254 Sir Wrt11am Rowan Hamirton’s Researches respecting Quaternions. 
A,..-Ds which have been already given in the equations (261). . (264). 
We may perceive that they would conduct also to the relations (267), (268) 
between those coefficients, and to the formula (266) for the decomposition of a pro- 
duct of three sums, containing each four squares, by eliminating the modulus p* 
of the quaternion (282) between two equations analogous to (251), namely, the 
two following : : 
» She ha Paes (286) 
where p, ’, #” have the significations (250), and where 
ee =m tm tm +m, &’=e@w+h+ee+a. (287) 
Addition and Subtraction of Numeral Sets ; Non-commutative Character of 
Quaternion Multiplication. 
31. Any two numeral sets may be added to each other, by adding their 
respective constituent numbers, primary to primary, secondary to secondary, and 
soforth ; and on a similar plan may swbtraction of such sets be performed; thus, 
for any two numeral quaternions we may write, 
(mj, mi, ms, m5) = (m,, m,, M,, M;) 
= (mem, m +m, m= m, m= m;); (288) 
and generally, by using = and A as the characteristics of sum and difference, and 
employing those signs of numeral separation which were proposed in the twenty- 
first article, we may write formule for sums and differences of numeral sets, which 
are analogous to, and may be considered as depending upon those marked (116), 
for the addition and subtraction of ordinal sets; namely, the following : 
Nre2g = =N,g; N-Ag = Anzg. (289) 
For the multiplication of numeral sets, we have already established principles 
and formule which involve, generally, the distributive and the associative pro- 
perties of the operation of the same name, as performed on single numbers ; but 
which do not retain, in general, the commutative property of that ordinary ope- 
ration upon numbers. Thus we may write, 
zy’ X tg = E(q' XQ), (290) 
Uxdqa=V¢xqa=¢¢y (291) 
and also, 
ee oe ae one 
