(20) δ{ ∫^{P₂}_{P₁} _ds_ } = 0
From this equation, we can in a wellknown way deduce 4 total
differential equations which define the geodetic line; this deduction is
given here for the sake of completeness.
Let λ, be a function of the co-ordinates _x__{ν}; this defines a series
of surfaces which cut the geodetic line sought-for as well as all
neighbouring lines from P₁ to P₂. We can suppose that all such curves
are given when the value of its co-ordinates _x__{ν} are given in terms
of λ. The sign δ corresponds to a passage from a point of the geodetic
curve sought-for to a point of the contiguous curve, both lying on the
same surface λ.