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We present a spectroscopic analysis of Eridanus IV (Eri IV) and Centaurus I
(Cen I), two ultra-faint dwarf galaxies of the Milky Way. Using IMACS/Magellan
spectroscopy, we identify 28 member stars of Eri IV and 34 member stars of Cen
I. For Eri IV, we measure a systemic velocity of $v_{sys} =
-31.5^{+1.3}_{-1.2}\:\mathrm{km\:s^{-1}}$ and velocity dispersion $\sigma_{v}=
6.1^{+1.2}_{-0.9}\:\mathrm{km\:s^{-1}}$. Additionally, we measure the
metallicities of 16 member stars of Eri IV. We find a metallicity of
$\mathrm{[Fe/H]}=-2.87^{+0.08}_{-0.07}$ and resolve a dispersion of
$\sigma_{\mathrm{[Fe/H]}} = 0.20\pm0.09$. The mean metallicity is marginally
lower than all other known ultra-faint dwarf galaxies, making it one of the
most metal-poor galaxies discovered thus far. Eri IV also has a somewhat
unusual right-skewed metallicity distribution. For Cen I, we find a velocity
$v_{sys} = 44.9\pm0.8\:\mathrm{km\:s^{-1}}$ and velocity dispersion $\sigma_{v}
= 4.2^{+0.6}_{-0.5} \:\mathrm{km\:s^{-1}}$. We measure the metallicities of 27
member stars of Cen I, and find a mean metallicity $\mathrm{[Fe/H]} =
-2.57\pm0.08$ and metallicity dispersion $\sigma_{\mathrm{[Fe/H]}} =
0.38^{+0.07}_{-0.05}$. We calculate the systemic proper motion, orbit, and the
astrophysical J-factor for each system, the latter of which indicates that Eri
IV is a good target for indirect dark matter detection. We also find no strong
evidence for tidal stripping of Cen I or Eri IV. Overall, our measurements
confirm that Eri IV and Cen I are dark matter-dominated galaxies with
properties largely consistent with other known ultra-faint dwarf galaxies. The
low metallicity, right-skewed metallicity distribution, and high J-factor make
Eri IV an especially interesting candidate for further followup.