and determine the conditions on a polynomial F (x, y) in order that (1) can have a continuous strictly monotone (say increasing) solution. If F(x, y) is x + y or xy, we have the classical cases of Cauchy's functional equations. The cases when F(x, y) is a general linear function,1 and also the case when it is x + y + nxy, have been treated.2 In this paper the case when F(x, y) is a polynomial of degree greater than unity is settled, as follows: