Passing invariant for unequal point-force spheres ($r(\delta r+3D/2){\sin }^2\theta =\mathrm{constant}$)

In the leading interaction of two sedimenting spheres, put $c=3D/4$ and let $\theta$ be the angle from the upward vertical. The equations are $\dot{r}=-(\delta +2c/r)\cos \theta$ and $r\dot{\theta }=(\delta +c/r)\sin \theta$. Their ratio separates toso $r(\delta r+2c){\sin }^2\theta$ is invariant. When $\delta ,c$ have the same nonzero sign, choose labels so both are positive. For a noncollinear encounter the invariant is positive, ensuring a strictly positive closest separation. Meanwhile $\dot{z}=-\delta -c(1+{\cos }^2\theta )/r<0$, so the vertical order reverses and eventually $r\ge |z|\to \infty$. Exactly collinear contact is outside this passing argument.