Sir Wiit1am Rowan Hamirron’s Researches respecting Quaternions. 223 
Multiplication of an ordinal Set by a Number. 
14. With these preparations it is easy to attach a perfectly clear conception 
to the act or process of multiplying any single ordinal relation, a, or any ordinal 
set, q, by any positive or negative number, m. For having already agreed to 
regard 1q and q, as well as 1a and a, as being symbols equivalent to each other, 
so that we have identically, or by definition, 
es ilty 10) = 1l0)9 (121) 
and adopting also from common Arithmetic, which may itself be regarded as a 
branch of the Science of Pure Time, since it involves the conception of succes- 
sion between things or thoughts as counted, the abbreviations 2, 3, &c., for the 
symbols 1 +1, 1+ 1-+1, &c., we shall have an analogous system of abbreviated 
symbols to denote the composition of any number of similar ordinal relations, 
whether those components be simple, as a, or complex, as q; namely, the fol- 
lowing : 
ata=2a ata+ta=a3a, &c.; 
q+q=2q, 4+q+q=3,, &e. 
We may also agree to write, at pleasure, 2 <a, 3 Xq, &c., instead of 2a, 3q, &c. ; 
and with this use of elementary notations, the distributive and associative pro- 
perties of multiplication offer themselves in the present theory, under the well- 
known and elementary forms, 
: 
’ 122 
j (122) 
m(a’ + a) = ma’ + ma; (m’ + m)a=m'a+ ma; (123) 
(m'm) X a =m! X (ma); (m' + m) X ma= ma; (124) 
in each of which each symbol a or a’ of a simple ordinal relation may be changed 
to the corresponding symbol q or q’ of an ordinal set, and in which we may, at 
Jirst, suppose that m,m’, m’ — m, and m’ + m, denote positive whole numbers. 
Then writing (as usual), 
<a 0 Oa — 0; (125) 
we shall be able, with the help of the interpretations in the last article, to remove 
the last mentioned restriction, and to suppose that m, m’, m’ +m, m’ — m, 
m' x m(=m'm), and m' + m(=™), denote any numbers, whole or fractional, 
VOL. XXI. 2H 
