Here we would like to describe the Monte Carlo procedure we use
for determining the corrections for reddening and underlying
stellar absorption in the Hydrogen lines. Beginning with an
observed line flux
*F*(), and an
equivalent width
*W*(), we can
parameterize the correction for underlying stellar
absorption as

(A1) |

The parameter *a*_{HI} is expected to be relatively
insensitive to wavelength because all of the Balmer lines should be
saturated in the stars which are producing the underlying continuum.
As described in section 2, the reddening
correction is applied to determine the intrinsic line intensity
*I*() relative to
*H*

(A2) |

We assume the intrinsic Balmer line ratios calculated by
Hummer & Storey (1987),
and we use the reddening function,
*f*(),
normalized at H, from the
Galactic reddening law of
Seaton (1979),
as parameterized by
Howarth (1983),
assuming a value of
R A_{V} / E_{B -
V} = 3.2. By comparing
*X*_{R}() to
theoretical values,
*X*_{T}(), we
determine the parameters *a*_{HI} and
*C*(*H*) self
consistently, and
run a Monte Carlo over the input data to test the robustness of the
solution and to determine the systematic uncertainty associated with
these corrections.

For definiteness, we list here the theoretical ratios we use:

(A3) |

where the temperature is
*T*_{4} = *T*/10^{4}K. The values for
*f*() we take are

(A4) |

We begin therefore with four input fluxes
*F*() along with their
associated observational (statistical) uncertainties and four equivalent
widths (and their observational uncertainties). In addition, the
theoretical ratios are temperature dependent, so we must add the
temperature and its
uncertainty as an additional observational input. From these, the ratios,
*X*_{R}() for
*H*,
*H*, and
*H* are obtained.
The ^{2} statistic is
defined by

(A5) |

where _{XR}() is the derived uncertainty in *X*_{R}.
The minimization of ^{2},
allows us to determine the values of *a*_{HI} and
*C*(*H*). The
uncertainties in the two outputs are
determined by varying the solution so that
^{2}(*a* ±
_{a}) -
^{2}(*a*) = 1.
_{C(H)} is similarly determined.

As we indicated above, we further test this solution and its robustness
by running a Monte Carlo on the input data. This also allows us to better
determine the systematic uncertainty in the output parameters
*a*_{HI}
and *C*(*H*). The
data for the Monte Carlo are generated from the input data,
*F*(),
*W*() and temperature
and the uncertainties
in these quantities. A Gaussian distribution of input values centered on
the observed values with a spread determined by their observational
uncertainties. A new set of data is then randomly
generated by picking input values from these Gaussian distributions.
Consequently, after running the
^{2} minimization
procedure, new values for the output parameters *a*_{HI} and
*C*(*H*) are found. Our
Monte Carlo produces 1000 randomly generated data sets from which we can
produce a distribution of solutions for *a*_{HI} and
*C*(*H*). One
would expect that the mean of the solutions for *a*_{HI}
and *C*(*H*)
tracks the original solution based on the actual observational data. The
spread in these allows us to test the systematic uncertainty associated
with these quantities.