  The Tracy-Widom distribution functions involve integrals of a Painlev\'e II
function starting from positive infinity. In this paper, we express the
Tracy-Widom distribution functions in terms of integrals starting from minus
infinity. There are two consequences of these new representations. The first is
the evaluation of the total integral of the Hastings-McLeod solution of the
Painlev\'e II equation. The second is the evaluation of the constant term of
the asymptotic expansions of the Tracy-Widom distribution functions as the
distribution parameter approaches minus infinity. For the GUE Tracy-Widom
distribution function, this gives an alternative proof of the recent work of
Deift, Its, and Krasovsky. The constant terms for the GOE and GSE Tracy-Widom
distribution functions are new.
