 Solution

For wall-induced motion of a planar potential dipole, the complex velocity convention is $F^{\prime }(z)=v_x-i{v}_y$. A singularity transported by the surrounding flow therefore satisfies $\dot{\bar{a}}=F_{\mathrm{reg}}^{\prime }(a)$, where its own term $m/(z-a)$ is removed before differentiating. The images at a given instant depend on that instantaneous position, but the spatial derivative treats their parameters as fixed.

For the circular wall, the derivative of the regular image is $\bar{m}/(1-\bar{a}z{)}^2$. For the quadrant, differentiate its three image terms. For the sloping wall, differentiate its single image term. This gives respectivelyThese are equations for $d{\bar{z}}_1/dt$, as printed. Conjugating gives equations for $d{z}_1/dt$ if desired; no self-induced singular velocity is included.