  In this paper we study for a given azimuthal quantum number $\kappa$ the
eigenvalues of the Chandrasekhar-Page angular equation with respect to the
parameters $\mu:=am$ and $\nu:=a\omega$, where $a$ is the angular momentum per
unit mass of a black hole, $m$ is the rest mass of the Dirac particle and
$\omega$ is the energy of the particle (as measured at infinity). For this
purpose, a self-adjoint holomorphic operator family $A(\kappa;\mu,\nu)$
associated to this eigenvalue problem is considered. At first we prove that for
fixed $|\kappa|\geq{1/2}$ the spectrum of $A(\kappa;\mu,\nu)$ is discrete and
that its eigenvalues depend analytically on $(\mu,\nu)\in\C^2$. Moreover, it
will be shown that the eigenvalues satisfy a first order partial differential
equation with respect to $\mu$ and $\nu$, whose characteristic equations can be
reduced to a Painleve III equation. In addition, we derive a power series
expansion for the eigenvalues in terms of $\nu-\mu$ and $\nu+\mu$, and we give
a recurrence relation for their coefficients. Further, it will be proved that
for fixed $(\mu,\nu)\in\C^2$ the eigenvalues of $A(\kappa;\mu,\nu)$ are the
zeros of a holomorphic function $\Theta$ which is defined by a relatively
simple limit formula. Finally, we discuss the problem if there exists a closed
expression for the eigenvalues of the Chandrasekhar-Page angular equation.
