\sqrt
{\varepsilon}\,\omega \,a/c$ remains finite in the limit $\varepsilon \to \infty$, provided the imaginary part of the frequency ($\omega ''$) tends to zero. The modes are referred to as ”exterior” when the product $k\,a/\sqrt{\varepsilon}=
\omega\,a/c$ remains finite with growing $\omega''$. In the first case the frequency equations for a dielectric sphere (\[3-3\]) and (\[3-4\]) tend to Eqs. (\[3-13\]) and (\[3-14\]) and in the second case they tend to Eqs. (\[2-7\]) and (\[2-8\]). The order of the root obtained will be denoted by the index $r$ for interior modes and by $r'$ for exterior modes. Thus $\text{TE}_{\,lr}$ and $\text{TM}_{\,lr}$ denote the interior TE- and TM-modes, respectively, while $\text{TE}_{\,lr'}$ and $\text{TM}_{\,lr'}$ stand for the exterior TE- and TM-modes.

For fixed $l$ the number of the modes of exterior type is limited because the frequency equations for exterior oscillations of a perfectly conducting sphere (\[2-7\]) and (\[2-8\]) have finite number of solutions (see the preceding Section). In view of this, the number of exterior TE- and TM-modes is given by the following rule. For even $l$ there are $l/2$ exterior TE-modes and $l/2$ exterior TM-modes, for odd $l$ the number of the modes $\text{TE}_{\,l\,r'}$ is $(l+1)/2$ and the number of the modes $\text{TM}_{\,l\,r'}$ 