  E727 in the Enestrom index. This is a translation from the Latin original
"Accuratior evolutio problematis de linea brevissima in superficie quacunque
ducenda" (1779). Given a surface $pdx+qdy+rdz=0$, Euler wants to develop
equations that give the geodesics on this surface. I am new to the calculus of
variations, so it is not clear to me what steps follow from results that are
previously known (like the Euler-Lagrange equation in the calculations) and
what steps follow from earlier in this paper. I would appreciate comments from
any readers who are familiar with calculus of variations.
