Analytical formulae for the Kirchhoff-Routh path function in multiply connected domains

Explicit formulae for the Kirchhoff-Routh path functions (or Hamiltonians) governing the motion of N-point vortices in multiply connected domains are derived when all circulations around the holes in the domain are zero. The method uses the Schottky-Klein prime function to find representations of the hydrodynamic Green's function in multiply connected circular domains. The Green's function is then used to construct the associated Kirchhoff-Routh path function. The path function in more general multiply connected domains then follows from a transformation property of the path function under conformal mapping of the canonical circular domains. Illustrative examples are presented for the case of single vortex motion in multiply connected domains.