262 Sir Wittram Rowan Hamitton’s Researches respecting Quaternions 
On the Operations of submultiplying, and of taking the Reciprocal of a numeral 
Set. 
35. As it has been found necessary to distinguish, in general, between two 
modes of multiplication of one numeral set by another, with different arrange- 
ments of the factors, so is it also necessary in this theory to distinguish generally 
between ¢wo inverse operations, namely, between the operation of division, and 
another closely connected operation, which may be called sab-multiplication. 
Vor if this last-named operation be now defined to be the returning to the multi- 
plicand, when the product and the multiplier are given, it will then be evidently 
distinct, in general, or, at least, for the case of quaternions, from the operation 
of division, which has been already defined to be the returning to the multiplier, 
when the multiplicand and product are given; because these two factors, the 
multiplier and the multiplicand, when regarded as numeral sets (at least if those 
sets be quaternions ), cannot generally change places with each other, without alter- 
ing the value of the product. To denote conveniently this new operation of szb- 
multiplication, or of returning from the set q’¢ to the set g, when the set ¢’ is 
given, we shall now introduce the conception of a reciprocal set, which may 
be denoted by any one of the three symbols, 
1 5 
Lal Weg eae (316) 
and of which the characteristic property is, that it satisfies generally the two reci- 
procal conditions, 
q'xm=a 9X¢'qd=4q, (317) 
of which the second follows from the first, and which may be more concisely 
written thus: 
a 9a (318) 
Thus, whether a numeral set g be multiplied or premultiplied by its reciprocal 
set q', the product in each case is unity; and when these two reciprocal sets 
are employed to operate, as successive multipliers, on any ordinal or numeral set 
as a multiplicand, they neutralize the effects of each other. It follows hence, 
that to submultiply by any numeral set is equivalent to multiplying by the reci- 
procal of that set ; so that we may write generally, for such sets, the formula of 
submultiplication (as in ordinary algebra) thus : 
1 
Fi V9V=T 99 =4 (319) 
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