineteenth article of the present 
paper, and in the earlier Essay there referred to, yet, for the case of numeral 
quaternions, the relations (8) between the products of the symbols 7, 7, k, give 
results opposed to the commutative formula, namely, the following : 
j=—ji, jk=—kj, ki= —ik. 
In fact, by (149), or by (209), to justify generally this commutative formula 
of multiplication, as applied to numeral sets of the order m, it would be necessary 
that the x’ coefficients of multiplication should be connected with each other by 
the relations included in the type, 
n,, r,s = n+, 7,2)° (292) 
Now these relations have, indeed, been established in our theory of numeral 
couples, since, in the abridged notation of the nineteenth article, and with the 
values there adopted, we have the equations, 
va = & ) ihe = g'; or, Novo = Noo 3 Non = Myo 5 (293) 
but they do not hold good in our theory of numeral quaternions, since we have 
been led to adopt values for the coefficients of multiplication, which give, on the 
contrary, 
Myo3 = — Nyy i Mag, = — N35 NggQ = — Myp- (294) 
Thus, if we still adopt the system of values of the coefficients of quaternion mul- 
tiplication assigned in the twenty-third article, we must reject the commutative 
property; and may establish a formula which is opposite in its character to the 
equation (292), namely, the following : 
Me can Tg Dial ute oa etla >: (): (295) 
VOL. XXI. 2m 
