The vector wave equation in prolate spheroidal coordinates $\xi$, $\eta$, $\phi$ is set up, the variables are separated, and the characteristic values (eigenvalues) and characteristic functions (eigenfunctions) of the resulting ordinary differential equations are obtained in series which converge rapidly in the neighborhood of resonance for spheroids of eccentricity near to unity. The coefficients of the two independent primitives in the linear combinations representing diverging and converging waves at infinity are calculated. The zeros of certain of the characteristic values are investigated.

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