  The geometric formulation of fiducial probability employed in this paper is
an improvement over the usual pivotal quantity formulation. For a single
parameter and single observation, the new formulation is based on the geometric
properties of an ordinary two variable function and its surface representation.
  The following theorem is proved: A fiducial distribution for the continuous
parameter $\theta$ exists if and only if (i) the continuous random probability
distributions of $x$ for different $\theta$'s are non-intersecting, and (ii)
the random distributions are complete, i.e. at the extreme values of $\theta$
the limiting probability distributions are zero and one for all $x$.
  The proof yields also a complete characterization of random distributions
that lead to fiducial distributions.
  The paper also treats intersecting distributions and non-intersecting
incomplete distributions. The latter, which are frequently encountered in a
null hypothesis, are shown to be associated with intersecting "composite"
distributions.
  An appendix compares the pivotal and geometric formulations.
