In the spherical limit, , this is readily integrated to give , where . is the longitude difference, corresponding the arc-length on the auxiliary sphere.
Expanding in powers of and integrating term by term, we thus
Vincenty has again rearranged a subset of the resulting equations into nested forms more suitable for computation:
As in Eqn. (22), the origins of , and
have been shifted from the equator to the initial point
(1). is then given by