  We study the effect of the chemical potential on the results of the BCS
theory of superconductivity. We assume that the pairing interaction is
manifested between electrons of single-particle energies in an interval $[\mu -
\hbar\omega_c, \mu + \hbar\omega_c]$, where $\mu$ and $\omega_c$ are parameters
of the model--$\mu$ needs not be equal to the chemical potential of the system,
denoted here by $\mu_R$. The BCS results are recovered if $\mu = \mu_R$. If
$\mu \ne \mu_R$ the physical properties change significantly: the energy gap
$\Delta$ is smaller than the BCS gap, a population imbalance appears, and the
superconductor-normal metal phase transition is of the first order. The
quasiparticle imbalance is an equilibrium property that appears due to the
asymmetry with respect to $\mu_R$ of the single-particle energy interval in
which the pairing potential is manifested.
  For $\mu_R - \mu$ taking values in some ranges, the equation for $\Delta$ may
have more than one solution at the same temperature, forming branches of
solutions when $\Delta$ is plotted vs $\mu_R-\mu$ at fixed $T$. The solution
with the highest energy gap, which corresponds to the BCS solution when $\mu =
\mu_R$, cease to exist if $|\mu-\mu_R| \ge 2\Delta_0$ ($\Delta_0$ is the BCS
gap at zero temperature). Therefore the superconductivity is conditioned by the
existence of the pairing interaction and also by the value of $\mu_R - \mu$.
