In this section, we will prove that the roots of the Lamé functions are: (i) real, (ii) unequal, and (iii) belong to the interval [–*h*_{2}, *h*_{2}]. We will prove the results for the variable *x*, representing any one of the ellipsoidal variables *ρ, μ, ν.*

**Proposition 5.1** *If k*(*x*), *l*(*x*), *m*(*x*), *n*(*x*) *are Lamé functions that belong to the Lamé classes K, L, M, N, respectively, then the functions*

*do not vanish at the points x* = ± *h*_{3} and *x* = ± *h*_{2}. *That is, the polynomial part of the Lamé functions have no roots at the points* ±*h*_{3} *and* ±*h*_{2}.

*Proof* It is straightforward to show that ...

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