For pt.I see ibid., vol.17, pt.1 (1984). The transverse Cartesian components By and Bz of the magnetic field of cylindrical saddle-shaped coils are evaluated in a series expansion. The terms of the series are written in the form Almnxlymzn, where the coefficients Almn are functions of the coil size and shape parameters R0, Z0, Phi 0 (R0 and 2Z0 are the radius and height of the cylinder defining the geometry of the saddle-shaped coil, Phi 0 is the angle of the arcs). The y component of the magnetic field is determined to the fourth order (l+m+n=4); the z component was evaluated including the sixth-order terms. The existence or non-existence of the second-order terms determine the limiting properties of By and Bz for r to 0. the analysis of the second-order terms as a function of the coil parameters reveals some general properties of the magnetic field at the coil centre.