256 Sir Wit1am Rowan Hamivton’s Researches respecting Quaternions. 
General Division of one numeral Set by another: Combination of the Opera- 
tions of Division and Multiplication of Quaternions. 
32. The general division of one numeral set by another, if regarded as the 
operation of returning to the multiplier, from the product and the multiplicand, 
involves no theoretical difficulty, since it depends on the solution, by elimination 
or otherwise, of a finite system of ordinary equations of the first degree, between 
the sought numerical constituents of the quotient ; and it has been already exem- 
plified, for couples and quaternions, in the nineteenth and twenty-seventh articles. 
But it is of essential importance to observe that, if division of numeral sets be 
thus defined by the formula, 
(+9) xq=q"s (296) 
in which, as in all other cases, we conceive the symbol of the multiplier to be 
placed at the left hand, and which is analogous to (129), we shall then not 
have, generally, for numeral sets, as for numbers, this other usual equation : 
gx (q+ g=q". 
In fact, if we were to assume, for example, that this latter and usual equation, 
though true for numbers and for numeral couples, was generally true for numeral 
quaternions also, we should then, in consequence of the definitional formula 
(296), which fixes the correlation of the signs X and +, with respect to numeral 
sets, be virtually assuming, also, that equation of commutative multiplication, 
q'7 = 947; which, for the case of quaternions at least, we have already seen reason 
to reject. Hence follows the important consequence that, in this case of quater- 
nions, the first member, g x (q’’ + 7), of the lately rejected equation, is the 
symbol of a new quaternion, distinct in general from the operand quaternion, 
¢', which has been first divided and afterwards multiplied by ene common ope- 
rator quaternion, q; these two operations, thus performed, having not generally 
neutralized each other, on account of the generally noncommutative character 
of the multiplication of numeral quaternions. It is, therefore, already an object 
of interest in this theory, and will be found to be a problem of which the geome- 
trical and physical applications are in a high degree important, ¢o determine the 
constituents of that new quaternion, q,, distinct from 9’’ 
sented by the symbol ¢ x (q’+ q), or which satisfies the equation 
IX (UF )=4 (297) 
, which is thus repre- 
a 
