The electromagnetic self-force of a Lorentz-contractible spherical shell of radius $R$ in rectilinear arbitrary motion: the terms of order $1\mathrm{/}R$ and $R^0$

We write the electromagnetic self-force of a Lorentz-contractible spherical shell of radius $R$ in arbitrary rectilinear motion as a series expansion in powers of $R$, and calculate the first two terms of this series. The method we use, which is based on the description of the particle in terms of a velocity-dependent volume charge density, avoids some important difficulties of the previous approaches from the literature. We compare our results with the results obtained by other authors. The whole calculation is done in the laboratory frame of reference.