to yield qualitatively similar results, although interesting differences in the details can be expected to emerge, which will be considered latter. In the above three equations we have introduced the constants $A=(c^{4}/2GM\bar{r}^{3})^{1/2}$ and $B=(2GM/\bar{r}^{3})^{1/2}$.

To make further progress we can attempt a solution through the method of separation of variables, proposing a solution of the form: $V_{1}=V_{r} V_{\theta}$, $U_{1}=U_{r} U_{\theta}$, $\rho_{1}=\rho_{r} \rho_{\theta}$. This ansatz yields two independent systems of three equations each, one for the radial, and one for the angular dependences of the perturbations. The radial equations become:

$$\frac{d \left( V_{r} /r^{1/2} \right)}{d r} = C_{r} r^{3/2} \frac{d \rho_{r}}{d r},$$

$$\frac{d(r U_{r})}{d r} = C_{\theta} r^{2} \rho_{r},$$

$$\frac{d \left( r^{1/2} V_{r} \right)}{d r} -\left( \frac{B C_{r}}{A}\right)\frac{d\left(r^{3/2} \rho_{r}\right)}{d r}=
C_{c} \frac{U_{r}}{r^{1/2}}.$$

While the angular ones result in:

$$V_{\theta} g(\theta) =\left( \frac{A}{C_{r}} \right) \rho_{\theta},$$

$$\frac{d \rho_{\theta}}{d \theta} = \left( \frac{C_{\theta}}{A} \right) g(\theta) U_{\theta},$$

$$\frac{d\left( sin(\theta) g(\theta) U_{\theta}\right)}{d \theta} =-C_{c} sin(\theta) g(\theta) V_{\theta}.$$

In splitting the radial and angular dependences of the perturbed continuity equation, eq.(24), we have used the result of the angular equation of the perturbed radial Euler equation, eq.(30). The constants $C_{r}$, $C_{\theta}$ and $C_{c}$ are the separation constants of the problem. Firstly we turn to the angular system, 