  Using the relation established by Johnson--Zumbrun between Hill's method of
aproximating spectra of periodic-coefficient ordinary differential operators
and a generalized periodic Evans function given by the $2$-modified
characteristic Fredholm determinant of an associated Birman--Schwinger system,
together with a Volterra integral computation introduced by Gesztesy--Makarov,
we give an explicit connection between the generalized Birman--Schwinger-type
periodic Evans function and the standard Jost function-type periodic Evans
function defined by Gardner in terms of the fundamental solution of the
eigenvalue equation written as a first-order system. This extends to a large
family of operators the results of Gesztesy--Makarov for scalar Schr\"odinger
operators and of Gardner for vector-valued second-order elliptic operators, in
particular recovering by independent argument the fundamental result of Gardner
that the zeros of the Evans function agree in location and (algebraic)
multiplicity with the periodic eigenvalues of the associated operator
