Division.-From the **commutative** law for multiplication, which shows that 3 X 4d.

This is, of course, on the usual assumption that the sign of a product is changed when that of any one of its factors is changed, - which merely means that-1 is **commutative** with all other quantities.

This is included in the preceding, but it is simpler in that the various operations are **commutative**.

The method is similar in some respects to the treatment of non **commutative** algebras.

The effect of these definitions is that the sum and the product of two quaternions are also quaternions; that addition is associative and **commutative**; and that multiplication is associative and distributive, but not **commutative**.