Summary. In an earlier nsner a theorem on the density of the density ordinate itself was obtained and used to provide an alternative proof of the Box-Muller method for the generation of Normal deviates. A more general theorem on what may be called vertical density representation is given here, The Box-Muller method requires using two uniform random numbers to produce two Normal deviates, If a method using only one uniform random number may be regarded as an ultimate method, then evidence is provided here that this limit cannot be achieved via such methods