Gauss gave a formula for the linking number of a two-component link in Euclidean 3-space in terms of an integral. The integrand has the property that it is invariant under rigid motions of the link in 3-space; i.e., the integrand is geometrically natural. I will present an analogous formula for three-component links: a formula which computes their Milnor triple linking number in the event that pairwise linking numbers vanish, using an integral whose integrand is also geometrically natural.