  The $T$-matrix, often obtained with Waterman's extended boundary condition
method (EBCM), is a widely-used tool for fast calculations of electromagnetic
scattering by particles. Here we investigate the quasistatic or long-wavelength
limit of this approach, where it reduces to an electrostatics problem. We first
present a fully electrostatic version of the EBCM/T-matrix method (dubbed
ES-EBCM). Explicit expressions are then given to quantitatively express the
long-wavelength limit of the EBCM matrix elements in terms of those of the
ES-EBCM formalism. From this connection we deduce a number of symmetry
properties of the ES-EBCM matrices. We then investigate the matrix elements of
the ES-EBCM formalism in the special case of prolate spheroids. Using the
general electrostatic solution in spheroidal coordinates, we derive fully
analytic expressions (in the form of finite sums) for all matrix elements.
Those can be used for example for studies of the convergence of the $T$-matrix
formalism. We illustrate this by discussing the validity of the Rayleigh
hypothesis, where analytical expressions highlight clearly the link with
analytical continuation of series.
