(\[a-27\])). As a result, the eigenfrequencies of electromagnetic field for this configuration are determined [@Stratton] by the frequency equation for the TE-modes $$\label{3-3} \Delta^{\text{TE}}_l(a\omega)\equiv
\sqrt{\varepsilon_1\mu_2}\,\hat j_l'(k_1a)\,\hat h_l(k_2a)-
\sqrt{\varepsilon_2\mu_1}\,\hat j_l(k_1a)\,{\hat h_l}'(k_2a)=0\,{}$$ and by the analogous equation for the TM-modes $$\label{3-4} \Delta^{\text{TE}}_l(a\omega)\equiv
\sqrt{\varepsilon_2\mu_1}\,\hat j_l'(k_1a)\,\hat h_l(k_2a)-
\sqrt{\varepsilon_1\mu_2}\,\hat j_l(k_1a)\,{\hat
h_l}'(k_2a)=0\,{,}$$ where $k_i=\sqrt{\varepsilon_i\mu_i}\,\omega/c,\quad i=1,2$ are the wave numbers inside and outside the sphere, respectively, and $\hat j_l(z)$ and $\hat h_l(z)$ are the Riccati-Bessel functions [@AS] $$\label{3-5} \hat j_l(z)=z\,j_l(z)=\sqrt{\frac{\pi
z}{2}}\,J_{l+1/2}(z)\,{,}\quad \hat
h_l(z)=z\,h_l^{(1)}(z)=\sqrt{\frac{\pi
z}{2}}\,H^{(1)}_{l+1/2}(z)\,{.}$$ In Eqs. (\[3-3\]) and (\[3-4\]) the orbital momentum $l$ assumes the values $1,2,\ldots $, and prime stands for the differentiation with respect of the arguments $k_1a$ and $k_2a$ of the Riccati-Bessel functions.

The frequency equations for a dielectric sphere of permittivity $\varepsilon $ placed in vacuum follow from Eqs. (\[3-3\]) and (\[3-4\]) after putting there $$\label{3-12} \varepsilon_1=\varepsilon, \quad
\varepsilon_2=\mu_1=\mu_2=1\,{.}$$ The roots of these equations have been studied in the Debye paper [@Debye] by making use of an approximate method. As the starting solution the eigenfrequencies of a perfectly conducting sphere were used. These frequencies are different for electromagnetic oscillations inside and outside sphere. Namely, inside sphere they are given by the roots of the following equations $(l\geq 1)$ $$\begin{aligned}
j_l\left ( \frac{\omega}{c}\,a \right)&=&0 