  There exists a paradox in quantum field theory: substituting a field
configuration which solves a subset of the field equations into the action and
varying it is not necessarily equivalent to substituting that configuration
into the remaining field equations. We take the $S^4$ and Freund-Rubin-like
instantons as two examples to clarify the paradox. One must match the
specialized configuration field variables with the corresponding boundary
conditions by adding appropriate Legendre terms to the action. Some comments
are made regarding exceptional degenerate cases.
