Integral Equation Methods in Potential Theory. II

where p, q are vector variables specifying points on L and where dq stands for the arc differential at q. As has already been pointed out (part I), this equation maynot exhibit a solution for L, but the difficulty can always be trivially obviated by a change of scale. If so, o-(q) generates potentials Oq(P), q5(P) defined by (5) and (6) of part I (Jaswon i963), where P denotes an interior or exterior pole as the case may be. Physical interpretations can be given to q$(P). It defines, for instance, the electrostatic potential generated by an equilibrium charge distribution on the cylindrical conductor having D as cross-section. Alternatively, it defines the stream function describing potential fluid motion round the cylinder, characterized by zero velocity at infinity; if so, 2fTo-(q) is the tangential fluid velocity at q, and the circulation is 2irs. However, apart from these physical considerations, equation (1) plays an important role in our numerical analysis because its numerical solution for any contour can always be checked, both directly and indirectly, to a fuller extent than possible with the more general equation