260 Sir Winrtam Rowan Hamiwton’s Researches respecting Quaternions. 
Thus, to multiply the numeral set g, as a multiplicand, by the numeral set q’, as 
a multiplier, comes to be regarded as being equivalent to the operations of mul- 
tiplying some single standard ordinal relation, a, or some ordinal set, q first by 
the given multiplicand set, g, and afterwards by the given multiplier set, q' ; 
and of then finding that third set, ¢, namely, the product q' x q, or qq, which, 
acting as a single multiplier, would produce the same final result, and would, 
therefore, serve, by its single operation, to replace this twofold process. In this 
view of the multiplication of one numeral set by another, the set proposed as a 
multiplicand is itself a previous multiplier, and may, therefore, be called a pre- 
muiltiplicator, or, more familiarly, a premultiplier. And thus, instead of saying 
that the product q’ X q, or q’q, is obtained by multiplying ¢ by gq’, we may be 
permitted occasionally to say that the same product results from premultiplying 
q’ by qs the symbol of the premultiplier being placed towards the right hand, as 
that of the multiplier is placed towards the left. 
With this phraseology, and with the definitional formula (296), which easily 
gives also this other connected formula, 
(Yxg)+q=7%, (312) 
division and premultiplication are mutually inverse operations ; that is to say, 
a numeral set, g’, remains, upon the whole, unchanged, when it is both divided 
and premultiplied, or both premultiplied and divided, by any other numeral set, 
g (of the same order). We may also agree to express the same results by sym- 
bols of fractional forms, a fraction being defined to be the quotient which is 
obtained when the numerator is divided by the denominator, so that we shall 
adopt here, as a definition, the formula 
4 Aa 
rik a (313) 
for then we may say that a fraction gives its numerator as the product, when it 
is premultiplied by its denominator; though it does not always, at least for the 
case of quaternions, produce that numerator when it is mu/tiplied by that de- 
nominator (the order of the factors being then different). In symbols, the 
equations 
0 
qe = (314) 
i oe ee 
a ees 
ae ee oy 
