Sir Witrt1am Rowan Haminron’s Lesearches respecting Quaternions. 261 
are here regarded as ¢dentical ; whereas these other usual equations, 
u / 
ee = aA = 4% 
of which the first is only an abridged way of writing a formula already rejected, 
while the second is connected therewith, are not generally true (or, at least, not 
universally so) /or numeral sets ; because the order of the factors in multiplica- 
tion is, in the present theory of such sets, not generally unimportant to the 
result. We have seen, for example, in the foregoing article, that the quaternion 
which may now be denoted by the symbol piu or by this other symbol, e, or 
by qq + q, instead of being generally equal to the quaternion gy”, is equal, in 
general, to another quaternion, q,, distinct from the former, though having 
several simple relations thereto, which will be found to be connected, in their 
geometrical and physical applications, with questions respecting the transforma- 
tion of rectangular coordinates in space, and the rotation ofa solid body. It 
may, therefore, be not useless to remark expressly here, that the following usual 
equations continue true in the present theory of numeral sets, as well as in 
common algebra: 
q q Wh ‘a7 
or, in words, that a fraction is multiplied by a numeral set when its mewmerator is 
multiplied thereby ; and that the va/we of a fraction, regarded as representing a 
numeral set, remains wachanged, or represents the same set as before, when its 
numerator and its denominator are both premultiplied, or both divided, by any 
common set (of the same order); both which results depend on the associative 
property of multiplication, and on the principle that two numeral sets cannot 
generally give equal products, when operating as multipliers on one common 
multiplicand (different from zero), unless they be themselves equal sets. These 
general remarks will become more clear by their future applications; meanwhile, 
we may here agree to use occasionally, for convenience and variety, another form 
of expression, consistent with the foregoing principles, and to say that, in the 
product q’q, the left hand factor, q’, is multiplied ixfo the right hand factor, g, 
as the latter has been said to be multiplied by the former, and as that former 
factor again has been said to be premultiplied by the latter. 
