Smeyers' procedure (1986) for the determination of linear, isentropic
oscillations of the incompressible MacLaurin spheroids is extended to
the compressible MacLaurin spheroids. It is shown that the solutions c
an be constructed by a direct integration of a finite set of different
ial equations written in spherical coordinates. Oblate spheroidal coor
dinates are used with regard to the boundary conditions that must be s
atisfied at the surface of the MacLaurin spheroid. For compressible Ma
cLaurin spheroids with eccentricities e varying from zero to unity, th
e modes are determined that stem from the fundamental radial mode and
the second-harmonic Kelvin modes in the non-rotating equilibrium spher
e with uniform mass density. The modes obtained agree with the modes d
etermined earlier by Chandrasekhar and Lebovitz (1962a, 1962b) by mean
s of the second-order tensor virial equations. Next, four axisymmetric
modes are determined that stem from the first radial overtone, the se
cond-harmonic p(1)- and g(1)(-)-mode, and the fourth-harmonic Kelvin m
ode in the non-rotating equilibrium sphere with uniform mass density.
The g(1)(-)-mode becomes dynamically stable at the eccentricity e = 0.
7724 and again dynamically unstable at e = 0.9952.