that is, (g'h' ) (g"h") is (g'h"') or 0 according as h' is or is not the same as g". Each class forms a sub-algebra of the algebra, the class (11) containing the idempotent basis. The class (22) may or may not contain an idempotent. If (11) contains two distinct idempotents, the process may be repeated, giving classes which we may represent by (11), (12), (13), (21), (22), (23), (31), (32), (33), with the analogous multiplication table. This division can be carried on in the same manner, giving multiplication tables of the form of perfect quadrates. The process stops when all independent idempotents have been found and isolated, so that every class on the main diagonal contains at most one idempotent, and otherwise only nilpotents. The classes not appearing on the main diagonal contain skew numbers only, alnd their squares vanish. Some of the