\quad
(\text{TE-modes})\,{,} \label{3-13}\\ \frac{d}{dr}
\left(r\,j_l\left ( \frac{\omega}{c}\,a \right)\right)&=&0\,{,}
\quad r=a \quad (\text{TM-modes})\,{,} \label{3-14}\end{aligned}$$ while outside sphere they are determined by Eqs. (\[2-7\]) and (\[2-8\]). The frequency equations for perfectly conducting sphere (\[2-7\]), (\[2-8\]) and (\[3-13\]), (\[3-14\]) can be formally derived by substituting (\[3-12\]) into frequency equations (\[3-3\]) and (\[3-4\]) and taking there the limit $\varepsilon \to \infty$.

Approximate calculation of the eigenfrequencies of a dielectric sphere without using computer [@Debye] didn’t allow one to reveal the characteristic features of the respective eigenfunctions (quasi-normal modes). The computer analysis of this spectral problem was accomplished in the work [@Gastine] where the experimental verification of the calculated frequencies was accomplished also by making use of radio engineering measurements.

These studies enable one to separate all the dielectric sphere modes into the [*interior*]{} and [*exterior*]{} modes and, at the same time, into the [*volume*]{} and [*surface*]{} modes. It is worth noting that all the eigenfrequencies are complex $$\label{3-15} \omega=\omega '-i\,\omega''\,{.}$$ Thus we are dealing with “leaky modes”.

The classification of the modes as the interior and exterior ones relies on the investigation of the behaviour of a given eigenfrequency in the limit $\varepsilon \to \infty$. The modes are called ”interior” when the product $k\,a= 