  A first-order relativistic wave equation is constructed in five dimensions.
Its solutions are eight-component spinors, which are interpreted as
single-particle fermion wave functions in four-dimensional spacetime. Use of a
``cylinder condition'' (the removal of explicit dependence on the fifth
coordinate) reduces each eight-component solution to a pair of degenerate
four-component spinors obeying the Dirac equation. This five-dimensional method
is used to obtain solutions for a free particle and for a particle moving in
the Coulomb potential. It is shown that, under the cylinder condition, the
results are the same as those from the Dirac equation. Without the cylinder
condition, on the other hand, the equation predicts some interesting new
phenomena. It implies the existence of a scalar potential, and for zero-mass
particles it leads to a four-dimensional fermionic equation analogous to
Maxwell's equation with sources.
