Although the synchrotron emission from a single electron moving with an ultrarelativistic velocity βc in a magnetic field B0 is elliptically polarized, the characteristics of the polarization ellipse being determined by the position of the direction of observation n relative to the cone swept out by the direction u of the velocity vector about the direction of B0, the resultant emission from a distribution of such gyrating electrons is, to a first approximation, linearly polarized in the direction perpendicular to the projection of B0 on to the plane transverse to n. The reasons for this are that for a single electron the emission is effectively confined to within a small angular distance O(ξ) of u, where ξ = √ (1 −β2) ≪ 1 and that (i) the fourth Stokes parameter is to this approximation an odd function of the angle ip between n and u at its closest approach, and (ii) the number of electrons passing within equal angular distances ψ = ±O(ξ) are equal. It follows that if ξ is large enough it is possible for the next approximation to the fourth Stokes parameter for a distribution of electrons to be significant, so that the resultant emission is elliptically polarized.