Stieltjes' Theorem

The $m+1$ Ellipsoidal Harmonics when $\kappa_1$, $\kappa_2$, and $\kappa_3$ are given can be arranged in such a way that the $r$th function has $r-1$ zeros between $-a^2$ and $-b^2$ and the remaining $m+r-1$ zeros between $-b^2$ and $-c^2$ (Whittaker and Watson 1990).

See also Ellipsoidal Harmonic

References

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 560-562, 1990.