  This report models the refractive index above a telescope site by an
atmosphere with exponential decay of the refractive index (susceptibility) as a
function of altitude. The air is represented as a spherical hull around the
surface. We compute (i) the differential zenith angle -- the difference between
the actual zenith angle at arrival of rays at the telescope and a hypothetical
zenith angle without the atmosphere -- and (ii) the optical path length
distribution of rays at arrival in the entrance pupil as a function of the
distance to the pupil center. The key technique in this work is to expand some
integrals -- that depend on the refractive index profile along the curved path
of each ray from some virtual plane in the direction of the star up to the
entrance pupil -- in power series of small parameters.
