AbstractIt would be reasonable to suppose that if the formulae given by Rainsford in [1] were extended still further, there would be terms of greater than fifth order, elliptic and otherwise; and that of these, the elliptic would be of less importance. Now it is easy to see that all such terms that are not elliptic, obtained by putting the eccentricity = 0, could be provided by“closed”spherical formulae. Rainsford's extension was tested at 1000 miles in [2], and showed differences of up to 16·7 metres in distance and 0·70″in azimuth. The“Spherical”Gauss system came in to 0·0 metres in length and 0·00″in forward and back azimuths for all of the nine test lines. The numerical results are given later in detail.