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In § 2 of this paper these equations are abstractly formulated as functional equations. In § 3 it is proved that any function satisfying Equations I and II also satisfies HI, provided certain underlying conditions A and B are fulfilled. In § 4 a number system Sft is introduced whose elements are the numbers [£? r] where £ and r are real numbers. Two operations, addition and multiplication, are introduced, and it is shown that these operations obey all the ordinary laws of algebra except the associative law of addition. This number system J£ is then used in the discussion of the complete independence t of Equations I, II and III.

On the complete independence of the functional equations of involution