A comprehensive method is given for describing in reduced terms pendent drops, emergent bubbles, sessile drops and captive bubbles of a fluid $\alpha $ within a fluid $\beta $ as figures produced by revolution of their meridian curves (X,Z). Boundary conditions at a solid surface are: (a) with a constant contact radius [Note: Equation omitted. See the image page 81 for this equation.]; or (b) with a constant contact angle $\theta $. A new curve of maximum pendent-drop or emergent-bubble volume is given for case (b); that for case (a) was described in part I (Boucher & Evans 1975). The stability of the various capillary system and boundary condition combinations is discussed in detail. In particular, pressure maxima for boundary condition (a) are analysed and their applicability to actual systems, involving pistons or syphons to deliver fluid $\alpha $, is explained. A thermodynamic analysis has been carried out and shown to be appropriate for identifying features pertaining to pressure maxima and volume maxima. The relevance of the study to systems with contact-angle hysteresis, and to more complex geometrical shapes of interface is discussed.