  We consider the critical points (equilibria) of a planar potential generated
by $n$ Newtonian point masses augmented with a quadratic term (such as arises
from a centrifugal effect). Particular cases of this problem have been
considered previously in studies of the circular restricted $n$-body problem.
We show that the number of equilibria is finite for a generic set of
parameters, and we establish estimates for the number of equilibria. We prove
that the number of equilibria is bounded below by $n+1$, and we provide
examples to show that this lower bound is sharp. We prove an upper bound on the
number of equilibria that grows exponentially in $n$. In order to establish a
lower bound on the maximum number of equilibria, we analyze a class of
examples, referred to as ``ring configurations'', consisting of $n-1$ equal
masses positioned at vertices of a regular polygon with an additional mass
located at the center. Previous numerical observations indicate that these
configurations can produce as many as $5n-5$ equilibria. We verify analytically
that the ring configuration has at least $5n-5$ equilibria when the central
mass is sufficiently small. We conjecture that the maximum number of equilibria
grows linearly with the number of point masses. We also discuss some
mathematical similarities to other equilibrium problems in mathematical
physics, namely, Maxwell's problem from electrostatics and the image counting
problem from gravitational lensing.
