A neutral tangent plane is defined so that small isentropic and adiabatic displacements of a fluid parcel in this plane do not produce bouyant restoring forces on the parcel. This local stability argument can also be used to trace a neutral trajectory in space along some pre-determined path in latitude and longitude. If one integrates laterally around an entire ocean basin on a neutral trajectory, one can arrive at a different depth than that of the starting point on the original CTD cast. This means that the definition of a neutral surface is path-dependent. This is a real effect; it results from the complicated equation of state of seawater, and is not an artifact of errors in the lateral integration procedure. Dyed patches of fluid are mixed laterally by meso-scale eddies along these neutral helices, and at the same time they are smoothed and advected in the vertical direction by small-scale mixing processes. It is shown that, while a neutral surface is formally ill-defined mathematically (in the above path-dependent sense), this is of little importance for the purpose of constructing lateral maps of properties in the ocean, since the ambiguity in constructing a neutral surface is often less than the measurement accuracy of modern oceanographic instruments (≈0.003 kg m -3 in density). Path-dependence in the definition of a neutral surface occurs because α/β is a function of pressure (where α and β are the thermal expansion and haline contraction coefficients respectively). The local contribution to path-dependence is proportional to ▽p· ▽Sx▽Sxθ , representing the angle between an isobaric surface and the line of intersection of an isohaline surface and a surface of constant potential temperature. This, in turn, is proportional to ▽npx▽nθ where ▽n is the epineutral gradient operator for properties measured in a neutral tangent plane. That is, unless isobars and potential isotherms drawn in a neutral tangent plane are parallel, a neutral trajectory around this point will not lie in the plane, but will describe a helix in space. Although complete surfaces with the “neutral property” do not exist, the neutral tangent plane is everywhere well-defined. The lateral motion along helical neutral trajectories produces vertical advection in the ocean. A method is described in this paper of taking an approximately neutral surface and distributing the path-dependent effects over the surface in a least-squares sense. At each point in the ocean an error vector is found that represents the difference between the “best-fit” surface (which is a mathematically well-defined surface) and the local slope of the neutral tangent plane, providing a rational way of distributing the path-dependent vertical velocity on the surface. The vertical fluxes of heat, salt or tracer produced by the path-dependence of neutral surfaces do not have a signature in the dissipation rate of mechanical energy that can be measured with microstructure instrumentation.