Summary: A formula for the generalized Schwarz - Christoffel conformal mapping from a bounded multiply connected circular domain to an unbounded multiply connected polygonal domain is derived. The formula for the derivative of the mapping function is shown to contain a product of powers of Schottky - Klein prime functions associated with the circular preimage domain. Two analytical checks of the new formula are given. First, it is compared with a known formula in the doubly connected case. Second, a new slit mapping formula from a circular domain to the triply connected region exterior to three slits on the real axis is derived using separate arguments. The derivative of this independently-derived slit mapping formula is shown to correspond to a degenerate case of the new Schwarz - Christoffel mapping. The example of the mapping to the triply connected region exterior to three rectangles centred on the real axis is considered in detail.