Abstract The differential equation governing the free, transverse vibration of a thin membrane, the Helmholtz equation, is also the governing equation for problems of microwave propagation, stress waves and acoustics. Simple analytical solutions may be found in elliptic and parabolic co-ordinates. The paper is concerned with the nature of the fundamental and higher modes of membranes of arbitrary shape, and in particular with the question of whether the level lines (contours) of the modes do or do not intersect each other, i.e., whether the displacement has or has not saddle points. We announce a result proved elsewhere and discuss its implications: if the membrane is not circular, then any mode with only interior nodal lines has non-nodal saddle points.