  We report the radial-velocity discovery of a second planetary mass companion
to the K0 V star HD 37605, which was already known to host an eccentric, P~55
days Jovian planet, HD 37605b. This second planet, HD 37605c, has a period of
~7.5 years with a low eccentricity and an Msini of ~3.4 MJup. Our discovery was
made with the nearly 8 years of radial velocity follow-up at the Hobby-Eberly
Telescope and Keck Observatory, including observations made as part of the
Transit Ephemeris Refinement and Monitoring Survey (TERMS) effort to provide
precise ephemerides to long-period planets for transit follow-up. With a total
of 137 radial velocity observations covering almost eight years, we provide a
good orbital solution of the HD 37605 system, and a precise transit ephemeris
for HD 37605b. Our dynamic analysis reveals very minimal planet-planet
interaction and an insignificant transit time variation. Using the predicted
ephemeris, we performed a transit search for HD 37605b with the photometric
data taken by the T12 0.8-m Automatic Photoelectric Telescope (APT) and the
Microvariability and Oscillations of Stars (MOST) satellite. Though the APT
photometry did not capture the transit window, it characterized the stellar
activity of HD 37605, which is consistent of it being an old, inactive star,
with a tentative rotation period of 57.67 days. The MOST photometry enabled us
to report a dispositive null detection of a non-grazing transit for this
planet. Within the predicted transit window, we exclude an edge-on predicted
depth of 1.9% at >>10sigma, and exclude any transit with an impact parameter
b>0.951 at greater than 5sigma. We present the BOOTTRAN package for calculating
Keplerian orbital parameter uncertainties via bootstrapping. We found
consistency between our orbital parameters calculated by the RVLIN package and
error bars by BOOTTRAN with those produced by a Bayesian analysis using MCMC.
