for helium. We assume also that the molecular gas content in these late-type dwarfs is small. This gives us (except for the parameter $M/L$) the gradient in the radial force due to the halo. We next estimate the vertical force from the thickness and velocity dispersion of the [ <span style="font-variant:small-caps;">HI</span> ]{}, using the vertical Jeans equation. Since we evaluate the forces at $z = 100$ pc, this should be an excellent approximation, even though the equations used are not internally self-consistent as in the approach in the first paragraph. This strategy has previously been used by @OBrien2010D.

### Radial Tracer

The radial force gradient is calculated using a classic rotation curve decomposition performed at the mid-plane of the galaxy [@vanAlbada1985A]: $$v_\textrm{total}^2(R) = v_\textrm{gas}^2(R) + v_\textrm{stellar}^2(R) + v_\textrm{halo}^2(R)\,\,.\label{eqn:vresidual}$$ The total rotation $v_\textrm{total}(R)$ is the observed rotation curve of the [ <span style="font-variant:small-caps;">HI</span> ]{}gas, as we measured previously in Paper III. The theoretical rotation curves of the gas $v_\textrm{gas}(R)$ and stars $v_\textrm{stellar}(R)$ represent the contributions due to the stellar and gaseous mass components. We calculate the theoretical rotation curve of the stellar and gaseous components, using equation (A.17) of @Casertano83 $$v^2(R) = -8GR \int_0^\infty r \int_0^\infty \frac{\partial \rho(r,z)}{\partial r} \frac{K(p)-E(p)}{(Rrp)^{1/2}} \,dz\,dr 