  We present a resolved-star spectroscopic survey of 15 dwarf spheroidal (dSph)
satellites of the Andromeda Galaxy (M31) as part of the Spectroscopic and
Photometric Landscape of Andromeda's Stellar Halo (SPLASH) project. We filter
foreground contamination from Milky Way (MW) stars, noting that MW substructure
is evident in this contaminant sample. We also filter M31 halo field giant
stars, and identify the remainder as probable dSph members. We then use these
members to determine the kinematical properties of the dSphs. For the first
time, we confirm that And XVIII, XXI, and XXII show kinematics consistent with
bound, dark matter-dominated galaxies. From the velocity dispersions for the
full sample of dSphs we determine masses, which we combine with the size and
luminosity of the galaxies to produce mass-size-luminosity scaling relations.
With these scalings we determine that the M31 dSphs are fully consistent with
the MW dSphs, suggesting that the well-studied MW satellite population provides
a fair sample for broader conclusions. We also estimate dark matter halo masses
of the satellites, and find that there is no sign that the luminosity of these
galaxies depends on their dark halo mass, a result consistent with what is seen
for MW dwarfs. Two of the M31 dSphs (And XV, XVI) have estimated maximum
circular velocities smaller than 12 km/s (to 1sigma), which likely places them
within the lowest mass dark matter halos known to host stars (along with Bootes
I of the MW). Finally, we use the systemic velocities of the M31 satellites to
estimate the total mass of the M31 halo: within 139 kpc, M31's mass is 8 +4.1 -
3.7 x 10^11 corresponding to a virial mass for M31's dark matter halo of 1.2
+0.9 -0.7 x 10^12, consistent with previous results.
