The five **equalities** which stand first in the five pairs of **equalities** in ï¿½ 15 (2) may therefore be taken as the main types of a simple statement of **equality**.

Similarly the **equalities** 99 X I o I = 9999 = wow - I 98 X 102 = 999 6 = moo() - 4 97 X 10 3 =9991 =1 0000 - 9 lead up to (A - a) (A+a) = A 2 - a 2.

The symbol e 0 behaves exactly like i in ordinary algebra; Hamilton writes I, i, j, k instead of eo, el, e2, es, and in this notation all the special rules of operation may he summed up by the **equalities** = - I.

All this is analogous to the corresponding formulae in the barycentric calculus and in quaternions; it remains to consider the multiplication of two or more extensive quantities The binary products of the units i are taken to satisfy the **equalities** e, 2 =o, i ej = - eeei; this reduces them to.

4.1.1 (i) **Equalities** and Inequalities