  Symbiotic binaries are comprised of nebulae, whose densest portions have
electron concentrations of 1E+8 - 1E+12 cm-3 and extend to a few AU. They are
optically thick enough to cause a measurable effect of the scattering of
photons on free electrons. In this paper we introduce modelling the extended
wings of strong emission lines by the electron scattering with the aim to
determine the electron optical depth, tau(e) and temperature, T(e), of
symbiotic nebulae. We applied our profile-fitting analysis to the broad wings
of the OVI 1032, 1038 A doublet and HeII 1640 A emission line, measured in the
spectra of symbiotic stars AG Dra, Z And and V1016 Cyg. Synthetic profiles fit
well the observed wings. By this way we determined tau(e) and T(e) of the layer
of electrons, throughout which the line photons are transferred. During
quiescent phases, the mean tau(e) = 0.056 +/- 0.006 and T(e) = 19 200 +/- 2 300
K, while during active phases, mean quantities of both parameters increased to
tau(e) = 0.64 +/- 0.11 and T(e) = 32 300 +/- 2 000 K. During quiescent phases,
the faint electron-scattering wings are caused mainly by free electrons
from/around the accretion disk and the ionized wind from the hot star with the
total column density, N(e) <~ 1E+23 cm-2. During active phases, the large
values of tau(e) are caused by a supplement of free electrons into the binary
environment as a result of the enhanced wind from the hot star, which increases
N(e) to ~1E+24 cm-2.
